Dunham - Data Mining.pdf

It is the best book on data mining so far, and I would defln,(teJ�_.,tdiiPt
my course. The book is very C011Jprehensive and cove� all of
topics and algorithms of which I am aware. The depth of CO!Irer•liM
topic or method is exactly right and appropriate. Each a/grorirtmti �r�
in pseudocode that is s , icient for any interested readers to
working implementation in a computer language of their choice.
-Michael H Huhns, Umversity of �UDilCiii
Discussion on distributed, parallel, and incremental algorithms is outst:tlftfi!tr··· '��
-Z an Obradovic, Temple Univef'Sf1tv
Margaret Dunham offers the experienced data base professional or graduate
level Computer Science student an introduction to the full spectrum of Data
Mining concepts and algorithms. Using a database perspective throughout,
Professor Dunham examines algorithms, data structures, data types, and
complexity of algorithms and space. This text emphasizes the use of data
mining concepts in real-world applications with large database components.
KEY FEATURES:
.. Covers advanced topics such as Web Mining and Spatialrremporal mining
Includes succinct coverage of Data Warehousing, OLAP, Multidimensional
Data, and Preprocessing
Provides case studies
Offers clearly written algorithms to better understand techniques
Includes a reference on how to use Prototypes and DM products
Prentice Hall
Upper Saddle River, NJ 07458
www. prenhall.com
2517
227
1
Hail
roductoty nd Advanced Topics
MARGARE�f H. DUNHJ

Contents
Preface
Part One Introduction
1 Introduction
1.1 Basic Data Mining Tasks .
1.1.1 Classification . . .
1.1.
2 Regression . . . .
1
.1
.
3 Time Series Analysis.
1.1.4 Prediction. . .
1.1.5 Clustering . . . . .
1
.1.6 Summarization . .
1.1.7 Association Rules
1.1.8 Sequence Discovery
1.2 Data Mining Versus Knowledge Discovery in Databases.
1.2.1 The Development of Data Mining.
1.3 Data Mining Issues ........ .
1.4 Data Mining Metrics . . . . . . . . . . . .
1.5 Social Implications of Data Mining . . . .
1.6 Data Mining from a Database Perspective.
1.7 The Future .... .
1.8 Exercises ..... .
1.9 Bibliographic Notes.
2 Related Concepts
2.1 Database/OLTP Systems
2.2 Fuzzy Sets and Fuzzy Logic
2.3 Information Retrieval . . .
2.4 Decision Support Systems
2.5 Dimensional Modeling . . .
2.5.1 Multidimensional Schemas
2.5.2 Indexing
2.6 Data Warehousing
2.7 OLAP ....... .
2.8 Web Search Engines
2.9 Statistics . . . . . .
2.10 Machine Learning.
2.1 1 Pattern Matching
2.12 Summary . . . . .
2.13 Exercises . . . . .
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vi Contents
3 Data Mining Techniques
3
.1 Introduction . . . . .
3
.
2 A Statistical Perspective on Data Mining
3
.
2
.1 Point Estimation . . . ... . ..
3
.
2
.
2 Models Based on Summarization
3
.
2
.3 Bayes Theorem . . . . .. .
3
.
2
.4 Hypothesis Testing . . ...
3
.
2
.5 Regression and Correlation
3
.3 Similarity Measures.
3
.4 Decision Trees . . . . . ..
3
.5 Neural Networks . .. . . .
3
.5.1 Activation Functions
3
.6 Genetic Algorithms .
3
.7 Exercises . . . . ..
3
.8 Bib�iographic Notes.
Part Two Core Topics
4 Classification
4.1 Introduction ...........
4.1
.1 Issues in Classification .
4.
2 Statistical-Based Algorithms ..
4.2.1 Regression .......
4.
2
.
2 Bayesian Classification.
4.3 Distance-Based Algorithms..
4.3
.1 Simple Approach....
4.3
.
2 K Nearest Neighbors
4.4 Decision Tree-Based Algorithms
4.4.1 ID3 ......
4.4.
2 C4.5 and C5 .0 .... .
4.4.3 CART . . . . . . . . ..
4.4.4 Scalable DT Techniques
4.5 Neural Network-Based Algorithms
4.5.1 Propagation . . . . . . . ..
4.5
.
2 NN Supervised Learning . .
4.5.3 Radial Basis Function Networks.
4.5.4 Perceptrons . . . . . . . . ..
4.6 Rule-Based Algorithms . . . . . . . . ..
4.6.1 Generating Rules from a DT ..
4.6.
2 Generating Rules from a Neural Net
4.6
.3 Generating Rules Without a DT or NN
4.7 Combining Techniques
4.8 Summary . .....
4.9 Exercises . . . . ..
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5 Clustering
5.1 Introduction
5.2 Similarity and Distance Measures
5.3 Outliers . . . . . . . . . . . . . .
5.4 Hierarchical Algorithms . . . ..
5.4.1 Agglomerative Algorithms .
5.4.
2 Divisive Clustering . . ..
5.5 Partitional Algorithms . . . . . ..
5.5.1 Minimum Spanning Tree. .
5
.5.2 Squared Error Clustering Algorithm.
5.5.3 K-Means Clustering ... . .
5.5.4 Nearest Neighbor Algorithm .
5.5.5 PAM Algorithm . . .. . ..
5.5
.6 Bond Energy Algorithm . . .
5.5.7 Clustering with Genetic Algorithms .
5.5.8 Clustering with Neural Networks
5.6 Clustering Large Databases
5.6.1 BIRCH . .. . ..
5.6
.
2 DBSCAN ..... .
5.6
.3 CURE Algorithm . .
5.7 Clustering with Categorical Attributes .
5.8 Comparison . . . ..
5.9 Exercises . . . . ..
5.10 Bibliographic Notes .
6 Association Rules
6.1 Introduction .
6
.
2 Large Itemsets
6.3 Basic Algorithms
6.3.1 Apriori Algorithm
6.3
.
2 Sampling Algorithm
6.3.3 Partitioning . . . . .
6.4 Parallel and Distributed Algorithms
6.4.1 Data Parallelism
6.4.
2 Task Parallelism
6.5 Comparing Approaches.
6.6 Incremental Rules . . . .
6
.7 Advanced Association Rule Techniques
6.7
.1 Generalized Association Rules.
6.7
.
2 Multiple-Level Association Rules
6.7
.3 Quantitative Association Rules
6
.7
.4 Using Multiple Minimum Supports
6.7
.5 Correlation Rules ....
6.8 Measuring the Quality of Rules
6.9 Exercises . . . .. .
6.10 Bibliographic Notes . . ... ..
Contents vii
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viii Contents
Part Three Advanced Topics
7
8
Web Mining
7
.1 Introduction
7
.
2 Web Content Minirig
7
.
2
.1 Crawlers ..
7
.
2
.
2 Harvest System .
7
.
2
.3 Virtual Web View
7
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2
.4 Personalization
7
.3 Web Structure Mining
7.3.1 PageRank .
7
.3
.
2 Clever .....
7
.4 Web Usage Mining . .
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.4.1 Preprocessing .
7
.4.2 Data Structures .
7
.
4
.3 Pattern Discovery
7
.4.4 Pattern Analysis
7
.5 Exercises • 0 • 0 • •
Spatial Mining
8.1
8
.
2
8.3
8.4
8.5
8.6
8
.7
8
.8
8
.9
Introduction • • • • • 0
Spatial Data Overview
8
.
2
.1 Spatial Queries
8
.
2
.2 Spatial Data Structures.
8
.
2
.3 Thematic Maps .....
8
.
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.4 Image Databases ....
Spatial Data Mining Primitives
Generalization and Specialization
8
.4.1
8
.4.
2
8
.4.3
8
.4.4
Progressive Refinement
Generalization ..
Nearest Neighbor.
STING 0 • • • • •
Spatial Rules ..... . .
8
.5.1 Spatial Association Rules
Spatial Classification Algorithm
8
.6.1 ID3 Extension . .. .
8.6 .
2 Spatial Decision Tree
Spatial Clustering Algorithms .
8.7.1 CLARANS Extensions .
8.7
.
2 SD(CLARANS)
8.7.3 DBCLASD .
8
.7
.4 BANG ....
8
.7
.5 WaveCluster
8
.7
.6 Approximation
Exercises • • • 0 • •
Bibliographic Notes ..
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9 Temporal Mining
· 9
.1 Introduction . . ...... .
9
.
2 Modeling Temporal Events .
9
.3 Time Series . . . . .. . . .
9
.3
.1 Time Series Analysis.
9.3
.
2 Trend Analysis
9.3
.3 Transformation
9
.3
.4 Similarity .
9o3 o5 Prediction 0 0 0
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.4 Pattern Detection .. 0
9.4.1 String Matching
9.5 Sequences . . . . .
9
.5.1 AprioriAll.. o
9
.5.2 SPADE ... o
9.5.3 Generalization
9.5.4 Feature Extraction
9
.6 Temporal Association Rules
9
.6.1 Intertransaction Rules
9.6
.
2 Episode Rules ....
9
.6 03 Trend Dependencies
9.6.4 Sequence Association Rules
9
.6
.5 Calendric Association Rules .
907 Exercises 0 .. 0 0 0
9
.8 Bibliographic Notes .
APPENDICES
Contents ix
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A Data Mining Products
A.1 Bibliographic Notes . • 0 • • 0 • • • • • • • • • 0 0 • • • • • • • • 0 • • •
274
2 8 9
B Bibliography 290
Index 305
About the Author 315

Preface
Data doubles about every year, but useful information seems to be decreasing. The area
of data mining has arisen over the last decade to address this problem. It has become
not only an important research area, but also one with large potential in the real world.
Current business users of data mining products achieve millions of dollars a year in
savings by using data minifg techniques to reduce the cost of day to day business
operations. Data mining techniques are proving to be extremely useful in detecting and
predicting terrorism.
The purpose of this book is to introduce the reader to various data mining con
cepts and algorithms. The book is concise yet thorough in its coverage of the many
data mining topics. Clearly written algorithms with accompanying pseudocode are used
to describe approaches. A database perspective is used throughout. This means that I
examine algorithms, data structures, data types, and complexity of algorithms and space.
The emphasis is on the use of data mining concepts in real-world applications with large
database components.
Data mining research and practice is in a state similar to that of databases in the
1960s. At that time applications programmers had to create an entire database environ
ment each time they wrote a program. With the development of the relational data model,
query processing and optimization techniques, transaction management strategies, and ad
hoc query languages (SQL) and interfaces, the current environment is drastically differ
ent. The evolution of data mining techniques may take a similar path over the next few
decades, making data mining techniques easier to use and develop. The objective of this
book is to help in this process.
The intended audience of this book is either the expeiienced database professional
who wishes to learn more about data mining or graduate level computer science students
who have completed at least an introductory database course. The book is meant to
be used as the basis of a one-semester graduate level course covering the basic data
mining concepts. It may also be used as reference book for computer professionals and
researchers.
Introduction
I Chl Introduction 1-
I Ch2 Related Concepts I
Core Topics
rl Ch4 Classification I
I Ch3 Data Mining Techniques I
r-H Ch5 Clustering
Advanced Topics
H Ch6 Association Rules I
I Ch7 Web Mining 1-
I Ch8 Spatial Mining 1-r- Appendix
I Ch9 Temporal Mining 1- y Data Mining Products
xi

xii Preface
The book is divided into four major parts: Introduction, Core Topics, Advanced
Topics, and Appendix
. The introduction covers background information needed to under
stand the later material. In addition, it examines topics related to data mining such as
OLAP, data warehousing, information retrieval, and machine learning
. In the first chapter
of the introduction I provide a very cursory overview of data mining and how it relates
to the complete KDD process
. The second chapter surveys topics related to data min
ing
. While this is not crucial to the coverage of data mining and need not be read to
understand later chapters, it provides the interested reader with an understanding and
appreciation of how data mining concepts relate to other areas
. To thoroughly under
stand and appreciate the data mining algorithms presented in subsequent chapters, it is
important that the reader realize that data mining is not an isolated subject
. It has its basis
in many related disciplines that are equally important on their own
. The third chapter
in this part surveys some techniques used to implement data mining algorithms
. These
include statistical techniques, neural networks, and decision trees
. This part of the book
provides the reader with an understanding of the basic data mining concepts
. It also
serves as J standalone survey of the entire data mining area
.
The Core Topics covered are classification, clustering, and association rules
. I view
these as the major data mining functions
.Other data mining concepts (such as prediction,
regression, and pattern matching) may be viewed as special cases of these three
. In each
of these chapters I concentrate on coverage of the most commonly used algorithms of
each type
. Our coverage includes pseudocode for these algorithms, an explanation of
them and examples illustrating their use
.
The advanced topics part looks at various concepts that complicate data mining
applications. I concentrate on temporal data, spatial data, and Web mining
. Again, algo
rithms and pseudocode are provided.
In the appendix, production data mining systems are surveyed. I will keep a more
up to data list on the Web page for the book
. I thank all the representatives of the various
companies who helped me correct and update my descriptions of their products.
All chapters include exercises covering the material in that chapter. In addition to
conventional types of exercises that either test the student's understanding of the material
or require him to apply what he has learned. I also include some exercises that require
implementation (coding) and research
.A one-semester course would cover the core topics
and one or more of the advanced ones.
ACKNOWLEDGMENTS
Many people have helped with the completion of this book
. Tamer Ozsu provided initial
advice and inspiration. My dear friend Bob Korfhage introduced me to much of computer
science, including pattern matching and information retrieval
. Bob, I think of you often
.
I particularly thank my graduate students for contributing a great deal to some of
the original wording and editing. Their assistance in reading and commenting on earlier
drafts has been invaluable
. Matt McBride helped me prepare most of the original slides,
many of which are still available as a companion to the book
. Yongqiao Xiao helped
write much of the material in the Web mining chapter. He also meticulously reviewed
an earlier draft of the book and corrected many mistakes. Le Gruenwald, Zahid Hossain,
Yasemin Seydim, and Al Xiao performed much of the research that provided information
found concerning association rules
. Mario Nascimento introduced me to the world of
Preface xiii
temporal databases, and I have used some of the information from his dissertation in
the temporal mining chapter. Nat Ayewah has been very patient with his explanations
of hidden Markov models and helped improve the wording of that section. Zhigang Li
has introduced me to the complex world of time series and helped write the solutions
manual. I've learned a lot, but still feel a novice in many of these areas
.
The students in my CSE 8 3 3 1 class (Spring 1 9 9 9 , Fall 2000, and Spring 2002) at
SMU have had to endure a great deal
. I never realized how difficult it is to clearly word
algorithm descriptions and exercises until I wrote this book. I hope they learned something
even though at times the continual revisions necessary were, I'm sure, frustrating
. Torsten
Staab wins the prize for find�ng and correcting the most errors
. Students in my CSE8 3 3 1
class during Spring 2002 helped me prepare class notes and solutions to the exercises
. I
thank them for their input
.
My family has been extremely supportive in this endeavor. My husband, Jim, has
been (as always) understanding and patient with my odd work hours and lack of sleep
.
A more patient and supportive husband could not be found. My daughter Stephanie has
put up with my moodiness caused by lack of sleep. Sweetie, I hope I haven't been too
short-tempered with you (ILYMMTYLM)
. At times I have been impatient with Kristina
but you know how much I love you
. My Mom, sister Martha, and brother Dave as always
are there to provide support and love
.
Some of the research required for this book was supported by the National Science
Foundation under Grant No
. IIS-9 8 208 4 1
. I would finally like to thank the reviewers
(Michael Huhns, Julia Rodger, Bob Cimikowski, Greg Speegle, Zoran Obradovic,
T.Y. Lin, and James Buckly) for their many constructive comments
. I tried to implement
as many of these I could
.

CHAPTER 1
Introduction
1.1 BASIC DATA MINING TASKS
1.2 DATA MINING VERSUS KNOWLEDGE OISCOVERY IN DATABASES
1.3 DATA MINING ISSUES
1.4 DATA MINING METRICS
1.5 SOCIAL IMPLICATIONS OF DATA MINING
1.6 DATA MINING FROM A DATABASE PERSPECTIVE
1.7 THE FUTURE
1.8 EXERCISES
1.9 BIBLIOGRAPHIC NOTES
The amount of data kept in computer files and databases is growing at a phenomenal rate.
At the same time, the users of these data are expecting mo!l'e sophisticated information
from them. A marketing manager is no longer satisfied with a simple listing of marketing
contacts, but wants detailed information about customers' past purchases as well as pre
dictions of future purchases. Simple structured/query language queries are not adequate
to support these increased demands for information. Data mining steps in to solve these
needs. Data mining is often defined as finding hidden information in a database. Alterna
tively, it has been called exploratory data analysis, data driven discovery, and deductive
learning.
Traditional database queries (Figure 1.1), access a database using a well-defined
query stated in a language such as SQL. The output of tht: query consists of the data
from the database that satisfies the query. The output is usually a subset of the database,
but it may also be an extracted view or may contain aggregations. Data mining access
of a database differs from this traditional access in several ways:
• Query: The query might not be well formed or precisely stated. The data miner
might not even be exactly sure of what he wants to see.
• Data: The data accessed is usually a different version from that of the original
operational database. The data have been cleansed and modified to better support
the mining process.
• Output: The output of the data mining query probably is not a subset of the
database. Instead it is the output of some analysis of the contents of the database.
The current state of the art of data mining is similar to that of database query processing
in the late 1960s and early 1970s. Over the next decade there undoubtedly will be great
3

4 Chapter 1 Introduction
SQL
Q� IDBMS 1- (Ds}
Results
FIGURE 1.1: Database access.
strides in extending the state of the art with respect to data mining. We probably will
see the development of "query processing" models, standards, and algorithms targeting
the data mining applications. We probably will also see new data structures designed for
the storage of databases being used for data mining applications. Although data mining
is currently in its infancy, over the last decade we have seen a proliferation of mining
algorithms, applications, and algorithmic approaches. Example 1.1 illustrates one such
application.
EXAMPL�1.1
Credit card companies must determine whether to authorize credit card purchases. Sup
pose that based on past historical information about purchases, each purchase is placed
into one of four classes: (1) authorize, (2) ask for further identification before authoriza
tion, (3) do not authorize, and (4) do not authorize but contact police. The data mining
functions here are twofold. First the historical data must be examined to determine how
the data fit into the four classes. Then the problem is to apply this model to each new
purchase. Although the second part indeed may be stated as a simple database query, the
first part cannot be.
Data mining involves many different algorithms to accomplish different tasks. All
of these algorithms attempt to fit a model to the data. The algorithms examine the data
and determine a model that is closest to the characteristics of the data being examined.
Data mining algorithms can be characterized as consisting of three parts:
• Model: The purpose of the algorithm is to fit a model to the data.
• Preference: Some criteria must be used to fit one model over another.
• Search: All algorithms require some technique to search the data.
In Example 1.1 the data are modeled as divided into four classes. The search requires
examining past data about credit card purchases and their outcome to determine what
criteria should be used to define the class structure. The preference will be given to
criteria that seem to fit the data best. For example, we probably would want to authorize
a credit card purchase for a small amount of money with a credit card belonging to a
long-standing customer. Conversely, we would not want to authorize the use of a credit
card to purchase anything if the card has been reported as stolen. The search process
requires that the criteria needed to fit the data to the classes be properly defined.
As seen in Figure 1.2, the model that is created can be either predictive or descrip
tive in nature. In this figure, we show under each model type some of the most common
data mining tasks that use that type of model.
1.1
Predictive
Section 1.1
Data mining
Basic Data Mining Tasks 5
----
Descriptive
-------------�
Classification Regression Time series Prediction Clustering Summarization Association Sequence
analysis rules discovery
FIGURE 1.2: Data mining models and tasks.
A predictive model makes a prediction about values of data using known results
found from different data. Predictive modeling may be made based on the use of
other historical data. For example, a credit card use might be refused not because of
the user's own credit history, but because the current purchase is similar to earlier
purchases that were subsequently found to be made with stolen cards. Example 1.1
uses predictive modeling to predict the credit risk. Predictive model data mining tasks
include classification, regression, time series analysis, and prediction. Prediction may
also be used to indicate a specific type of data mining function, as is explained in
section 1.1.4.
A descriptive model identifies patterns or relationships in data. Unlike the predictive
model, a descriptive model serves as a way to explore the properties of the data examined,
not to predict new properties. Clustering, summarization, association rules, and sequence
discovery are usually viewed as descriptive in nature.
BASIC DATA MINING TASKS
In the following paragraphs we briefly explore some of the data mining functions. We
follow the basic outline of tasks shown in Figure 1.2. This list is not intended to be
exhaustive, but rather illustrative. Of course, these individual tasks may be combined to
obtain more sophisticated data mining applications.
1.1.1 i Classification
Classification maps data into predefined groups or classes. It is often referred to as
supervised learning because the classes are determined before examining the data. Two
examples of classification applications are determining whether to make a bank loan and
identifying credit risks. Classification algorithms require that the classes be defined based
on data attribute values. They often describe these classes by looking at the character
istics of data already known to belong to the classes. Pattern recognition is a type of
classification where an input pattern is classified into one of several classes based on
its similarity to these predefined classes. Example 1.1 illustrates a general classification
problem. Example 1.2 shows a simple example of pattern recognition.
EXAMPLE 1.2
An airport security screening station is used to determine: if passengers are potential
terrorists or criminals. To do this, the face of each passenger is scanned and its basic
pattern (distance between eyes, size and shape of mouth, shape of head, etc.) is identified.

1.1.2
This pattern is compared to entries in a database to see if it matches any patterns that
are associated with known offenders.
Regression
Regression is used to map a data item to a real valued prediction vari�ble. In ac�al
ity, regression involves the learning of the function that does t�is mappi�g. Regre�si?n
assumes that the target data fit into some known type of functiOn (e.g., linear, logistic,
etc.) and then determines the best function of this type that models the given data. �orne
type of error analysis is used to determine which function is "best."
.
standard hnear
regression, as illustrated in Example 1.3, is a simple example of regressiOn.
EXAMPLE 1.3
A college ptofessor wishes to reach a certain level of savings before
.
her retirement.
Periodically, she predicts what her retirement savings will be based on Its �urre�t value
and several past values. She uses a simple linear regression fo�ula to
.
predict this
.
value
by fitting past behavior to a linear function and then using this functiOn to ?redict the
values at points in the future. Based on these values, she then alters her mvestment
portfolio.
1.1.3 Time Series Analysis
With time series analysis, the value of an attribute is examined as it varies over time. The
values usually are obtained as evenly spaced time points (daily, weeki�, hourly, etc.). A
time series plot (Figure 1.3), is used to visualize the time series. In this figure you can
easily see that the plots for Y and Z have similar behavior, while X appears to have less
volatility. There are three basic functions performed in time series
.
analysis
:
In on� case,
distance measures are used to determine the similarity between different tlme senes. In
the second case, the structure of the line is examined to determine (and perhaps classi�y)
its behavior. A third application would be to use the historical time series plot to predict
future values. A time series example is given in Example 1.4.
EXAMPLE 1.4
Mr. Smith is trying to determine whether to purchase stock from Companies X, Y,
or z. For a period of one month he charts the daily stock price for ea�h co�pany.
Figure 1.3 shows the time series plot that Mr. Smith ha� gene�ated. Usmg this and
similar information available from his stockbroker, Mr. Sllllth decides to purchase stock
X because it is less volatile while overall showing a slightly larger relative amount of
growth than either of the other stocks. As a matter of fact, the �to
.
cks �or Y and Z have
a similar behavior. The behavior of Y between days 6 and 20 IS Identical to that for Z
between days 13 and 27.
Section 1.1
FIGURE 1.3: Time series plots.
1.1.4 Prediction
Basic Data Mining Tasks 7
---o-X
__.,___ y
--z
Many real-world data mining applications can be seen as predicting future data states
based on past and current data. Prediction can be viewed as a type of classification. (Note:
This is a data mining task that is different from the prediction model, although the pre
diction task is a type of prediction model.) The difference is that prediction is predicting
a future state rather than a current state. Here we are referring to a type of application
rather than to a type of data mining modeling approach, as discussed earlier. Prediction
applications include flooding, speech recognition, machine learning, and pattern recog
nition. Although future values may be predicted using time series analysis or regression
techniques, other approaches may be used as well. Example 1.5 illustrates the process.
EXAMPLE 1.5
Predicting flooding is a difficult problem. One approach uses monitors placed at various
; points in the river. These monitors collect data relevant to flood prediction: water level,
' rain amount, time, humidity, and so on. Then the water level at a potential flooding point
in the river can be predicted based on the data collected by the sensors upriver from this
point. The prediction must be made with respect to the time the data were collected.
1.1.5 Clustering
Clustering is similar to classification except that the groups are not predefined, but rather
defined by the data alone. Clustering is alternatively referred to as unsupervised learn
ing or segmentation. It can be thought of as partitioning or segmenting the data into
groups that might or might not be disjointed. The clustering is usually accomplished by
determining the similarity among the data on predefined attributes. The most similar data
are grouped into clusters. Example 1.6 provides a simple clustering example. Since the
clusters are not predefined, a domain expert is often required to interpret the meaning of
the created clusters.

EXAMPLE 1.6
A certain national department store chain creates special catalogs targeted to various
demographic groups based on attributes such as income, location, and physical charac
teristics of potential customers (age, height, weight, etc
.
)
. To determine the target mailings
of the various catalogs and to assist in the creation of new, more specific catalogs, the
company performs a clustering of potential customers based on the determined attribute
values
. The results of the clustering exercise are then used by management to create
special catalogs and distribute them to the correct target population based on the cluster
for that catalog
.
A special type of clustering is called segmentation. With segmentation a database
is partitioned into disjointed groupings of similar tuples called segments. Segmentation
is often viewed as being identical to clustering
. In other circles segmentation is viewed
as a specilic type of clustering applied to a database itself
. In this text we use the two
terms, clustering and segmentation, interchangeably
.
1.1.6 Summarization
Summarization maps data into subsets with associated simple descriptions
. Summariza
tion is also called characterization or generalization. It extracts or derives representative
information about the database. This may be accomplished by actually retrieving portions
of the data
. Alternatively, summary type information (such as the mean of some numeric
attribute) can be derived from the data
. The summarization succinctly characterizes the
contents of the database
. Example 1.7illustrates this process
.
EXAMPLE 1.7
One of the many criteria used to compare universities by the U.S. News & World Report
is the average SAT or AC T score [GM99]. This is a summarization used to estimate the
type and intellectual level of the student body.
1.1.7 Association Rules
Link analysis, alternatively referred to as affinity analysis or association, refers to the
data mining task of uncovering relationships among data
. The best example of this
type of application is to determine association rules
. An association rule is a model that
identifies specific types of data associations. These associations are often used in the retail
sales community to identify items that are frequently purchased together
. Example 1.8
illustrates the use of association rules in market basket analysis
. Here the data analyzed
consist of information about what items a customer purchases. Associations are also used
in many other applications such as predicting the failure of telecommunication switches
.
EXAMPLE 1.8
A grocery store retailer is trying to decide whether to put bread on sale
. To help determine
the impact of this decision, the retailer generates association rules that show what other
Section 1.2 Data Mining Versus Knowledge Discovery in Databases 9
products are frequently purchased with bread
. He finds that 60% of the time that bread is
sold so are pretzels and that 70% of the time jelly is also sold
. Based on these facts, he
tries to capitalize on the association between bread, pretzels, and jelly by placing some
pretzels and jelly at the end of the aisle where the bread is placed. In addition, he decides
not to place either of these items on sale at the same time.
Users of association rules must be cautioned that these are not causal relation
ships
. They do not represent any relationship inherent in the actual data (as is true with
functional dependencies) or in the real world. There probably is no relationship between
bread and pretzels that causes them to be purchased together. And there is no guarantee
that this association will apply in the future
. However, association rules can be used to
assist retail store management in effective advertising, marketing, and inventory control
.
1.1.8 Sequence Discovery
Sequential analysis or sequence discovery is used to determine sequential patterns in data
.
These patterns are based on a time sequence of actions
. These patterns are similar to
associations in that data (or events) are found to be related, but the relationship is based
on time.Unlike a market basket analysis, which requires the items to be purchased at
the same time, in sequence discovery the items are purchased over time in some order.
Example 1.9 illustrates the discovery of some simple patterns
.A similar type of discovery
can be seen in the sequence within which data are purchased
. For example, most people
who purchase CD players may be found to purchase CDs within one week
. As we will
see, temporal association rules really fall into this category.
EXAMPLE 1.9
The Webmaster at the XYZ Corp. periodically analyzes the Web log data to determine
how users of the XYZ's Web pages access them
. He is interested in determining what
sequences of pages are frequently accessed
. He determines that 70 percent of the users
of page A follow one of the following patterns of behavior: (A, B, C) or (A, D, B, C)
or (A, E, B, C). He then determines to add a link directly from page A to page C.
1.2 DATA M I NI NG VERSUS KNOWLEDGE DISCOVERY I N DATABASES
The terms knowledge discovery in databases (KDD) and data mining are often used
interchangeably. In fact, there have been many other names given to this process of
discovering useful (hidden) patterns in data: knowledge extraction, information discovery,
exploratory data analysis, information harvesting, and unsupervised pattern recognition
.
Over the last few years KDD has been used to refer to a process consisting of many
steps, while data mining is only one of these steps. This is the approach taken in this
book
. The following definitions are modified from those found in [FPSS96c, FPSS96a].
DEFINITION 1.1. Knowledge discovery in databases (KDD) is the process of
finding useful information and patterns in data.
DEFINITION 1.2. Data mining is the use of algorithms to extract the information
and patterns derived by the KDD process.

The KDD process is often said to be nontrivial; however, we take the larger view that
KDD is an all-encompassing concept. A traditional SQL database query can be viewed
as the data mining part of a KDD process. Indeed, this may be viewed as som�what
simple and trivial. However, this was not the case 30 years ago. If we were to advance
30 years into the future, we might find that processes thought of today as nontrivial and
complex will be viewed as equally simple. The definition of KDD includes the keyword
useful. Although some definitions have included the term "potentially useful," we believe
that if the information found in the process is not useful, then it really is not information.
Of course, the idea of being useful is relative and depends on the individuals involved.
KDD is a
'
process that involves many different steps. The input to this process is
the data, and the output is the useful information desired by the users. However, the
objective may be unclear or inexact. The process itself is interactive and may require
much elapsed time. To ensure the usefulness and accuracy of the results of the process,
interaction throughout the process with both domain experts and technical experts might
be needed. Figure 1.4 (modified from [FPSS96c]) illustrates the overall KDD process.
frhe KDD process consists of the following five steps [FPSS96c]:
• Selection: The data needed for the data mining process may be obtained from
many different and heterogeneous data sources. This first step obtains the data
from various databases, files, and nonelectronic sources.
• Preprocessing: The data to be used by the process may have incorrect or miss
ing data. There may be anomalous data from multiple sources involving different
data types and metrics. There may be many different activities performed at this
time. Erroneous data may be corrected or removed, whereas missing data must be
supplied or predicted (often using data mining tools).
• Transformation: Data from different sources must be converted into a common
format for processing. Some data may be encoded or transformed into more usable
formats. Data reduction may be used to reduce the number of possible data values
being considered.
• Data mining: Based on the data mining task being performed, this step applies
algorithms to the transformed data to generate the desired results.
• Interpretation/evaluation: How the data mining results are presented to the users
is extremely important because the usefulness of the results is dependent on it.
Various visualization and GUI strategies are used at this last step.
Transformation techniques are used to make the data easier to mine and more use
ful, and to provide more meaningful results. The actual distribution of the data may be
0S•l�tion 0 Prepro=&og
O
"'"'form•tioo
D
D•t. mhU�lot<or><ot.tion 0
Initial Target Preprocessed Transformed Model Knowledge
data data data data
FIGURE 1.4: KDD process (modified from [FPSS96c]).
Section 1.2 Data Mining Versus Knowledge Discovery in Databases 11
modified to facilitate use by techniques that require specific types of data distributions.
Some attribute values may be combined to provide new values, thus reducing the com
plexity of the data. For example, current date and birth date could be replaced by age.
One attribute could be substituted for another. An example would be replacing a sequence
of actual attribute values with the differences between consecutive values. Real valued
attributes may be more easily handled by partitioning the values into ranges and using
these discrete range values. Some data values may actually be removed. Outliers, extreme
values that occur infrequently, may actually be removed. The data may be transformed
by applying a function to the values. A common transformation function is to use the log
of the value rather than the value itself. These techniques make the mining task easier by
reducing the dimensionality (number of attributes) or by reducing the variability of the
data values. The removal of outliers can actually improve the quality of the results. As
with all steps in the KDD process, however, care must be used in performing transfor
mation. If used incorrectly, the transformation could actually change the data such that
the results of the data mining step are inaccurate.
Visualization refers to the visual presentation of data. The old expression "a picture
is worth a thousand words" certainly is true when examining the structure of data. For
example, a line graph that shows the distribution of a data variable is easier to understand
and perhaps more informative than the formula for the corresponding distribution. The use
of visualization techniques allows users to summarize, extra.ct, and grasp more complex
results than more mathematical or text type descriptions of the results. Visualization
techniques include:
• Graphical: Traditional graph structures including bar charts, pie charts, histograms,
and line graphs may be used.
• Geometric: Geometric techniques include the. box plot and scatter diagram
techniques.
• Icon-based: Using figures, colors, or other icons can improve the presentation of
the results.
• Pixel-based: With these techniques each data value is shown as a uniquely colored
pixel.
• Hierarchical: These techniques hierarchically divide the display area (screen) into
regions based on data values.
• Hybrid: The preceding approaches can be combined into one display.
Any of these approaches may be two-dimensional or three-dimensional. Visualization
tools can be used to summarize data as a data mining technique itself. In addition,
visualization can be used to show the complex results of data mining tasks.
The data mining process itself is complex. As we will see in later chapters, there
are many different data mining applications and algorithms. These algorithms must be
carefully applied to be effective. Discovered patterns must be correctly interpreted and
properly evaluated to ensure that the resulting information is meaningful and accurate.

Databases
Algorithms
Information
retrieval
Statistics
Machine
learning
FIGURE 1 .5: Historical perspective of data mining.
1 .2.1 The Development of Data Mining
The current evolution of data mining functions and products is the result of years of
influence from many disciplines, including databases, information retrieval, statistics,
algorithms, and machine learning (Figure 1 .5). Another computer science area that has
had a major impact on the KDD process is multimedia and graphics. A major goal of KDD
is to be able to describe the results of the KDD process in a meaningful manner. Because
many different results are often produced, this is a nontrivial problem. Visualization
techniques often involve sophisticated multimedia and graphics presentations. In addition,
data mining techniques can be applied to multimedia applications.
Unlike previous research in these disparate areas, a major trend in the database
community is to combine results from these seemingly different disciplines into one
unifying data or algorithmic approach. Although in its infancy, the ultimate goal of this
evolution is to develop a "big picture" view of the area that will facilitate integration of
the various types of applications into real-world user domains.
Table 1 . 1 shows developments in the areas of artificial intelligence (AI), information
retrieval (IR), databases (DB), and statistics (Stat) leading to the current view of data
mining. These different historical influences, which have led to the development of the
total data mining area, have given rise to different views of what data mining functions
actually are (RG99]:
• Induction is used to proceed from very specific knowledge to more general infor
mation. This type of technique is often found in AI applications.
• Because the primary objective of data mining is to describe some characteristics
of a set of data by a general model, this approach can be viewed as a type of com
pression . Here the detailed data within the database are abstracted and compressed
to a smaller description of the data characteristics that are found in the model.
• As stated earlier, the data mining process itself can be viewed as a type of querying
the underlying database. Indeed, an ongoing direction of data mining research is
Section 1 .2 Data Mining Versus Knowledge Discovery in Databases 13
TAB LE 1 . 1 : Time Line of Data Mining Development
Time Area
Late 1 700s Stat
Early 1900s Stat
Early 1920s Stat
Early 1940s AI
Early 1950s
Early 1 950s
Late 1 950s AI
Late 1950s Stat
Early 1960s AI
Early 1960s DB
Mid 1960s
Mid 1 960s Stat
IR
IR
Stat
Late 1 960s DB
Early 1 970s IR
Mid 1 970s AI
Late 1970s Stat
Late 1 970s Stat
Early 1980s AI
Mid 1 980s AI
Early 1 990s DB
1 990s DB
1 990s DB
Contribution
Bayes theorem of probability
Regression analysis
Maximum likelihood estimate
Neural networks
Nearest neighbor
Single link
Perceptron
Resampling, bias reduction, jackknife estimator
ML started
Batch reports
Decision trees
Linear models for classification
Similarity measures
Clustering
Exploratory data analysis (EDA)
Relational data model
SMART IR systems
Genetic algorithms
Estimation with incomplete data (EM algorithm)
K-means clustering
Kohonen self-organizing map
Decision tree algorithms
Association rule algorithms
Web and search engines
Data warehousing
Online analytic processing (OLAP)
Reference
[Bay63]
[Fis2 1 ]
[MP43]
[FJ5 1 ]
[FLP+51]
[Ros58]
[FF63]
[HMS66]
[Nil65]
[Cod70]
[Sal7 1 ]
[Hol75]
[DLR77]
(Koh82]
[Qui86]
how to define a data mining query and whether a query language (like SQL) can
be developed to capture the many different types of data mining queries.
• �escrib�ng a lar�e database can be viewed as using ap
proximation to help uncover
hidden mformatwn about the data.
• When dealing with large databases, the impact of size and efficiency of developing
an abstract model can be thought of as a type of search problem.
It
.
is int�resting to t�nk about the various data mining problems and how each may be
VIewed m several different perspectives based on the viewpoint and background of the
rese
.
�cher
.
or d�;eloper. "W_e �ention these different perspectives only to give the reader
the
.
f�ll picture of data rmmng. Often, due to the varied backgrounds of the data mining
partic
.
Ipants
:
we find that the s�me problem� (and perhaps even the same solutions) are
descnbed differently. Indeed, different terrmnologies can lead to misunderstandings and

apprehension among the different players. You can see statisticians voice co�cern over
the compounded use of estimates (approximation) with results being generalized when
they should not be. Database researchers often voice concern about the inefficiency of
many proposed AI algorithms, particularly when used on very large databases. IR and
those interested in data mining of textual databases might be concerned about the fact
that many algorithms are targeted only to numeric data. The approach taken in this book
is to examine data mining contributions from all these different disciplines together.
There are at least two issues that characterize a database perspective of examining
data mining concepts: efficiency and scalability. Any solutions to data mining problems
must be able to perform well against real-world databases. As part of the efficiency, we
are concerned about both the algorithms and the data structures used. Parallelization may
be used to improve efficiency. In addition, how the proposed algorithms behave as the
associated database is updated is also important. Many proposed data mining algorithms
may work well against a static database, but they may be extremely inefficient as changes
are made to the database. As database practitioners, we are interested in how algorithms
perform against very large databases, not "toy" problems. We also usually assume that
the data are stored on disk and may even be distributed.
1 .3 DATA MINING ISSUES
There are many important implementation issues associated with data mining:
1. Human interaction: Since data mining problems are often not precisely stated,
interfaces may be needed with both domain and technical experts. Technical experts
are used to formulate the queries and assist in interpreting the results. Users are
needed to identify training data and desired results.
2. Overfitting: When a model is generated that is associated with a given database
state it is desirable that the model also fit future database states. Overfitting occurs
whe� the model does not fit future states. This may be caused by assumptions that
are made about the data or may simply be caused by the small size of the training
database. For example, a classification model for an employee database may be
developed to classify employees as short, medium, or tall. If the training database
is quite small, the model might erroneously indicate that a short person is anyone
under five feet eight inches because there is only one entry in the training database
under five feet eight. In this case, many future employees would be erroneously
classified as short. Overfitting can arise under other circumstances as well, even
though the data are not changing.
3. Outliers: There are often many data entries that do not fit nicely into the derived
model. This becomes even more of an issue with very large databases. If a model
is developed that includes these outliers, then the model may not behave well for
data that are not outliers.
4. Interpretation of results: Currently, data mining output may require experts to
correctly interpret the results, which might otherwise be meaningless to the average
database user.
5. Visualization of results: To easily view and understand the output of data mining
algorithms, visualization of the results is helpful.
Section 1.4 Data Mining Metrics 15
6. Large datasets: The massive datasets associated with data mining create problems
when applying algorithms designed for small datasets. Many modeling applica
tions grow exponentially on the dataset size and thus are too inefficient for larger
datasets. Sampling and parallelization are effective tools to attack this scalability
problem.
7. High dimensionality: A conventional database schema may be composed of many
different attributes. The problem here is that not all attributes may be needed to
solve a given data mining problem. In fact, the use of some attributes may interfere
with the correct completion of a data mining task. The use of other attributes may
simply increase the overall complexity and decrease the efficiency of an algorithm.
This problem is sometimes referred to as the dimensionality curse, meaning that
there are many attributes (dimensions) involved and it is difficult to determine
which ones should be used. One solution to this high dimensionality problem is
to reduce the number of attributes, which is known as dimensionality reduction.
However, determining which attributes not needed is not always easy to do.
8. Multimedia data: Most previous data mining algorithms are targeted to traditional
data types (numeric, character, text, etc.). The use of multimedia data such as is
found in GIS databases complicates or invalidates many proposed algorithms.
9. Missing data: During the preprocessing phase of KDD, missing data may be
replaced with estimates. This and other approaches to handling missing data can
lead to invalid results in the data mining step.
10. Irrelevant data: Some attributes in the database might not be of interest to the
data mining task being developed.
11. Noisy data: Some attribute values might be invalid or incorrect. These values are
often corrected before running data mining applications.
12. Changing data: Databases cannot be assumed to be static. However, most data
mining algorithms do assume a static database. This requires that the algorithm be
completely rerun anytime the database changes.
13. Integration: The KDD process is not currently integrated into normal data pro
cessing activities. KDD requests may be treated as special, unusual, or one-time
needs. This makes them inefficient, ineffective, and not general enough to be used
on an ongoing basis. Integration of data mining functions into traditional DBMS
systems is certainly a desirable goal.
14. Application: Determining the intended use for the information obtained from the
data mining function is a challenge. Indeed, how business executives can effectively
use the output is sometimes considered the more difficult part, not the running of
the algorithms themselves. B ecause the data are of a type that has not previously
been known, business practices may have to be modified to determine how to
effectively use the information uncovered.
These issues should be addressed by data mining algorithms and products.
1 .4 DATA MINING METRICS
Measuring the effectiveness or usefulness of a data mining approach is not always
straightforward. In fact, different metrics could be used for different techniques and

also based on the interest level. From an overall business or usefulness perspective, a
measure such as return on investment (ROI) could be used. ROI examines the difference
between what the data mining technique costs and what the savings or benefits from
its use are. Of course, this would be difficult to measure because the return is hard to
quantify. It could be measured as increased sales, reduced advertising expe�d
.
iture, or
both. In a specific advertising campaign implemented via targeted �a�alog mmlmgs, �e
percentage of catalog recipients and the amount of
.
�urchase per rectptent would provtde
one means to measure the effectiveness of the mmhngs.
In this text, however, we use a more computer science/database perspective to
measure various data mining approaches. We assume that the business management
has determined that a particular data mining application be made. They subsequently
will determine the overall effectiveness of the approach using some ROI (or related)
strategy
. Our objective is to compare different alternatives to implementing a spe�ific
data mining task. The metrics used include the traditional met�cs of s?ace
.
and ttme
based on complexity analysis
. In some cases, such as accuracy m classtficatwn, more
specific �etrics targeted to a data mining task may be used
.
1 .5 SOCIAL IMPLICATIONS OF DATA MINI NG
The integration of data mining techniques into normal day-to
.
-�ay activities
.
has become
commonplace
. We are confronted daily with targeted adverttsmg, and busmesses have
become more efficient through the use of data mining activities to reduce costs
. Data
mining adversaries, however, are concerned that this informati�n is being obtained �t
the cost of reduced privacy. Data mining applications can denve m�ch d�mographtc
information concerning customers that was previously not known or �dden �� the dat�.
The unauthorized use of such data could result in the disclosure of mformat10n that ts
deemed to be confidential.
We have recently seen an increase in interest in data mining techniques tm�ge�ed
to such applications as fraud detection, identifying criminal suspects, and predtctwn
of potential terrorists
. These can be viewed as types of classifica�ion problems.
.
�he
approach that is often used here is one of "profiling" the ty�ical �e�av10r or ch:rractenstlcs
involved. Indeed, many classification techniques work by tdenttfymg the attnbute va
.
lues
that commonly occur for the target class. Subsequent records will be the
.
n cl�ssified
based on these attribute values. Keep in mind that these approaches to classificatiOn are
imperfect. Mistakes can be made
. Just because an individual �akes a series of credit
card purchases that are similar to those often made when a card IS stolen does not mean
that the card is stolen or that the individual js a criminal
.
Users of data mining techniques must be sensitive to these issues and must not
violate any privacy directives or guidelines
.
1 .6 DATA MINI NG FROM A DATABASE PERSPECTIVE
Data mining can be studied from many different perspectives
. An IR researcher p�o�a?ly
would concentrate on the use of data mining techniques to access text data; a statistiCian
might look primarily at the historical techniques, includi�g time �eries an�l�sis, �ypoth
esis testing, and applications of Bayes theorem; a machme learrung sp�cialist might be
interested primarily in data mining algorithms that learn; and an algonthms researc�er
would be interested in studying and comparing algorithms based on type and complexity.
Section 1.7 The Future 1 7
The study of data mining from a database perspective involves looking at all types
of data mining applications and techniques. However, we are interested primarily in those
that are of practical interest. While our interest is not limited to any particular type of
algorithm or approach, we are concerned about the following implementation issues:
• Scalability: Algorithms that do not scale up to perform well with massive real
world datasets are of limited application. Related to this is the fact that techniques
should work regardless of the amount of available main memory.
• Real-world data: Real-world data are noisy and have many missing attribute
values
. Algorithms should be able to work even in the presence of these problems.
• Update: Many data mining algorithms work with static datasets. This is not a
realistic assumption.
• Ease of use: Although some algorithms may work well, they may not be well
received by users if they are difficult to use or understand.
These issues are crucial if applications are to be accepted a:nd used in the workplace
.
Throughout the text we will mention how techniques perforn1 in these and other imple
mentation categories.
Data mining today is in a similar state as that of databases in the early 1960s. At
that time, each database application was implemented independently even though there
were many similarities between different applications
. In the mid 1960s, an abundance
of database management systems (DBMS) like tools (such as bill of material systems
including DBOMP and CFMS) emerged. While these made the development of applica
tions easier, there were still different tools for different applications. The rise of DBMS
occurred in the early 1970s. Their success has been due partly to the abstraction of data
definition and access primitives to a small core of needed requirements. This abstraction
process has yet to be performed for data mining tasks. Each task is treated separately
.
Most data mining work (to date) has focused on specific algorithms to realize each indi
vidual data mining task
. There is no accepted abstraction to a small set of primitives
.
One goal of some database researchers is the development of such an abstraction.
One crucial part of the database abstraction is query processing support. One reason
relational databases are so popular today is the development of SQL. It is easy to use
(at least when compared with earlier query languages such as the DBTG or IMS DML)
and has become a standard query language implemented by all major DBMS vendors.
SQL also has well-defined optimization strategies
.·Although there currently is no corre
sponding data mining language, there is ongoing work in the area of extending SQL to
support data mining tasks.
1.7 THE FUTURE
The advent of the relational data model and SQL were milestones in the evolution of
database systems.Currently, data mining is little more than a set of tools that can be used
to uncover previously hidden information in a database.While there are many tools to aid
in this process, there is no all-encompassing model or approach
. Over the next few years,
not only will there be more efficient algorithms with better interface techniques, but also
steps will be taken to develop an all-encompassing model for data mining
. While it may

1 8 Chapter 1 Introduction
not look like the relational model, it probably will include similar items: �l�orithms,
data model, and metrics for goodness (like normal forms). Current data �rung tools
re uire much human inter�ction not only to define the r�quest, bu� also to m�erpret �he
re;ults. As the tools become better and more integrated, �his extensive burna� mteractwn
is likely to decrease. The various data mining applic�twns
.
are of man� diverse types,
so the development of a complete data mining model ts desrrable. A maJor �e:'elopment
will be the creation of a sophisticated "query �anguage" that include� tradittona
t s:;-L
functions as well as more complicated requests such as those found m OLAP on ne
analytic processing) and data mining applications. .
A data mining query language (DMQL) based on SQL has been proposed. Unlike
SQL where the access is assumed to be only to relational databases,
.
DMQL �llows
acce�s to background information such as concept hierarchies. Anot�er difference IS that
the retrieved data need not be a subset or aggregate of data from relatiOns.
.
Thus, a �MQL
statement must indicate the type of knowledge to be mined. Another dtfference IS
.
that
a DMQL statement can indicate the necessary importance dt threshol? that any mmed
informatfon should obey. A BNF statement of DMQL (from [Za199]) Is:
{DMQL) : : =
USE DATABASE {database_name)
{USE HIERARCHY {hierarchy_name)
{rule_spec)
RELATED TO {attr_or_agg_li st)
FROM {relation ( s ) )
[WHERE (condition)]
FOR {attribute)}
[ORDER BY {order-li st)]
{WITH [{kinds_of)] THRESHOLD {threshold_value)
[FOR {attribute ( s ) )] }
The heart of a DMQL statement is the rule specification portion. This is where the
true data mining request is made. The data mining request can be one of the follow-
• A generalized relation is obtained by generalizing data from input data.
.
ing [HFW+96]:
1
• A characteristic rule is a condition that is satisfied by almost all records m a target
class.
• A discriminate rule is a condition that is satisfied by a target class but is different
from conditions satisfied in other classes.
• A classification rule is a set of rules that are used to classify data.
The term knowledge and data discovery management system
_
(KDDMS) has been
coined to describe the future generation of data mining systems that mclu�e not o�ly data
mining tools but also techniques to manage the underlying dat�, ensur� Its constst�ncy
d
and provide concurrency and recovery features. A KDD.M_S will provide access vta a
hoc data mining queries that have been optimized for effictent access.
A KDD process model, CRISP-DM (CRoss-Industry Standard Processf
or Data
Mining)��:arisen and is applicable to many different applications. The model addresses
Section 1 .9 Bibl iographic Notes 1 9
all steps in the KDD process, including the maintenance of the results of the data mining
step. The CRISP-DM life cycle contains the following steps: business understanding,
data understanding, data preparation, modeling, and evaluation deployment. The steps
involved in the CRISP-DM model can be summarized as the "the 5As:" assess, access,
analyze, act, and automate.
1 .8 EXERCISES
1. Identify and describe the phases in the KDD process. How does KDD differ from
data mining?
2. Gather temperature data at one location every hour starting at 8:00 A.M. for 12
straight hours on 3 different days. Plot the three sets of time series data on the
same graph. Analyze the three curves. Do they behave in the same manner? Does
there appear to be a trend in the temperature during the day? Are the three plots
similar? Predict what the next temperature value would have been for the next hour
in each of the 3 days. Compare your prediction with the actual value that occurred.
3. Identify what work you performed in each step of the KDD process for exercise
2. What were the data mining aCtivities you completed?
4. Describe which of the data mining issues you encountered while completing exer
cise 2.
5. Describe how each of the data mining issues discussed in section 1 .3 are com
pounded by the use of real production databases.
6. (Research) Find two other definitions for data mining. Compare these definitions
with the one found in this chapter.
7. (Research) Find at least three examples of data mining applications that have
appeared in the business section of your local newspaper or other news publication.
Describe the data mining applications involved. ·
Although many excellent books have been published that examine data mining and
knowledge discovery in databases, most are high-level books that target users of data
mining techniques and business professionals. There have been, however, some other
technical books that examine data mining approaches and a1gorithms. An excellent text
that is written by one of the foremost experts in the area is Data Mining Concepts and
T
echniques by Jiawei Han and Micheline Katnber [HKOl ] . 1bis book not only examines
data mining algorithms, but also includes a thorough coverage of data warehousing,
OLAP, preprocessing, and data mining language developments. Other books that provide
a technical survey of portions of data mining algorithms include [AdaOO] and [IiMSOl ] .
There have been several recent surveys and overviews of data mining, including
special issues of the Communications o
f the ACM in November 1996 and November
1999, IEEE Transactions on Knowledge and Data Engineering in December 1 996, and
Computer in August 1999. Other survey articles can be found: [FPSS96c], [FPSS96b],
[GGR99a], [Man96], [Man97], and [RG99]. A popular tutorial booklet has been produced
by Two Crows Corporation [Cor99]. A complete discussion of the KDD process is found
in [BA96]. Articles that examine the intersection between databases and data mining
include [Cha97], [Cha98], [CHY96], [Fay98], and [HKMT95]. There have also been

several tutorials surveying data mining concepts: [Agr94], [Agr95], [Han96], and [RS99].
A recent tutorial [Kei97] provided a thorough survey of visualization techniques as well
as a comprehensive bibliography.
The aspect of parallel and distributed data mining has become an important research
topic. A workshop on Large-Scale Parallel KDD Systems was held in 1999 [ZHOO].
The idea of developing an approach to unifying all data mining activities has
been proposed in [FPSS96b], [Man96], and [Man97]. The term KDDMS was first pro
posed in [IM96]. A recent unified model and algebra that supports all major data mining
tasks has been proposed [JLNOO]. The 3W model views data as being divided into three
dimensions. An algebra, called the dimension algebra, has been proposed to access this
three-dimensional world.
DMQL was developed at Simon Fraser University [HFW+96].
There are several KDD and data mining resources. The ACM (Association for
Computing Machinery) has a special interest group, SIGKDD, devoted to the promotion
and dissemination of KDD information. SIGKDD Explorations is a free newsletter pro
duced by, ACM SIGKDD. The ACM SIGKDD home page contains a wealth of resources
concerni�g KDD and data mining (www.acm.org/sigkdd).
A vendor-led group, Data Mining Group (DMG), is active in the development of
data mining standards. Information about DMG can be found at www.dmg.org. The
ISO/IEC standards group has created a final committee draft for an SQL standard includ
ing data mining extensions [ComOl]. In addition, a project begun by a consortium of
data mining vendors and users resulted in the development of the data mining process
model, CRISP-DM (see: www.crisp-dm.org).
There currently are several research journals related to data mining. These include
IEEE Transactions on Knowledge and Data Engineering published l:>y IEEE Computer
Society and Data Mining and Knowledge Discovery from Kluwer Academic Publish
ers. International KDD conferences include the ACM SIGKDD International Conference
on Knowledge Discovery and Data Mining (KDD), the Conference on Information and
Knowledge Management (CIKM), the IEEE International Conference on Data Mining
(ICDM), the European Conference on Principles of Data Mining and Knowledge Dis
covery (PKDD), and the Pacific-Asia Conference on Knowledge Discovery and Data
Mining (PAKDD). KDnuggets News is an e-mail newsletter that is produced biweekly.
It contains a wealth of KDD and data mining information for practitioners, users, and
researchers. Subscriptions are free at www.kdnuggets.com. Additional KDD resources
can be found at Knowledge Discovery Central (www.kdcentral.com).
C H A P T E R 2
Related Concepts
2.1 DATABASE/OLTP SYSTEMS
2.2 FUZZY SETS AND FUZZY LOGIC
2.3 INFORMATION RETRIEVAL
2.4 DECISION SUPPORT SYSTEMS
2.5 DIMENSIONAL MODELING
2.6 DATA WAREHOUSING
2.7 OLAP
2.8 WEB SEARCH ENGINES
2.9 STATISTICS
2.10 MACHINE LEARNING
2.1 1 PATTERN MATCHING
2.12 SUMMARY
2.13 EXERCISES
Data mining applications have existed for thousands of years. For example, the classifi
cation of plants as edible or nonedible is a data mining task. The development of the data
mining discipline has its roots in many other areas. In this chapter we examine many
concepts related to data mining. We briefly introduce each concept and indicate how it
is related to data mining.
2.1 · DATABASE/OLTP SYSTEMS
A database is a collection of data usually associated with some organization or enterprise.
Unlike a simple set, data in a database are usually viewed to have a particular structure
or schema with which it is associated. For example, (/D, Name, Address, Salary, JobNo)
may be the schema for a personnel database. Here the schema indicates that each record
(or tuple) in the database has a value for each of these five attributes. Unlike a file, a
database is independent of the physical method used to store it on disk (or other media).
It also is independent of the applications that access it. A database management system
(DBMS) is the software used to access a database.
Data stored in a database are often viewed in a more abstract manner or data
model. This data model is used to describe the data, attributes, and relationships among
them. A data model is independent of the particular DBMS used to implement and
access the database. In effect, it can be viewed as a documentation and communication
tool to convey the type and structure of the actual data. A common data model is the
21

22 Chapter 2 Related Concepts
FIGURE 2 . 1 : ER model example
.
ER (entit�-relationship) data model. Although originally proposed in 1
9 7 6 , the ER �ata
model is still used today with many extensions and improvements to the first destgn
.
Example 2
.1 illustrates the use of an ER model with an associated ER diagram seen in
Figure 2. 1. The basic components of an ER model are the entities an� the r�latio�ship
_
s
.
An entity is ass ociated with a real-world object and has a key that umquely tdenttfies tt.
A relationship is used to describe as association that exists between entities
.
EXAMPLE 2.1
An employee database consists of employees and information concerning the j
_ob� that
they perform. An entity would be an Employee and the key could be the ID
. Stmtlarly,
different jobs can be associated with a job number so that we can think of the Job as an
entity with key JobNo
. In Figure 2
.1
, there is a rectangle for each entity. The diamond is
used to represent the relationship between the two entities
. Here the relationship HasJob
indicates that a specific employee with key ID has a particular job with key JobNo
.The
attributes associated with Employee are {ID, Name, Address, Salary} and the attributes
for Job are {JobNo, NobDesc, PayRange}.
The ER model is often used to abstractly view the data independent of DBMS
.
DBMS systems often view the data in a structure more like a table. This gives rise to
the relational model, where data are viewed as being composed of relations
. Taking a
mathematical perspective, a relation is a subset of a Cartesian product
. Imagine looking
at the domain, or set of values, associated with each atttibute in the Employee example
.
A relation R could then be viewed as a subset of the product of the domains:
R £; dom(ID ) x dom(Name) x dom(Address) x dom(Salary) x dom(JobNo) (2
.1)
Access to a relation can be performed based on operations in the traditional set algebra
such as union and intersection
. This extended group of set operations is referred to as
relational algebra. An equivalent set based on first-order predicate calculus is called
relational calculus. Access to databases is usually achieved via a query language. This
query language may be based on relational algebra or calculus. · Although many query
languages have been proposed, the standard language used by most DBMSs is SQL
.
Section 2.2
SELECT Name
FROM R
WHERE Salary > 100,000
Fuz:zy Sets and Fuzzy Logic 23
FIGURE 2.2: SQL example
.
Figure 2.2 shows a sample SQL statement issued against the relation R, which lists the
names of all employees with a salary greater than $ 1 00, 00
0.
Users' expectations for queries have increased, as have the amount and sophisti
cation of the associated data
. In the early days of database (DB) and online transac
tion processing (OLTP) systems, simple selects were enough
. Now queries are complex,
involving data distributed over many sites, and they use complicated functions such as
joins, aggregates, and views. Traditional database queries usually involve retrieving data
from a database based on a well-defined query
. As shown in Figure 2.2, a user may
ask to find all employees who earn over $ 100
, 000. This could be viewed as a type of
classification application as we segment the database into two classes: those who have
salaries satisfying the predicate and those who do not. A simple database application
is not thought of as a data mining task, however, because the queries are well defined
with precise results. Data mining applications, conversely, are often vaguely defined with
imprecise results. Users might not even be able to precisely define what they want, let
alone be able to tell if the results of their request are accurate. A database user usually
can tell if the results of his query are not correct
. Thus, it is usually assumed that a
DBMS returns the correct results for a query
. Metrics (instead of quality) often include
such things as response time and throughput
.
When viewed as a query sys tem, data mining queries extend database concepts
.
Data mining problems are often ill-posed with many different solutions. Judging the
effectiveness of the result of a data mining request is often difficult
. A major difference
between data mining queries and those of database systems is the output
. Basic database
queries always output either a subset of the database or aggregates of the data. A data
mining query outputs a KDD object. A KDD object is either a rule, a classification, or a
cluster
. These objects do not exist before executing the query, and they are not part of the
database being queried. Aggregation operators have existed in SQL for years. They do
not return objects existing in the database, but return a model of the data. For example,
an average operator returns the average of a set of attribute values rather than the values
themselves
.This is a simple type of data mining operator.
2.2 FUZZY SETS AND FUZZY LOGIC
A set is normally thought of as a collection of objects. It can be defined by enumerating
the set
F = { l , 2, 3 , 4 , 5 } (2.2)
or by indicating the set membership requirement
F = {x I X E z+ and X :::: 5} (2. 3 )
A fuzzy set is a set, F, in which the set membership function, f, is a real valued (as
opposed to boolean) function with output in the range [ 0, 1]: An element x is said

to belong to F with probability f(x) and simultaneously to be in -.F with probability
1 - f(x). In actuality, this is not a true probability, but rather the degree of truth associated
with the statement that x is in the set. To show the difference, let us look at a fuzzy set
operation. Suppose the membership value for Mary being tall is 0.7 and the value for
her being thin is 0.4. The membership value for her being both is 0.4, the minimum of
the two values. If these were really probabilities, we would look at the product of the
two values.
Fuzzy sets have been used in many computer science and database areas. In the
classification problem, all records in a database are assigned to one of the predefined
classification areas. A common approach to solving the classification problem is to assign
a set membership function to each record for each class. The record is then assigned to
the class that has the highest membership function value. Similarly, fuzzy sets may be
used to describe other data mining functions. Association rules are generated given a
confidence value that indicates the degree to which it holds in the entire database. This
can be thought of as a membership function.
Qqeries can be thought of as defining a set. With traditional database queries,
however, the set membership function is boolean. The set of tuples in relation R that
satisfy the SQL statement in Figure 2.2 can be defined as
{x I x E R and x.Salary > 100, 000} (2.4)
Here x.Salary refers to the Salary attribute within the tuple X. Some queries, however, do
not have a membership function that is boolean. For example, suppose that we wished
to find the names of employees who are tall:
{x I x E R and x is tall} (2.5)
This membership function is not boolean, and thus the results of this query are fuzzy. A
good example of queries of this type are searches performed on the Web.
Figure 2.3 shows the real difference between traditional and fuzzy set membership.
Suppose there are three sets (short, medium, and tall) to which a person can be classified
based on his height. In Figure 2.3(a) the traditional (or crisp) set membership values are
shown. Part (b) shows the triangular view of set membership values. Notice that there
is a gradual decrease in the set membership value for short; there is a gradual increase
and decrease for set membership in the medium set; there is a gradual increase in the set
membership value for tall.
Height
(a) Crisp sets
Short Medium
Height
(b) Fuzzy sets
FIGURE 2.3: Fuzzy vs. traditional set membership.
Tall
Section 2.2 Fuzzy Sets and Fuzzy Logic 25
Fuzzy logic is reasoning with uncertainty. That is, instead of a two valued logic
(true and false), there are multiple values (true, false, maybe). Fuzzy logic has been
used in database systems to retrieve data with imprecise or missing values. In this case,
the membership of records in the query result set is fuzzy. As with traditional boolean
logic, fuzzy logic uses operators such as -., 1, and v. Assuming that x and y are fuzzy
logic statements and that mem(x) defines the membership value, the following values are
commonly used to define the results of these operations:
mem(-.x)
mem(x 1 y)
mem(x v y)
1 - mem(x)
= min(mem(x), mern(y))
max(mem(x), mem(y))
(2.6)
(2.7)
(2.8)
Fuzzy logic uses rules and membership functions to estimate a continuous function. Fuzzy
logic is a valuable tool to develop control systems for such things as elevators, trains,
and heating systems. In these cases, instead of providing a crisp on-off environment, the
fuzzy controller provides a more continuous adjustment.
Most real-world classification problems are fuzzy. This is illustrated by Figure 2.4.
In this figure we graphically show the threshold for approving a loan based on the
income of the individual and the loan amount requested. A loan officer may make the
loan decision by simply approving any loan requests on or above the line and rejecting
any requests that fall below the line [Figure 2.4(a)]. This type of decision would not
be fuzzy. However, this type of decision could lead to elToneous and perhaps costly
decisions by the loan officer. From a data mining perspective, this application is a clas
sification problem; that is, classify a loan application into the approval or reject class.
There are many other factors (other than income) that should be used to predict the
classification problem (such as net worth and credit rating). Even if all the associated
predictors could be identified, the classification problem is not a black-and-white issue.
It is possible that two individuals with exactly the same predictor values should be
placed in two different classes. This is due to the fuzzy nature of this classification. This
is shown by the shading around the line in Figure 2.4(b). We could perhaps classify
Loan
amount
(a) Simplistic loan approval
Loan
amount
Income
(b) Loan approval is not precise
FIG URE 2.4: Fuzzy classification.

2.3
the individuals into multiple classes: approve, reject, unknown, probably approve, and
probably reject. This approach attacks the fuzzy miture of the classification by flagging
the applications requiring more analysis into the three new fuzzy classes. Procedural
policies could .then �e used to determine the ultimate classification of these cases into
the final approve or reject classes. This is one of many possible approaches to defuzzify
the classification problem.
IN FORMATION RETRIEVAL
Information retriewil (IR) (and more recently digit�! libraries and Internet searching)
involves retrieving desired information from textual data. The historical development of
IR was based on effective use of libraries. So a typical 1R request would be to find all
library documents related to a particular subject, for example "data mining." This is,
in fact, a classification task because the set of documents in the library is divided into
classes based on the keywords involved. In IR systems, documents are represented by
document surrogates consisting of data, such as identifiers, title, authors, dates, abstracts,
extracts, reviewg, and keywords. As can be seen, the data consist of both formatted and
unformatted (text) data. The retrieval of documents is based on calculation of a similarity
measure showing how close each document is to the desired results (i.e., the stated query).
Similarity measures are also used in classification and clustering problems.
An IR system consists of a set of documents D = {D1 , . . . , Dn }. The input is a
query, q, often stated as a list of keywords. The similarity between the query and ea�h
document is then calculated: sim(q, D;). This similarity measure is a set membership
function describing the likelihood that the document is of interest (relevant) to the user
based on the user's interest as stated by the query. The effectiveness of the system in
processing the query is often measured by looking at precision and recall:
Precision =
Recall =
1 Relevant and Retrieved I
(2_9)
I Retrieved I
Relevant and Retrieved I
I Relevant I
(2.10)
Precision is used to answer the question: "Are all documents retrieved ones that I am
interested in?" Recall answers: "Have all relevant documents been retrieved?" Here a
document is relevant if it should have been retrieved by the query. Figure 2.5 illustrates
the four possible query results available with IR queries. Of these four quadrants, two
represent desirable outcomes: relevant and retrieved or not relevant and not retrieved.
The other two quadrants represent error situations. Documents that are relevant and not
retrieved should have been retrieved but were not. Documents that are not relevant and
retrieved should not have been retrieved but were. Figure 2.6 illustrates the basic structure
of a conventional information retrieval query.
Many similarity measures have been proposed for use in information retrieval. As
stated earlier, sim (q , D;) 1 s i s n is used to determine the results of a query q applied
to a set of documents D = {D1 , . . . , Dn }. Similarity measures may also be used to clus
ter or classify documents by calculating sim (D; , DJ) for all documents in the database.
Thus, similarity can be used for document-document, query-query, and query-document
measurements. The inverse document frequency (IDF) is often used by similarity mea
sures. IDF assumes that the importance of a keyword in calculating similarity measures is
Section 2.3 Information Retrieval 27
Relevant
Retrieved
Relevant
Not retrieved
Not relevant Not relevant
Retrieved Not retrieved
FIGURE 2.5: IR query result measures.
Keywords
Q::::: G-0Documents
Documents
FIGURE 2.6: Information retrieval query.
Domestic Lion
�
Feline -- Cat
'
Cheetah Tiger
Mix Persian Siamese Tabby Burmese
�
Siberian White Indochi
nese Sumatran South Chinese
FIGURE 2.7: Concept hierarchy.
inversely proportional to the total number of documents that contain it. Given a keyword,
k, and n documents, IDF can be defined as
n
IDFk = lg - + 1
I documents containing k 1
(2. 1 1)
Concept hierarchies are often used in information retrieval systems to show the
relationships between various keywords (concepts) as related to documents. Suppose
you wish to find all documents about cats. Figure 2.7 illustrates a concept hierarchy that
could be used to answer this query. This figure shows that feline and cat are similar
terms. In addition, a cat may be domestic or tiger or lion or cheetah. In turn, a tiger
may be a Siberian, White, Indochinese, Sumatran, or South Chinese. Nodes lower in the
tree represent more specific groups of tigers. When a user requests a book on tigers, this
query could be modified by replacing the keyword "tiger" with a keyword at a higher

Tall Tall
Classified tall Classified
not tall
20 10
45 25
Not tall Not tall
Classified tall Classified
not tall
FIGURE 2.8: Precision and recall applied to classification.
level in the tree, suQh as "cat." Although this would result in a higher reca
11, the precision
would decrease. A concept hierarchy may actually be a DAG (directed acyclic graph)
rather than a tree.
�.
has had
.
a �a
jor impact on the development of data mining. Much of the
dat� nunmg classification and clustering approaches had their origins in the document
retrieval problems of lib�ary scie�ce and
.
information retrieval. Many of the similarity
m�a
.
sures de
.
vel�ped f�r mformatwn retneval have been applied to more general data
numn� �pphcatl?ns
:
Similarly, the precision and recall measures are often applied to
data �rung �pplicatw�s,
.
as illust�ate? in Example 2.2. Concept hierarchies are frequently
used
.
m
.
spatial data numn� apph��tlons
:
Data mining consists of many more types of
applicatwns than are found m tradltwnal mformation retrieval. The linking and predictive
tasks have no real counterpart in IR, for example.
EXAMPLE 2.2
The accuracy of a predictive modeling technique can be described based on precision
and recall. Suppose 100 college students are to be classified based on height. In actuality,
there are 30 tall students and 70 who are not tall. A classification technique classifies 65
student� as
.
tall and 35 as not tall. The precision and recall applied to this problem are
shown m Figure 2.8. The precision is 20/65, while the recall is 20/30. The precision is
low because so many students who are not tall are classified as such.
2.4 DECISION SUPPORT SYSTE MS
Decisio� support systems (DSS) are comprehensive computer systems and related tools
that assist managers in making decisions and solving problems. The goal is to I
·m
th d · · ki
. . . . prove
e eclSlon-ma ng process by providmg specific mformation needed by manag t
Th d
.
" f
. . emen .
ese s�stems tuer �om �adihon�l database management systems in that more ad
hoc quenes and custonuzed mformatwn may be provided Recently the te ·
. .
· , rms executzve
mformatwn systems (E/S) and executive support systems (ESS) have evolv d 11
Th II · d 1 ·
e as we .
ese systems a rum at eve opmg the business structure and comput t h ·
er ec mques to
2.5
Section 2.5 Dimensional Modeling 29
better provide information needed by management to make effective business decisions.
Data mining can be thought of as a suite of tools that assist in the overall DSS process;
that is, DSS may use data mining tools.
In many ways the term DSS is much more broad than the term data mining. While
a DSS usually contains data mining tools, this need not be so. Likewise, a data min
ing tool need not be contained in a DSS system. A decision support system could be
enterprise-wide, thus allowing upper-level managers the data needed to make intelligent
business decisions that impact the entire company. A DSS typically operates using data
warehouse data. Alternatively, a DSS could be built around a single user and a PC. The
bottom line is that the DSS gives managers the tools needed to make intelligent decisions.
DIMENSIONAL MODELING
Dimensional modeling is a different way to view and interrogate data in a database.
This view may be used in a DSS in conjunction with data mining tasks. Although not
required, for efficiency purposes the data may be stored using different data structures
as well. Decision support applications often require that information be obtained along
many dimensions. For example, a sales manager may want to obtain information about
the amount of sales in a geographic region, particular time frame, and by-product type.
This query requires three dimensions. A dimension is a collection of logically related
attributes and is viewed as an axis for modeling the data. The time dimension could
be divided into many different granularities: millennium, century, decade, year, month,
day, hour, minute, or second. Within each dimension, these entities form levels on which
various DSS questions may be asked. The specific data stored are called the f
acts and
usually are numeric data. Facts consist of measures and context data. The measures are
the numeric attributes about the facts that are queried. DSS queries may access the facts
from many different dimensions and levels. The levels in each dimension facilitate the
retrieval of facts at different levels. For example, the sales information could be obtained
for the year 1999, for the month of February in the year 2000, or between the times
of 1 0 and 1 1 A.M. on March 1 , 2000. The same query could be formulated for a more
general level, roll up, or for a more specific level, drill down.
Table 2.1 shows a relation with three dimensions: Products, Location, and Date.
Determining a key for this relation could be difficult because it is possible for the same
product to be sold multiple times on the same day. In this case, product 150 was sold at
two different times in Dallas on the same day. A finer granularity on the time (perhaps
down to the minute rather than date as is here) could make a key. However, this illustrates
that choice of key may be difficult. The same multidimensional data may also be viewed
as a cube. Figure 2.9 shows a view of the data from Table 2. 1 as a cube. Each dimension
is seen as an axis for the cube. This cube has one fact for each unique combination of
dimension values. In this case, we could have 8 * 7 * 5 = 230 facts stored (even though
the relation showed only 10 tuples). Obviously, this sparse amount of data would need
to be stored efficiently to reduce the amount of space required.
The levels of a dimension may support a partial order or a total order and can be
viewed via a directed path, a hierarchy, or a lattice. To be consistent with earlier uses of
the term, we use aggregation hierarchy, even though it may be a lattice, to refer to the
order relationship among different levels in a dimension. We use < to represent this order
relationship. X < Y if X and Y are levels in the same dimension and X is contained

TABLE 2. 1 : Relational View of Multidimensional Data
ProdiD LociD Date
123 Dallas 022900
1 23 Houston 020100
1 50 Dallas 03 1500
150 Dallas 03 1500
150 Fort Worth 021000
150 Chicago 012000
200 Seattle 030100
300 Rochester 021500
500 Bradenton 022000
500 Chicago 012000
Seattle
�--1--+--+-+---V
Rochester e.--+---+---+-f---V
Houston
1----l----+----+-1---V
FortWorth
e.--+---+---+-f---V
Dallas
e.--+---+---+-f---1"
Chicago
1----l----+----+-1--V
Bradenton L-...L__L---L_L-__v
123 150 200 300 500
Products
Quantity
5
10
1
5
5
20
5
200
15
10
FIGURE 2.9: Cube.
UnitPrice
25
20
100
95
80
75
50
5
20
25
in Y. Figure 2.1 0(a) shows a total order relationship among levels in the Product dimen
sion from Figure 2.9. Here Product < Type < Company. The two facts that we are
using in this example are Quantity and UnitPrice. When this order relationship is satis
fied between two levels, there is an aggregate type of relationship among the facts. Here
the Quantity of products sold of a particular type is the sum of the quantities for all
products within that type. Similarly, the quantity of products sold for a company is the
sum of all products sold across all product types. The aggregate operation is not always
the sum, however. When looking at the unit price, it would be reasonable to look at
Year
1
· Month Season Planet
1
Day Country Continent
1 .1
Company Hour AM/PM State Region
I I 1
Product type Minute Zip Code County
I I
Product Second City
(a) Product dimension (b) T
ime dimension (c) Location dimension
FIGURE 2.10: Aggregation hierarchies.
such aggregate operations as average, maximum, and minimum prices. Figure 2.10(b)
shows a hierarchical relationship among levels in the time dimension, and Figure 2. 10(c)
shows a lattice for the location dimension. Here Day < Month but Day f:. Season. The
aggregation can be applied only to levels that can be found in the same path as defined
by the < relationship. When levels for a dimension satisfy tlJ.is structure, the facts along
these dimensions are said to be additive. If we add the sales data for all 24 hours in
a day, we get the sales data for that day. This is not always the case. Looking at the
location dimension, if we were to sum up the sales data for all zip codes in a given
county, however, we would not get the sales data for the county. Thus, these dimensions
are not additive. This is due to the fact that zip codes may span different counties. The
use of nonadditive dimensions complicate the roll up and drill down applications.
2.5.1 Multidimensional Schemas
Specialized schemas have been developed to portray multidimensional data. These in
clude star schema, snowflake schema, and fact constellation schema.
A star schema shows data as a collection of two types: facts and dimensions.
Unlike a relational schema, which is flat, a star schema is a graphical view of the data.
At the center of the star, the data being examined, the facts, are shown in f
act tables
(sometimes called major tables). On the outside of the facts, each dimension is shown
separately in dimension tables (sometimes called minor tables). The simplest star schema
has one fact table with multiple dimension tables. In tlJ.is case each fact points to one
tuple in each of the dimensions. The actual data being accessed are stored in the fact
tables and thus tend to be quite large. Descriptive information about the dimensions
is stored in the dimensions tables, which tend to be smaller. Figure 2.1 1 shows a star
schema based on the data in Figure 2.9. Here one extra dimension, division, is shown.
The facts include the quantity and price, while the dimensions are the product, time,

Product ID
1
Day iD
Description
1/
Day
'JYpe Month
T
ype Description Quarter
Product ID
Year
Product
Day iD
Day
I
Salesman ID
Location ID
�
Quantity
Unit Price
Salesman ID
Location iD
Dept
Sales Zip Code
Dept Desc
State
Div
City
Div Desc
Location
Division
FIGURE 2.1 1 : Star schema.
location, and division. Descriptive information about a product includes the description,
type, and type description. Access to the fact table from a dimension table can be accom
plished via a join between a dimension table and the fact table on particular dimension
values. For example, we could access all locations in Dallas by doing the following
SQL query:
SELECT Quant i t y , Price
FROM Facts , Locat ion
Where ( Fact s . Locat ioniD = Locat i on . Locat i oniD)
and
( Locat ion . City = ' ' Dallas ' ' )
Here the LocationiD is a foreign key from the fact table to the Location dimension table.
The primary key for the fact table is a collection of foreign keys that point to the dimen
sion tables. Although this example shows only one fact table, there may be several. In
addition, a dimension table may itself point to another dimension table.
A star schema view can be obtained via a relational system where each dimension
is a table and the facts are stored in a fact table. The facts can be accessed relatively
efficiently through the creation of indices for the dimensions. However, the sheer volume
of the data involved, exacerbated by the need to summarize the fact information at
different levels across all the dimensions, complicates the process. For example, we may
wish to see all sales in all regions for a particular day. Or we may want to see all sales in
May 2000 for the salesmen in a specific department. To ensure efficiency of access, facts
may be stored for all possible aggregation levels. The fact data would then be extended
to include a level indicator.
We assume that the data are stored as both fact tables and dimension tables. Data
in the fact table can be viewed as a regular relation with an attribute for each fact to be
stored and the key being the values for each dimension. There are four basic approaches
to the storage of data in dimension tables [PB99]. Each dimension table can
_
be stored in
one of these four manners. Figure 2.12 illustrates these four approaches w1�h the
_
sal�s
data The first technique, the flattened technique, stores the data for each d1mens�on m
exa�tly one table. There is one row in the table for each row in the l�west le:el m. the
dimensional model. The key to the data are the attributes for all levels m th�t d1mens1?n.
With the flattened approach, a roll up is accomplished by a SUM aggregatwn operatwn
over the appropriate tuples. Even though this approach suffers f
rom space problems as the
D Location ID1 Quantity, Unit Price)
sales ( Product ID , Day ID , Salesman I ,
Product ( Product ID, Description , Type, Type Description)
Day (� Month, Quarter, Year)
Division ( Salesman ID , Dept , Dept Desc , Div, Div Desc)
Locat ion (Location ID , Zip Code , State , City)
(a) Flattened
ID' Salesman ID , Location ID" Quantity, Unit Price)
sales ( Product ID, Da
y
Product ( Product ID, Desc ription , Type )
Types (� Type Description)
Day (Day ID , Month)
Months ( Month, Quarter)
Quarters ( Quarter , Year)
Years ( Year)
salesman ( Salesman ID, Dept )
Depts (Dept, Dept Desc, Div)
Divs (Div, Div Des c)
Location (Location ID , Zip Code)
Zip ( Zip Code , City)
Cities ( State , City)
states ( State)
(b) Normalized
Sales ( Product ID , Da
y ID, salesman ID, Location ID, Quantity, Unit Price)
Product ( Product ID, Description, Type, Type Description)
Types (� Type Description)
Day (Day ID, Month, Quarter, Year)
Months (Month, Quarter, Year)
Quarters (Quarter, Year)
Years (Year)
salesman ( Salesman ID , Dept , Dept Desc , Div, Div Desc)
Depts (Dep
t , Dept Desc , Div, Div Desc)
Divs ( Div, Div Desc)
Location (Location ID , Zip Code , State , City)
Zip ( Z ip Code, State , City)
Cities ( State, City)
states (�)
(c) Expanded
Sales ( Product ID, Da
y ID, Salesman ID , Location I�, �uantity, Unit
Product ( Product ID, Description, Type , Type Descr�pt�on, Level No)
Day (� Month, Quarter , Year, Level �o)
.
Divis ion (Salesman ID, Dept , Dept Desc , D�v, D�v Desc , Level No)
Location (Location ID, Zip Code , State , City, Level No)
(d) Levelized
FIGURE 2 . 1 2: Options to implement star schema.
Price)

number of attributes grows with the number of levels, it does facilitate the simple
implementation of many DSS applications via the use of traditional SQL aggregation
operations.
The second technique to store a dimension table is called the normalized technique,
where a table exists for each level in each dimension. Each table has one tuple for every
occurrence at that level. As with traditional normalization, duplication is removed at the
expense of creating more tables and potentially more expensive access to factual data
due to the requirement of more joins. Each lower level dimension table has a foreign
key pointing to the next higher level table.
Using expanded dimension tables achieves the operational advantages of both the
flattened and the normalized views, while actually increasing the space requirements
beyond that of the flattened approach. The number of dimension tables is identical to
that in the normalized approach, and the structure of the lowest level dimension table
is identical to that in the flattened technique. Each higher level dimension table has, in
addition to the attributes existing for the normalized structure, attributes from all higher
level dimensiol}S.
The levelized approach has one dimension table as does the flattened technique.
However, the aggregations have already been performed. There is one tuple for each
instance of each level in the dimension, the same number existing in all normalized
tables combined. In addition, attributes are added to show the level number.
An extension of the star schema, the snowflake schema facilitates more complex
data views. In this cas�. the aggregation hierarchy is shown explicitly in the schema itself.
An example of a snowflake schema based on the sales data is shown in Figure 2.13. A
snowflake schema can be viewed as a partially normalized version of the correspond
ing star schema. The division and location dimension tables have been normalized in
th�s figure.
2.5.2 Indexing
With multidimensional data, indices help to reduce the overhead of scanning the extrem
ely large tables. Although the indices used are not defined specifically to support multi
dimensional data, they do have inherent advantages in their use for these types of data.
With bitmap indices each tuple in the table (fact table or dimension table) is repre
sented by a predefined bit so that a table with n tuples would be represented by a vector
of n bits. The first tuple in the table is associated with the first bit, the second with the
second bit, and so on. There is a unique bit vector for each value in the domain. This
vector indicates which associated tuples in the table have that domain value. To find the
precise tuples, an address or pointer to each tuple would also have to be associated with
each bit position, not each vector. Bitmap indices facilitate easy functions such as join
and aggregation through the use of bit arithmetic operations. Bitmap indices also save
space over more traditional indices where pointers to records are maintained.
Join indices supportjoins by precomputing tuples from tables that join together and
pointing to the tuples in those tables. When used for multidimensional data, a common
approach �s to 'cteate a join index between a dimension table and a fact table. This
facilitates the efficient identification of facts for a specific dimension level and/or value.
Join indices can be created for multiway joins across multiple dimension tables. Join
indices can be constructed using bitmaps as opposed to pointers.
Section 2.6 Data Warehousing 35
Product ID
1
DayiD
Description
I
Day
Type Month
Quarter
Type Description Product ID
Year
Product Day iD
Salesman ID
Day
Dept Location ID
�Location IDI
Dept Desc Quantity
Div I Salesman ID r Unit Price
Div Desc � Dept Sales Zip Code f--- Zip Code
Department Salesman Location State
City
Zip Codes
FIGURE 2 . 1 3: Snowflake schema.
Traditional B-tree indices may be constructed to access each entry in the fact table.
Here the key would be the combination of the foreign keys to the dimension tables.
2.6 DATA WAREHOUSING
Decision support systems (DSS) are subject-oriented, integ:rated, time-variant, and non
volatile. The term data warehouse was first used by William Inmon in the early 1980s. He
defined data warehouse to be a set of data that supports DSS and is "subject-oriented,
integrated, time-variant, nonvolatile" [Inm95]. With data warehousing, corporate-wide
data (current and )listorical) are merged into a single repository. Traditional databases
contain operational data that represent the day-to7day needs of a company. Traditional
business data proces&!ng (such as billing, inventory control, payroll, and manufactur
ing support) support online transaction processing and. batch reporting applications. A
data warehouse, however, COJ1tains inf
ormational data, which are used to support other
functions suet> as planning and forecasting. Although much of the content is similar
between the operational and informational data, much is different. As a matter of fact,
the operational data are transformed into the informational data. Example 2.3 illustrates
the difference between the two.
EXAMPLE 2.3
The ACME Manufacturing Company maintains several operational databases: sales,
billing, employee, manufacturing, and warehousing. These are used to support the day
to-day functions such as writing paychecks, placing orders for supplies needed in the
manufacturing process, billing customers, and so on. The president of ACME, Stephanie
Eich, wishes to streamline manufacturing to concentrate production on the most profitable
products. To perform this task; she asks several "what if' questions, does a projection of
current sales into the future, and examines data at different geographic and time dimen
sions. All the data that she needs to perform this task can be found in one or more of
the existing databases. However, it is not easily retrieved in the exact format that she
desires. A data warehouse is created with exactly the sales information she needs by

locati
.
on and time. OLAP retrieval tools are provided to facilitate quick response to her
questions at any and all dimension granularities.
.
The data w�eh
.
ouse market supports such diverse industries as manufacturing,
retat�, telec�mmumcatw
.
ns, and health care. Think of a personnel database for a company
that ts co�tm�ally mo?tfied as personnel are added and deleted. A personnel database
that contams t�formatwn about the current set of employees is sufficient. However, if
management wtshes t� analyze trends with respect to employment history, more data are
ne�d�d. They may wtsh to determine if there is a problem with too many employees
qmttmg. To analyze
.
this problem, they would need to know which employees have
left, when they left, why they left, and other information about their employment. For
man�gement to make these types of high-level business analyses, more historical data
(notJUst the curre�t s�apshot that is typically stored) and data from other sources (perhaps
employment applicatiOns and results of exit interviews) are required. In addition, some of
the data in the pe�sonnel database, such as address, may not be needed. A data warehouse
provides just thistinformation. In a nutshell, a data warehouse is a data repository used
to support decision support systems.
.The basic motivation for this shift to the strategic use of data is to increase business
pro�t�bili�. Tradi�i�nal data processing applications support the day-to-day clerical and
admmtstrative dectswns, while data warehousing supports long-term strategic decisions.
� 1996 report by
.
International Data Corporation (IDC) stated that an average return on
mvestment (RO!) m data warehousing reached 401% [BS97, 292],
Figure 2.!4, adapted from [BS97, Figure 6.1] shows a simple view of a data ware
house. The baste components of a data warehousing system include data migration, the
warehouse, and access too
.
ls. The data are extracted from operational systems, but must
be reformatted, cleansed, mtegrated, and summarized before being placed in the ware
house. M�ch of the operational data are not needed in the warehouse (such as employee
addresse� m
.
E�ample 2.3) andare removed duringthis conversion process. This migration
proc�ss
.
Is stmtlar to that needed for data mining applications except that data mining
apphcatt�ns
.
need not necessarily be performed on summarized or business-wide data.
The a?plicatwns that are shown in Figure 2.14 to access a warehouse include traditional
querymg, OLAP
: �nd data mining. Since the warehouse is stored as a database, it can be
accessed by traditiOnal query languages.
/
QQue<y tooffi
� Transformation 0 �/l
�
� OLAP tools
Opet>tion•l '"' D•t> W><<ho� n
� Data mining tools
FIGURE 2. 1 4: Data wat·ehouse.
Section 2.6 Data Warehousing 37
TABLE 2.2: Comparing Operational Database to Data Warehouse
Operational Data Data Warehouse
Application OLTP OLAP
Use Precise queries Ad hoc
Temporal Snapshot Historical
Modification Dynamic Static
Orientation Application Business
Data Operational values Integrated
Size Gigabits Terabits
Level Detailed Summarized
Access Often Less often
Response Few seconds Minutes
Data schema Relational Star/snowflake
Table 2.2 summarizes the differences between operational data stored in traditional
databases and the data stored in a data warehouse. The traditional database applications
are related to OLTP where the users' requests are stated in a high-level query language
(such as SQL) and the results are subsets of the relationships. Data warehouse applica
tions are directly related to business decisions and analysis of the data, OLAP. While
operational data usually represent facts concerning a snapshot in time (the current time),
a warehouse has historical data as well. Data in the warehouse are not modified as fre
quently as data in a conventional database. Updates are hatched and merged into the
warehouse at weekly or monthly intervals. Although this means that the warehouse data
are not completely up-to-date, this usually is not a problem with the normal decision
support applications. A conventional database usually is related to one application type.
This is a fallout of the normalization design process used. The warehouse is associated
with the business enterprise, not with an application. Traditional databases may be on
the order of megabytes or gigabytes, whereas a warehouse may be terabytes in size. The
fact that conventional data are stored in many diverse fo:rmats and locations makes it
inefficient to support decision support applications. OLTP users expect to get a response
in a few seconds. As a result of the complexity of OLAF' application, their users may
have to wait minutes for a response to their query.
The data transformation process required to convert operational data to informa
tional involves many functions including:
• Unwanted data must be removed.
• Converting heterogeneous sources into one common schema. This problem is the
same as that found when accessing data from multiple heterogeneous sources. Each
operational database may contain the same data with different attribute names. For
example, one system may use "Employee ID," while another uses "EID" for the
same attribute. In addition, there may be multiple data types for the same attribute.
• As the operational data is probably a snapshot of the data, multiple snapshots may
need to be merged to create the historical view.

• Summarizing data is performed to provide a higher level view of the data. This
summarization may be done at multiple granularities and for different dimensions.
• New derived data (e.g., using age rather than birth date) may be added to better
facilitate decision support functions.
• Handling missing and erroneous data must be performed. This could entail replac
ing them with predicted or default values or simply removing these entries.
The portion of the transformation that deals with ensuring valid and consistent data is
sometimes referred to as data scrubbing or data staging.
There are many benefits to the use of a data warehouse. Because it provides an
integration of data from multiple sources, its use can provide more efficient access of
the data. The data that are stored often provide different levels of summarization. For
example, sales data may be found at a low level (purchase order), at a city level (total of
sales for a city), or at higher levels (county, state, country, world). The summary can be
provided for different types of granularity. The sales data could be summarized by both
salesman and department. These summarizations are provided by the conversion process
instead of being calculated when the data are accessed. Thus, this also speeds up the
processing of the data for decision support applications.
The data warehouse may appear to increase the complexity of database manage
ment because it is a replica of the operational data. But keep in mind that much of the
data in the warehouse are not simply a replication but an extension to or aggregation
of the data. In addition, because the data warehouse contains historical data, data stored
there probably will have a longer life span than the snapshot data found in the oper
ational databases. The fact that the data in the warehouse need not be kept consistent
with the current operational data also simplifies its maintenance. The benefits obtained
by the capabilities (e.g., DSS support) provided usually are deemed to outweigh any
disadvantages.
A subset of the complete data warehouse, data mart, may be stored and accessed
separately. The level is at a departmental, regional, or functional level. These separate
data marts are much smaller, and they more efficiently support narrower analytical types
of applications.
A virtual warehouse is a warehouse implemented as a view from the operational
data. While some of this view may actually be materialized for efficiency, it need not
all be.
There are several ways to improve the performance of data warehouse applications.
• Swnmarization: Because many applications require summary-type information,
data that are known to be needed for consolidation queries should be presum
marized before storage. Different levels of summarization should be included to
improve performance. With a 20 to 100% increase in storage space, an increase in
performance of 2 to 10 times can be achieved [Sin98, p. 302].
• Denormalization: Traditional normalization reduces such problems as redundancy
as well as insert, update, and deletion anomalies. However, these improvements
are achieved at the cost of increased processing time due to joins. With a data
warehouse, improved performance can be achieved by storing denormalized data.
Section 2.7 OLAP 39
Since data warehouses are not usually updated as frequently as operational data
are, the negatives associated with update operations are not an issue.
• Partitioning: Dividing the data warehouse into smaller fragments may reduce
processing time by allowing queries to access small data sets.
The relationship between data mining and data warehousing can be viewed as
symbiotic [Inm96]. Data used in data mining applications are often slightly modified
from that in the databases where the data permanently reside. The same is true for data
in a data warehouse. When data are placed in a warehouse, they are extracted from the
database, cleansed, and reformatted. The fact that the data are derived from multiple
sources with heterogeneous formats complicates the proble:m. In addition, the fact that
the source databases are updated requires that the warehouse be updated periodically or
work with stale data. These issues are identical to many of those associated with data
mining and KDD (see Figure 1 .4). While data mining and data warehousing are actually
orthogonal issues, they are complementary. Due to the types of applications and massive
amount of data in a data warehouse, data mining applications can be used to provide
meaningful information needed for decision support systems. For example, management
may use the results of classification or association rule applications to help determine
the target population for an advertising campaign. In addition, data mining activities can
benefit from the use of data in a data warehouse. However, its use is not required. Data
warehousing and data mining are sometimes thought of as the same thing. Even though
they are related, they are different and each can be used without the other.
2.7 OLAP
Online analytic processing (OIAP) systems are targeted to provide more complex query
results than traditional OLTP or database systems. Unlike database queries, however,
OLAP applications usually involve analysis of the actual dtata. They can be thought of
as an extension of some of the basic aggregation functions available in SQL. This extra
analysis of the data as well as the more imprecise nature of the OLAP queries is what
really differentiates OLAP applications from traditional database and OLTP applications.
OLAP tools may also be used in DSS systems.
OLAP is performed on data warehouses or data marts. The primary goal of OLAP
is to support ad hoc querying needed to support DSS. The multidimensional view of data
is fundamental to OLAP applications. OLAP is an application view, not a data structure
or schema. The complex nature of OLAP applications requires a multidimensional view
of the data. The type of data accessed is often (although not a requirement) a data
warehouse.
OLAP tools can be classified as ROLAP or MOLAP. With MOIAP (multidimen
sional OIAP), data are modeled, viewed, and physically stored in a multidimensional
database (MDD). MOLAP tools are implemented by specialized DBMS and software
systems capable of supporting the multidimensional data directly. With MOLAP, data
are stored as an n-dimensional array (assuming there are n dimensions), so the cube view
is stored directly. Although MOLAP has extremely high storage requirements, indices
are used to speed up processing. With ROIAP (relational OIAP), however, data are
stored in a relational database, and a ROLAP server (middleware) creates the multi
dimensional view for the user. As one would think, the ROLAP tools tend to be less

L-
o
c �,,
,
a . ��
:.
T
/
t . 'm
i
0 �
IZJ
n <y
'b-
Roll up
�
-
Drill down
Products
(a) Single cell (b) Multiple cells (c) Slice (d) Dice
FIGURE 2 . 1 5: OLAP operations.
complex, but also less efficient. MDD systems may presummarize along all dimensions.
A third approach, hybrid OLAP (HOLAP), combines the best features of ROLAP and
MOLAP. Queries are stated in multidimensional terms. Data that are not updated fre
quently will be stored as MDD, whereas data that are updated frequently will be stored
as RDB. 1
As seen in Figure 2.15, there are several types of OLAP operations supported by
OLAP tools:
• A simple query may look at a single cell within the cube [Figure 2. 15(a)].
• Slice: Look at a subcube to get more specific information. This is performed by
selecting on one dimension. As seen in Figure 2. 15(c), this is looking at a portion
of the cube.
• Dice: Look at a subcube by selecting on two or more dimensions. This can be
performed by a slice on one dimension and then rotating the cube to select on a
second dimension. In Figure 2. 15(d), a dice is made because the view in (c) is
rotated from all cells for one product to all cells for one location.
• Roll up (dimension reduction, aggregation): Roll up allows the user to ask ques
tions that move up an aggregation hierarchy. Figure 2. 15(b) represents a roll up
from (a). Instead of looking at one single fact, we look at all the facts. Thus, we
could, for example, look at the overall total sales for the company.
• Drill down: Figure 2.15(a) represents a drill down from (b). These functions allow
a user to get more detailed fact information by navigating lower in the aggregation
hierarchy. We could perhaps look at quantities sold within a specific area of each
of the cities.
• Visualization: Visualization allows the OLAP users to actually "see" results of an
operation.
To assist with roll up and drill down operations, frequently used aggregations can be
precomputed and stored in the warehouse. There have been several different definitions
for a dice. In fact, the term slice and dice is sometimes viewed together as indicating
that the cube is subdivided by selecting on multiple dimensions.
8 WEB SEARCH ENGINES
2.
Section 2.8 Web Search Engi nes 41
As a result of the large amount of data on the Web and the fact that it is continually
growing, obtaining desired information can be challenging. Web search engines are used
to access the data and can be viewed as query systems much like IR systems. As with
IR queries, search engine queries can be stated as keyword, boolean, weighted, and so
on. The difference is primarily in the data being searched, pages with heterogeneous data
and extensive hyperlinks, and the architecture involved.
Conventional search engines suffer from several problems [RS99]:
• Abundance: Most of the data on the Web are of no interest to most people. In
other words, although there is a lot of data on the Web, an individual query will
retrieve only a very small subset of it.
• Limited coverage: Search engines often provide results from a subset of the Web
pages. Because of the extreme size of the Web, it is impossible to search the entire
Web any time a query is requested. Instead, mos� search engines create indices
that are updated periodically. When a query is requested, often only the index is
directly accessed.
• Limited query: Most search engines provide access based only on simple key
word-based searching. More advanced search engines may retrieve or order pages
based on other properties such as the popularity of pages.
• Limited customization: Query results are often determined only by the query
itself. However, as with traditional IR systems, the desired results are often depen
dent on the background and knowledge of the user as well. Some more advanced
search engines add the ability to do customization using user profiles or historical
information.
Traditional IR systems (such as Lexis-Nexis) may actually be tailored to a specific
domain. The Web, however, has information for everyone.
As discussed in Chapter 7, True Web mining consists of content, structure, and
usage mining. Web search engines are very simplistic examjJles of Web content mining.
2.9 STATISTICS
Such simple statistical concepts as determining a data distribution and calculating a mean
and a variance can be viewed as data mining techniques. Each of these is, in its own
right, a descriptive model for the data under consideration.
Part of the data mining modeling process requires searching the actual data. An
equally important part requires inferencing from the results of the search to a general
model. A current database state may be thought of as a sample (albeit large) of the real
data that may not be stored electronically. When a model is generated, the goal is to
fit it to the entire data, not just that which was searched. Any model derived should
be statistically significant, meaningful, and valid. This problem may be compounded
by the preprocessing step of the KDD process, which may actually remove some of
the data. Outliers compound this problem. The fact that most database practitioners

and users are not probability experts also complicates the issues. The tools needed t
make the
.
computationally diffic�lt problems tractable may actually invalidate the results
�
Assum?twns often
.
made about mdependence of data may be incorrect, thus leading to
errors m the resultmg model.
An often-used tool i� da�a mini�g and machine learning is that of sampling. Here a
subset �f �e total populatiOn IS exanuned, and a generalization (model) about the entire
�opulatwn IS made from this subset. In statistics this approach is referred to as statistical
mf
erence.
.
Of course, this generalization may lead to errors in the final model caused by
the samplmg process.
The term exploratmy data analysis was actually coined by statisticians to describe
the f�c� that t�e data can actually drive the creation of the model and any statistical char
actenstics. This seems contradictory to the traditional statistical view that one should not
be corrupt�d or
.
influenced by looking at the data before creating the model. This would
u�necessanly
.
bias any resulting hypothesis. Of course, database practitioners would never
thini<: of creating a model of the data without looking at the data in detail first and then
creatmg a schema tq describe it.
.
Some data mining applications determine correlations among data. These relation
ships, however, are not causal in nature. A discovered association rule may show that
60 percent of the time when customers purchased hot dogs they also bought beer. Care
must ?e taken when assigning any significance to this relationship. We do not know why
these Items were purchased toget
.
her. �erhaps the beer was on sale or it was an extremely
hot day. There may be no relatwnship between these two data items except that they
were often purchased together. There need not be any probabilistic inference that can be
deduced.
.
Statis�ics �esearch has produced many of the proposed data mining algorithms. The
difference lies m the goals, the fact that statisticians may deal with smaller and more
form�tted data sets, and the emphasis of data mining on the use of machine learning
techniques. Howeve�, �t
.
is often the ca
.
se that the term data mining is used in a deroga
tory mann�: by statisticians as d
.
ata mming is "analysis without any clearly formulated
hypothes�s
.
[Man96]. Indeed, this may be the case because data itself, not a predefined
hypothesis, Is the guide.
Pr?bability distributions can be used to describe the domains found for different
data
.
at�nbutes. Statistical inference techniques can be viewed as special estimators and
pred�ctw� �eth?ds. Use of these approaches may not always be applicable because the
p�ec�se �Istnbutwn o
.
f real data v�lues may not actually follow any specific probability
dist�b�twn, assumptiOns on the mdependence of attributes may be invalid, and some
heunstic-based estimator techniques may never actually converge.
I��as ?een stated that "the main difference between data mining and statistics is that
data numng I� �ea�t to ?e used by the business user-not the statistician" [BS97, 292]. As
such, data numng (particularly from a database perspective) involves not only modelin
but also the development of effective and efficient algorithms (and data structures) t
g
perform the modeling on large data sets.
0
2.10 MACHINE LEARNING
Artificial intelligence (AI) includes many DM techniques such as neural t k d
. . . ne wor s an
classificatiOn. However, AI IS more general and involves areas outside traditional data
Section 2 . 1 0 Machine Learning 43
numng. AI applications also may not be concerned with scalability as data sets may
be small.
Machine learning is the area of AI that examines how to write programs that can
leani. In data mining, machine learning is often used for prediction or classification.
With machine learning, the computer makes a prediction and then, based on feedback
as to whether it is correct, "learns" from this feedback. It learns through examples,
domain knowledge, and feedback. When a similar situation arises in the future, this
feedback is used to make the same prediction or to make a completely different pre
diction. Statistics are very important in machine learning programs because the results
of the predictions must be statistically significant and must perform better than a naive
prediction. Applications that typically use machine learning techniques include speech
recognition, training moving robots, classification of astronomical structures, and game
playing.
When machine learning is applied to data mining tasks, a model is used to represent
the data (such as a graphical structure like a neural network or a decision tree). During
the learning process, a sample of the database is used to train the system to properly
perform the desired task. Then the system is applied to the general database to actually
perform the task. This predictive modeling approach is divided into two phases. During
the training phase, historical or sampled data are used to create a model that represents
those data. It is assumed that this model is representative not only for this sample data,
but also for the database as a whole and for future data as well. The testing phase then
applies this model to the remaining and future data.
A basic machine learning application includes several major aspects. First, an
appropriate training set must be chosen. The quality of the training data determines
how well the program learns. In addition, the type of feedback available is important.
Direct feedback entails specific information about the results and impact of each possible
move or database state. Indirect feedback is at a higher level, with no specific informa
tion about individual moves or predictions. An important aspect is whether the learning
program can actually propose new moves or database states. Another major feature that
impacts the quality of learning is how representative the training set is of the overall final
system to be examined. If a program is to be designed to perform speech recognition, it
is hoped that the system is allowed to learn with a large sample of the speech patterns it
will encounter during its actually processing.
There are two different types of machine learning: supervised learning and unsu
pervised learning. A supervised approach learns by example. Given a training set of
data plus correct answers, the computational model successively applies each entry in
the training set. Based on its ability to correctly handle each of these entries, the model
is changed to ensure that it works better with this entry if it were applied again. Given
enough input values, the model will learn the correct behavior for any potential entry.
With unsupervised data, data exist but there is no knowledge of the correct answer of
applying the model to the data.
Although machine learning is the basis for many of the core data mining research
topics, there is a major difference between the approaches taken by the AI and database
disciplines. Much of the machine learning research has focused on the learning portion
rather than on the creation of useful information (prediction) for the user. Also, machine
learning looks at things that may be difficult for humans to do or concentrates on how
to develop learning techniques that can mimic human behavior. The objective for data

TAB LE 2.3: Relationship Between Topics [FPSM9 1]
Database Management
Database is an active, evolving entity
Records may contain erroneous or
missing data
Typical field is numeric
Database contains millions of records
AI should get down to reality
Machine Learning
Database is static
Databases are complete and noise-free
Typical feature is binary
Database contains hundreds of
instances
All database problems have been
solved
mining is to un<lover information that can be used to provide information to humans (not
take their place). These two conflicting views are summarized in Table 2.3, which orig
inally appeared in [FPSM91]. The items listed in the first column indicate the concerns
and views that are taken in this book. Many of the algorithms introduced in this text were
created in the AI community and are now being exported into more realistic data min
ing activities. When applying these machine learning concepts to databases, additional
concerns and problems are raised: size, complex data types, complex data relationships,
noisy and missy data, and databases that are frequently updated.
2. 1 1 PATTERN MATCHING
Pattern matching or pattern recognition finds occurrences of a predefined pattern in
data. �attern matching is used in many diverse applications. A text editor uses pattern
matching to find occurrences of a string in the text being edited. Information retrieval and
Web search engines may use pattern matching to find documents containing a predefined
pattern (perhaps a keyword). Time series analysis examines the patterns of behavior in
data obt�ned from two different time series to determine similarity. Pattern matching
can be VIewed as a type of classification where the predefined patterns are the classes
under consideration. The data are then placed in the correct class based on a similarity
between the data and the classes.
2 . 1 2 SUMMARY
When viewed as a query system, data mining queries extend both database and IR
�oncepts. Dat� mining problems are often ill-posed, with many different solutions. Judg
I�g the effectiveness of the result of a data mining request is often difficult. A major
c
hfference between data mining queries and those of earlier types is the output. Basic
database queries always output either a subset of the database or aggregates of the data.
A data mining query outputs a KDD object. A KDD object is either a rule, classifier,
or clustering [IM96]. Thes� objects
.
do not exist before executing the query and they are
not part �f the database bemg
.
que�ed. Table 2.4 compares the different query systems.
AggregatiOn operators have extsted m SQL for years. They do not return objects existing
Section 2. 1 4 Bibliographic Notes 45
TABLE 2.4: Relationship Between Topics
Area
DB/OLTP
IR
OLAP
DM
Query
Precise
Precise
Analysis
Vague
Data
Database
Documents
Multidimensional
Preprocessed
Results
Precise
Vague
Precise
Vague
Output
DB objects or aggregation
Documents
DB objects or aggregation
KDD objects
in the database, but return a model of the data. For example., an average operator returns
the average of a set of attribute values rather than the values themselves. This is a simple
type of data mining operator.
2. 13 EXERCISES
1. Compare and contrast database, information retrieval, and data mining queries.
What metrics are used to measure the performance of each type of query?
2. What is the relationship between a fuzzy set membership function and classifica
tion? lllustrate this relationship using the problem of assigning grades to students
in classes where outliers (extremely high and low grades) exist.
3. (Research) Data warehouses are often viewed to contain relatively static data.
Investigate techniques that have been proposed to provide updates to this data
from the operational data. How often should these updates occur?
There are many excellent database books, including [DatOO], [ENOO], [GMUW02], and
[0001]. There are several books that provide introductions to database query processing
and SQL, including [DD97] and [YM98].
Fuzzy sets were first examined by Lotti Zadeh in [Zad65]. They continue to be an
important research topic in all areas of computer science, with many introductory texts
available such as [KY95], [NW99], and [San98]. Some texts explore the relationship
between fuzzy sets and various data mining applications. Pedrycz and Gomide examine
fuzzy neural networks and fuzzy genetic algorithms. [PG98]
There are several excellent information retrieval texts, including [BYRN99] and
[SM83]. The ER (entity-relationship) data model was first proposed by Chen in 1976
[Che76].
An examination of the relationship between statistics and KDD can be found in
[IP96]. This article provides an historical perspective of the development of statistical
techniques that have influenced data mining. Much of the research concerning outliers
has been performed in the statistics community [BL94] and [Haw80].
There is an abundance of books covering of DSS, OLAP, dimensional modeling,
multidimensional schemas, and data warehousing, including [BE97], [PKB98], [Sin98],
and [Sin99].
An excellent textbook on machine learning is [Mit97]. The relationships between
machine learning and data mining are investigated in [MBK99].

C H A P T E R 3
Data Min i ng Techniques
3.1 INTRODUCTION
3.2 A STATISTICAL PERSPECTIVE ON DATA MINING
3.3 SIMILARITY MEASURES
3.4 DECISION TREES
3.5 NEURAL NETWORKS
3.6 GENETIC ALGORITHMS
3.7 EXERCISES
3.1 INTRODUCTION
There are many different methods used to perform data mining tasks. These techniques
not only require specific types of data structures, but also imply certain types of algorith
mic approaches. In this chapter we briefly introduce some of the common data mining
techniques. These will be examined in more detail in later chapters of the book as they
are used to perform specific data mining tasks.
Parametric models describe the relationship between input and output through the
use of algebraic equations where some parameters are not specified. These unspecified
parameters are determined by providing input examples. Even though parametric model
ing is a nice theoretical topic and can sometimes be used, often it is either too simplistic or
requires more knowledge about the data involved than is available. Thus, for real-world
problems, these parametric models may not be useful.
Nonparametric techniques are more appropriate for data mining applications. A
nonparametric model is one that is data-driven. No explicit equations are used to deter
mine the model. This means that the modeling process adapts to the data at hand. Unlike
parametric modeling, where a specific model is assumed ahead of time, the nonpara
metric techniques create a model based on the input. While the parametric methods
require more knowledge about the data before the modeling process, the nonparametric
technique requires a large amount of data as input to the modeling process itself. The
modeling process then creates the model by sifting through the data. Recent nonparamet
ric methods have employed machine learning techniques to be able to learn dynamically
as data are added to the input. Thus, the more data, the better the model created. Also, .
this dynamic learning process allows the model to be created continuously as the data
is input. These features make nonparametric techniques particularly suitable to database
46
Section 3.2 A Statistical Perspective on Data Mining 47
applications with large amounts of dynamically changing data. Nonparametric techniques
include neural networks, decision trees, and genetic algorithms.
3.2 A STATISTICAL PERSPECTIVE ON DATA MINING
There have been many statistical concepts that are the basis for data mining techniques.
We briefly review some of these concepts.
3.2.1 Point Estimation
Point estimation refers to the process of estimating a population parameter, 8, by an
estimate of the parameter, G. This can be done to estimate mean, variance, standard
deviation, or any other statistical parameter. Often the estimate of the parameter for
a general population may be made by actually calculating the parameter value for a
population sample. An estimator technique may also be used to estimate (predict) the
value of missing data. The bias of an estimator is the difference between the expected
value of the estimator and the actual value:
Bias = E(G) - 8 (3.1)
An unbiased estimator is one whose bias is 0. While point estimators for small data sets
may actually be unbiased, for larger database applications we would expect that most
estimators are biased.
One measure of the effectiveness of an estimate is the mean squared error (MSE),
which is defined as the expected value of the squared difference between the estimate
and the actual value:
MSE(G) = E(G - 8)2 (3.2)
The squared error is often examined for a specific prediction to measure accuracy rather
than to look at the average difference. For example, if the ttue value for an attribute was
10 and the prediction was 5, the squared error would be (5 ·- 10)2 = 25. The squaring is
performed to ensure that the measure is always positive and to give a higher weighting to
the estimates that are grossly inaccurate. As we will see, the MSE is commonly used in
evaluating the effectiveness of data mining prediction techniques. It is also important in
machine learning. At times, instead of predicting a simple point estimate for a parameter,
one may determine a range of values within which the true parameter value should fall.
This range is called a con
fidence interval.
The root mean square (RMS) may also be used to estimate error or as another
statistic to describe a distribution. Calculating the mean does not indicate the magnitude
of the values. The RMS can be used for this purpose. Given a set of n values X =
{x1 , . . . , xn}, the RMS is defined by
RMS = (3.3)
An alternative use is to estimate the magnitude of the error. The root mean square error
(RMSE) is found by taking the square root of the MSE.
A popular estimating technique is the jackknife estimate. With this approach, the
estimate of a parameter, e, is obtairled by omitting one value from the set of observed

48 Chapter 3 Data Mining Techniques
values. Suppose that there is a set of n values X = {x1 , . . . , Xn}. An estimate for the
mean would be i-1 n
I>j + L Xj
A
J=l J=i+l
/L(i) =
n - I
(3.4)
Here the subscript (i) indicates that this estimate is obtained by omitting the i1h value.
Given a set of jackknife estimates, ecil, these can in tum be used to obtain an overall
estimate
EXAMPLE 3.1
z=e(j)
A j=l
Bc.l = -
n
(3 .5)
Suppose that a coin is tosse9 in the air five times with the following results ( 1 indicates
a head and 0 indicates a tail): {1, 1, 1 , 1, 0}. If we assume that the coin toss follows the
Bernoulli distribution, we know that
(3 .6)
Assuming a perfect coin when the probability of 1 and 0 are both 1/2, the likelihood
is then
5
L(p I 1, 1 , 1 , 1, 0) = Jl 0.5 = 0.03 (3.7)
i=l
However, ifthe coin is not perfect but has a bias toward heads such that the probability
of getting a head is 0.8, the likelihood is
L(p 1 1 , 1 , 1 , 1, 0) = 0.8 X 0.8 X 0.8 X 0.8 X 0.2 = 0.08 (3.8)
Here it is more likely that the coin is biased toward getting a head than that it is not
biased. The general formula for likelihood is
5
L(p 1 XJ , • . • , x5) = Jlpx;(l - p)!-x; = pLT=1x;(l - p)5-I:T=1x; (3 .9)
i=l
By taking the log we get
l(p) = logL(p) = 't;x, log(p) + (s-'t;x. )log(! - p)
and then we take the derivative with respect to p
5
5 S - LXi
at(p) _
L
xi i=I
---a;-
-
i=l p
- 1 - p
(3.10)
(3. 1 1)
Setting equal to zero we finally obtain
i=l
p = --
5
(3.12)
For this example, the estimate for p is then p = � = 0.8. 'Thus, 0.8 is the value for p
that maximizes the likelihood that the given sequence of heads and tails would occur.
Another technique for point estimation is called the maximum likelihood estimate
(MLE). Likelihood can be defined as a value proportional to the actual probability that
with a specific distribution the given sample exists. So the sample gives us an estimate
for a parameter from the distribution. The higher the likelihood value, the more likely
the underlying distribution will produce the results observed. Given a sample set of
values X = {xJ, . . . , x11 } from a known distribution function f(xi I 8), the MLE can
estimate parameters for the population from which the sample is drawn. The approach
obtains parameter estimates that maximize the probability that the sample data occur for
the specific model. It looks at the joint probability for observing the sample data by
multiplying the individual probabilities. The likelihood function, L, is thus defined as
n
L(E> I XJ , . . . , Xn) = Dt(xi I 8) (3. 13)
i=l
The value of 8 that maximizes L is the estimate chosen. This can be found by taking
the derivative (perhaps after finding the log of each side to simplify the formula) with
respect to 8. Example 3.1 illustrates the use of MLE.
ALGORITHM 3.1
Input :
e = {th , . . . , Op}
X
obs = {xl, . . . , Xk}
X
miss = {Xk+l, . . . , Xn}
Output :
e
EM algorithm:
i := 0 ;
/ / Parameters t o be estimated
/ / Input database values observed
// Input database values mis s ing
//Estimates for 8
Obtain initial parameter MLE estimate , 0Ji ;
repeat
Estimate mi ss ing data I xi . ;
ffilSS
i++
Obtain next parameter estimate , Oi to
maximize l ikelihood ;
until estimate converges ;
Theexpectation-maximization (EM) algorithmis anapproachthat solves the estima
tion problemwithincomplete data. TheEMalgorithmfinds an MLEforaparameter(such

50 Chapter 3 Data M ining Techniques
as a mean) using a two-step process: estimation and maximization. The basic EM algo
rithm is shown in Algorithm 3.1. An initial set of estimates for the parameters is obtained.
Given these estimates and the training data as input, the algorithm then calculates a value
for the missing data. For example, it might use the estimated mean to predict a missing
value. These data (with the new value added) are then used to determine an estimate for
the mean that maximizes the likelihood. These steps are applied iteratively until succes
sive parameter estimates converge. Any approach can be used to find the initial parameter
estimates. In Algorithm 3.1 it is assumed that the input database has actual observed val
ues Xobs = {xi , . . . , xk} as well as values that are missing Xmiss = ·{xk+i , . . . , Xn }. We
assume that the entire database is actually X = Xobs U Xmiss · The parameters to be
estimated are 8 = {{h , . . . , ep}. The likelihood function is defined by
L(e 1 X) = OJ(x; 1 8) (3.14)
i=l
We are looking for the 8
,
that maximizes L. The MLE of 8 are the estimates that satisfy
1
a ln L(8 I X)
ae;
= 0 (3. 15)
The expectation part of the algorithm estimates the missing values using the current
estimates of e. This can initially be done by finding a weighted average of the observed
data. The maximization step then finds the new estimates for the e parameters that
maximize the likelihood by using those estimates of the missing data. An illustrative
example of the EM algorithm is shown in Example 3.2.
EXAMPLE 3.2
We wish to find the mean, f.L, for data that follow the normal distribution where the known
data are { 1 , 5, 10, 4} with two data items missing. Here n = 6 and k = 4. Suppose that
we initially guess {l0 = 3. We then use this value for the two missing values. Using this,
we obtain the MLE estimate for the mean as
k
LX; L X;
A I -
i=i
+
i=k+i
3 33
3
+
3
4 33
f.L -
-- --- = . + -- = .
n n 6
(3.1 6)
We now repeat using this as the new value for the missing items, then estimate the
mean as
k
l:x; L x;
A 2 i=i i=k+i
3 33
4.33 +4.33
f.L = -- + -- = . + = 4.77
n n 6
Repeating we obtain
k n
LXi L Xi
A 3 i=i i=k+l
3 33
4,
77
+4,77
f.L = -- + -- = . + = 4.92
n n 6
(3. 17
)
(3.18)
Section 3.2 A Statistical Perspective on Data M i n ing 51
and then
k n
L:x; L x;
fl4
=
i=l
+
i=k+l
= 3.33
+ 4.92 + 4.92
= 4
.97
n n 6
(3.19)
We decide to stop here because the last two estimates are only 0.05 apart. Thus, our
estimate is fl = 4.97.
One of the basic guidelines in estimating is Ockham 's Razor,1 which basically
states that simpler models generally yield the best results.
3.2.2 Models Based on Summarization
There are many basic concepts that provide an abstraction and summarization of the data
as a whole. The basic well-known statistical concepts such as mean, variance, standard
deviation, median, and mode are simple models of the underlying population. Fitting a
population to a specific frequency distribution provides an even better model of the data.
Of course, doing this with large databases that have multiple attributes, have complex
and/or multimedia attributes, and are constantly changing is not practical (let alone always
possible).
There are also many well-known techniques to display the structure of the data
graphically. For example, a histogram shows the distribution of the data. A box plot is a
more sophisticated technique that illustrates several different features of the population at
once. Figure 3.1 shows a sample box plot. The total range of the data values is divided
into four equal parts called quartiles. The box in the center of the figure shows the range
between the first, second, and third quartiles. The line in the box shows the median. The
lines extending from either end of the box are the values that are a distance of 1.5 of the
interquartile range from the first and third quartiles, respectively. Outliers are shown as
points beyond these values.
Smallest value within 1.5 Largest value within 1.5
interquartile range from 1st quartile interquartile range from 3rd quartile

Outliers
/ '··r"
�--
1 ---+-
1 --'---·---h---<:
:r:;��
-�-
J --�

1st quartile 3rd quartile
FIGURE 3. 1 : Box plot example.
1 Sometimes this is spelled Occum or Occam. It is named after William Ockham, who was a monk in the
late thirteenth and early fourteenth centuries. However, it was first used by Durand de Saint-Pourcain an earlier
French theologian.

3.2.3
. . .
..
..
.
FIGURE 3.2: Scatter diagram example.
Another visual technique to display data is called a scatter diagram. This is a
graph on a two-dimensional axis of points representing the relationships between x and
y values. By plotting the actually observable (x, y) points as seen in a sample, a visual
image of some derivable functional relationship between the x and y values in the total
population may be seen. Figure 3.2 shows a scatter diagram that plots some observed
values. Notice that even though the points do not lie on a precisely linear line, they do
hint that this may be a good predictor of the relationship between x and y.
Bayes Theorem
With statistical inference, information about a data distribution are inferred by examining
data that follow that distribution. Given a set of data X = {x1 , . . . , x11 }, a data mining
problem is to uncover properties of the distribution from which the set comes. B ayes rule,
defined in Definition 3.1, is a technique to estimate the likelihood of a property given
the set of data as evidence or input. Suppose that either hypothesis h1 or hypothesis h2
must occur, but not both. Also suppose that Xi is an observable event.
DEFINITION 3.1. Bayes Rule or Bayes Theorem is
(3.20)
Here P(h1 I Xi) is called the posterior probability, while P(h1) is the prior prob
ability associated with hypothesis h 1 . P(xi) is the probability of the occurrence of
data value Xi and P(xi I h1) is the conditional probability that, given a hypothesis,
the tuple satisfies it.
TABLE 3. 1 : Training Data for Example 3.3
ID Income Credit Class Xi
1 4 Excellent h1 X4
2 3 Good h1 X?
3 2 Excellent hi xz
4 3 Good h1 X?
5 4 Good ht xs
6 2 Excellent ht xz
7 3 Bad hz xu
8 2 Bad hz XlQ
9 3 Bad h3 xu
10 Bad h4 Xg
Where there are m different hypotheses we have:
Thus, we have
m
P(xi ) = L P(xi I hj)P(hj)
j=l
(3.21)
(3.22)
Bayes rule allows us to assign probabilities of hypotheses given a data value, P(hj I Xi).
Here we discuss tuples when in actuality each Xi may be an attribute value or other data
label. Each h1 may be an attribute value, set of attribute values (such as a range), or even
a combination of attribute values.
Example 3.3 uses the training data in Table 3.1 to illustrate the use o� �ayes rule.
Example 3.3 also illustrates that we may take advantage of other probabthty laws to
determine combinations of probabilities. For example, we may find P(h t) = P (l <
$10,000 1 Good) by instead finding P(l < $10,000) P(Good I I < $10,000).
EXAMPLE 3.3
Suppose that a credit loan authorization problem can be associated with fo�r hypo
theses: H = {h1 , h2, h3, h4} where h1 = authorize purchase, hz = authonze after
further identification, h3 = do not authorize, and h4 = do not authorize but contact
police. The training data for this example are shown in Table 3.1. From training data,
we find that P(h1) = 60%, P(hz) = 20%, P(h3) = 10%, and P(h4) = 10%. To
make our predictions, a domain expert has determined that the attributes we shoul� be
looking at are income and credit category. Assume that income, I, has been categonzed
by ranges [0, $10,000), [$10,000, $50,000), [$50,000, $100,000), and [$100,000
.
' oo).
These ranges are encoded and are shown in Table 3.1 as 1, 2, 3, and 4, respectively.
Suppose that credit is categorized as excellent, good, or bad. By com?inin� these, we
then have 12 values in the data space: D = {xi , . . . , XJ2}. The relationship between

TABLE 3.2: x; Assignments for Example 3.3
Excellent
Good
Bad
XJ
xs
Xg
2 3 4
X4
Xg
XJ2
these x; values and the two attributes is shown in Table 3.2. Using these values, the last
column in Table 3.1 shows the x; group into which that tuple falls. Given these, we can
then calculate P (x; I hJ) and P(x;). We illustrate how this is done with h I · There are
six tuples from the training set that are in h1 ; we use the distribution of these across the
x; to obtain: P (x7 I hJ) = y6, P (x4 I h1) = 1/6, P(x2 I hi) = 2/6, P (xg I hi) = 1/6,
and P(x; I h1) = 0 for all ather values of i. Suppose we wanted to predict the class for
X4. We thus need to find P (hJ I x4) for each hJ . We would then classify x4 to the class
with the largest value for h 1 . We find P(h1 1 x4) = (P(x41�1<�X(h1)) = O/�i0·6) = 1 . We
would thus classify X4 to the h 1 class.
This example illustrates some issues associated with sampling. First note that
Table 3.1 has no entries for XJ , x3, xs, X6, or XJ2· This makes it impossible to use
this training sample to determine how to make predictions for this combination of input
data. If indeed these combinations never occur, this would not be a problem. However,
in this case we certainly do not know this to be true. Another issue with this sample is its
size. Of course, a sample of this size is too small. But what constitutes a good sample?
Size certainly is not the only criterion. This is a crucial issue that impacts the quality
of any data mining technique that uses sampling. There is much work in the statistics
community on good sampling strategies, so we do not cover that topic in this text.
3.2.4 Hypothesis Testing
Hypothesis testing attempts to find a model that explains the observed data by first
creating a hypothesis and then testing the hypothesis against the data. This is in contrast
to most data mining approaches, which create the model from the actual data without
guessing what it is first. The actual data itself drive the model creation. The hypothesis
usually is verified by examining a data sample. If the hypothesis holds for the sample, it
is assumed to hold for the population in general. Given a population, the initial (assumed)
hypothesis to be tested, Ho, is called the null hypothesis. Rejection of the null hypothesis
causes another hypothesis, Ht , called the alternative hypothesis, to be made.
One technique to perform hypothesis testing is based on the use of the chi-squared
statistic. Actually, there is a set of procedures referred to as chi squared. These proce
dures can be used to test the association between two observed variable values and to
determine if a set of observed variable values is statistically significant (i.e., if it dif
fers from the expected case). A hypothesis is first made, and then the observed values
are compared based on this hypothesis. Assuming that 0 represents the observed data
and E is the expected values based on the hypothesis, the chi-squared statistic, x2, is
defined as:
(3.23)
When comparing a set of observed variable values to determine statistical signifi
cance, the values are compared to those of the expected case. This may be the uniform
distribution. Example 3.4 illustrates this process. We could look at the ratio of the differ
ence of each observed score from the expected value over the expected value. However,
since the sum of these scores will always be 0, this approach cannot be used to compare
different samples to determine how they differ from the expected values. The solution to
this is the same as we saw with the mean square error-square the difference. Here, if all
scores are as expected, the result would be 0. Statistical tables (found in most statistics
books) allow the actual value to be evaluated to determine its significance.
EXAMPLE 3.4
Suppose that there are five schools being compared based on students' results on a set
of standardized achievement tests. The school district expects that the results will be
the same for each school. They know that the total score for the schools is 375, so the
expected result would be that each school has an average score of 75. The actual average
scores from the schools are: 50, 93, 67, 78, and 87. The district administrators want to
determine if this is statistically significant. Or in simpler te1ms, should they be worried
about the distribution of scores. The chi-squared measure here is
2 - (50 - 75)2 (93 - 75)2 (67 - 75)2 (78 - 75)
2
(87 - 75)2
- 15 55
X -
75
+
75
+
75
+
75
+
75
- .
(3.24)
Examining a chi-squared significance table, it is found that tltis value is significant. With
a degree of freedom of 4 and a significance level of 95%, the critical value is 9.488.
Thus, the administrators observe that the variance between the schools' scores and the
expected values cannot be associated with pure chance.
3.2.5 Regression and Correlation
Both bivariate regression and correlation can be used to evaluate the strength of a
relationship between two variables. Regression is generally used to predict future values
based on past values by fitting a set of points to a curve. Correlation, however, is used
to examine the degree to which the values for two variables behave similarly.
Linear regression assumes that a linear relationship exists between the input data
and the output data. The common formula for a linear relationship is used in this model:
y = CQ + C)X[ + · · · + CnXn (3.25)
Here there are n input variables, which are called predictors or regressors; one output
variable (the variable being predicted), which is called the response; and n + 1 constants,
which are chosen during the modeling process to match the input examples (or sample).
This is sometimes called multiple linearregressionbecausethere ismorethanone predictor.
Example 3.5 is an example of the use of linear regression.

EXAMPLE 3.5
It is known that a state has a fixed sales tax, but it is not known what the amount happens
to be. The problem is to derive the equation for the amount of sales tax given an input
purchase amount. We can state the desired linear equation to be y = co + qx1 . So we
really only need to have two samples of actual data to determine the values of co and
q . Suppose that we know (10, 0.5) and (25, 1 .25) are actual purchase amount and tax
amount pairs. Using these data points, we easily determine that co = 0 and c1 = 0.05.
Thus, the general formula is y = 0.05 x; . This would be used to predict a value of y for
any known x; value.
Admittedly, Example 3.5 is an extremely simple problem. However, it illustrates
how we all use the basic classification and/or prediction techniques frequently. Figure 3.3
illustrates the more ge¢ral use of linear regression with one input value. Here we have
a sample of data that we wish to model using a linear model. The line generated by the
linear regression technique is shown in the figure. Note, however, that the actual data
points usually do not fit the linear model exactly. Thus, this model is an estimate of what
the actual input-output relationship is. We can use the generated linear model to predict
an output value given an input value, but unlike that for Example 3.5, the prediction
would be an estimate rather than the actual output value.
T
wo different data variables, X and Y, may behave very similarly. Correlation is
the problem of determining how much alike the two variables actually are. One standard
formula to measure linear correlation is the correlation coefficient r. Given two variables,
X and Y, the correlation coefficient is a real value r E [- 1 , 1]. A positive number indi
cates a positive correlation, whereas a negative number indicates a negative correlation.
Here negative correlation indicates that one variable increases while the other decreases
in value. The closer the value of r is to 0, the smaller the correlation. A perfect rela
tionship exists with a value of 1 or - 1 , whereas no correlation exists with a value of 0.
When looking at a scatter plot of the two variables, the closer the values ar� to a straight
FIG U RE 3.3: Simple linear regression.
3.3
Section 3.3 Similarity Measures 57
line, the closer the r value is to 1 or -1. The value for r is defined as
(3.26)
where X and Y are the means for X and Y, respectively. Suppose that X = (2, 4, 6, 8,
10) . If Y = X, then r = 1 . When Y = (1, 3, 5, 7, 9), r = 1 . If Y = (9, 7, 5, 3, 1),
r = - 1 .
When two data variables have a strong correlation, they are similar. Thus, the
correlation coefficient can be used to define similarity for clustering or classification.
SIMILARITY MEASURES
The use of similarity measures is well known to anyone who has performed Internet
searches using a search engine. In such a search, the set of all Web pages represents the
whole database, and these are divided into two classes: those that answer your query and
those that do not. Those that answer your query should be more like each other than
those that do not answer your query. The similarity in this case is defined by the query
you state, usually based on a keyword list. Thus, the retrieved pages are similar because
they all contain (to some degree) the keyword list you have specified.
The idea of similarity measures can be abstracted and applied to more general
classification problems. The difficulty lies in how the similarity measures are defined
and applied to the items in the database. Since most similarity measures assume numeric
(and often discrete) values, they may be difficult to use for more general data types. A
mapping from the attribute domain to a subset of the integers may be used.
DEFINITION 3.2. The similarity between two tuples t; and t1 , sim(t; , t1) , in a
database D is a mapping from D x D to the range [0, 1]. Thus, sim(t; , t1) E [0, 1].
The objective is to define the similarity mapping such that documents that are more
alike have a higher similarity value. Thus, the following are desirable characteristics of
a good similarity measure:
• Vt; E D, sim(t; , t;) = 1
• Vt; , t1 E D, sirn(t; , fJ) = 0 if t; and t1 are not alike at all
• Vt; , fJ , tk E D, sim(t; , fJ ) < sim(t; , tk) if t; is more like tk than it is like t1
So how does one define such a similarity mapping? This, of course, is the difficult part.
Often the concept of "alikeness" is itself not well defined. When the idea of similar
ity measure is used in classification where the classes are: predefined, this problem is
somewhat simpler than when it is used for clustering where the classes are not known
in advance. Again, think of the IR example. Each IR query provides the class definition
in the form of the IR query itself. So the classification problem then becomes one of
determining similarity not among all tuples in the database but between each tuple and
the query. This makes the problem an O (n) problem rather than an O (n2) problem.
Here are some of the more common similarity measures used in traditional IR
systems and more recently in Internet search engines:

L:k t- t .
• Overlap: sim(t; , t1) =
h-I zh 111
. ("k 2 "k 2 )
mm L...h=l tih ' L...h=l tJh
In these formulas it is assumed that similarity is being evaluated between two vectors
t; = (tiJ , . . . , t;k) and fJ = {fJ J , . . . , fJk), and vector entries usually are assumed to be
nonnegative numeric valwts. They could, for example, be a count of the number of times
an associated keyword appears in the document. If there is no overlap (i.e., one of the two
vectors always has a 0 in one of the two terms), the resulting value is 0. If the two are
identical, the resulting measure is 1 . The overlap measure, however, does not satisfy this
restriction. These formulas have their origin in measuring similarities between sets based
on the intersection between the two sets. Dice's coefficient relates the overlap to the
average size of the two sets together. Jaccard's coefficient is used to measure the overlap
of two sets as related to the whole set caused by their union. The cosine coefficient relates
the overlap to the geometric average of the two sets. The overlap metric determines the
degree to which the two sets overlap.
Distance or dissimilarity measures are often used instead of similarity measures.
As implied, these measure how "unlike" items are. Traditional distance measures may
be used in a two-dimensional space. These include
• Euclidean: dis(ti , lJ) = Jz=�=l (tih - t1h)2
• Manhattan: dis(t; , fJ) = L�=l I (t;h - fJh) I
To compensate for the different scales between different attribute values, the attribute
values may be normalized to be in the range [0, 1]. If nominal values rather than numeric
values are used, some approach to determining the difference is needed. One method is
to assign a difference of 0 if the values are identical and a difference of 1 if they
are different.
3.4 DECISION TREES
A decision tree is a predictive modeling technique used in classification, clustering, and
prediction tasks. Decision trees use a "divide and conquer" technique to split the problem
search space into subsets. It is based on the "Twenty Questions" game that children play,
as illustrated by Example 3.6. Figure 3.4 graphically shows the steps in the game. This
tree has as the root the first question asked. Each subsequent level in the tree consists
of questions at that stage in the game. Nodes at the third level show questions asked
Section 3.4 Decision Trees 59
Alive?
�
Ever alive? Person?
N
fes
�
Friend?
N
les
. . . FINISHED
FIGURE 3.4: Decision tree for Example 3.6.
at the third level in the game. Leaf nodes represent a successful guess as to the object
being predicted. This represents a correct prediction. Each question successively divides
the search space much as a binary search does. As with a binary search, questions
should be posed so that the remaining space is divided into two equal parts. Often young
children tend to ask poor questions by being too specific, such as initially asking "Is it
my Mother?" This is a poor approach because the search space is not divided into two
equal parts.
·
EXAMPLE 3.6
Stephanie and Shannon are playing a game of "1venty Questions." Shannon has in mind
some object that Stephanie tries to guess with no more than 20 questions. Stephanie's
first question is "Is this object alive?" Based on Shannon's answer, Stephanie then asks
a second question. Her second question is based on the answer that Shannon provides
to the first question. Suppose that Shannon says "yes" as her first answer. Stephanie's
second question is "Is this a person?" When Shannon responds "yes," Stephanie asks
"Is it a friend?" When Shannon says "no," Stephanie then asks "Is it someone in my
family?" When Shannon responds "yes," Stephanie then begins asking the names of
family members and can immediately narrow down the search space to identify the
target individual. This game is illustrated in Figure 3.4.
DEFINITION 3.3. A decision tree (DT) is a tree where the root and each internal
node is labeled with a question. The arcs emanating from each node represent each
possible answer to the associated question. Each leaf node represents a prediction
of a solution to the problem under consideration.

DEFINITION 3.4. A decision tree (DT) model2 is a computational model consisting
of three parts:
1. A decision tree as defined in Definition 3.3.
2. An algorithm to create the tree.
3. An algorithm that applies the tree to data and solves the problem under
consideration.
The building of the tree may be accomplished via an algorithm that examines
data from a training sample or could be created by a domain expert. Most decision tree
techniques differ in how the tree is created. We examine several decision tree technique
in later chapters of the book. Algorithm 3.2 shows the basic steps in applying a tuple to
the DT, step three in Definition 3.4. We assume here that the problem to be performed
is one of prediction, so the last step is to make the prediction as dictated by the final
leaf node in the tree. The complexity of the algorithm is straightforward to analyze. For
each tuple in the databas!, we search the tree from the root down to a particular leaf.
At each level, the maximum number of comparisons to make depends on the branching
factor at that level. So the complexity depends on the product of the number of levels
and the maximum branching factor.
ALGORITHM 3.2
Input :
T
D
/ /Decis ion t ree
/ / Input database
Output :
M //Model predi ction
DTProc algorithm:
/ / Simp l i s t ic algorithm to i l lus trate predi ct ion
technique us ing DT
for each t E D do
n = root node of T;
while n not leaf node do
Obtain answer to ques t ion on n app l ied to t ;
Ident ify arc from t , which contains correct answe r ;
n = node at end o f this arc ;
Make predi ct ion f or t based on labe l ing of n ;
We use Example 3.7 to further illustrate the use of decision trees.
EXAMPLE 3.7
Supposethat students in a particular university are to be classified as short, tall, or medium
based on their height. Assume that the database schema is {name, address, gender, height,
age, year, major}. To construct a decision tree, we must identify the attributes that are
important to the classification problem at hand. Suppose that height, age, and gender are
2Note that we have two separate definitions: one for the tree itself and one for the model. Although we
differentiate between the two here, the more common approach is to use the term decision tree for either.
3.5
Section :1.5 Neural Networks 61
Gender
=F =M
Height Height
<1.3 m >2 m
< = 2 m
Short Medium Tall Short Medium Tall
FIGURE 3.5: Decision tree for Example 3.7.
chosen. Certainly, a female who is 1.95 m in height is considered as tall, while a male
of the same height may not be considered tall. Also, a child 10 years of age may be tall
if he or she is only 1.5 m. Since this is a set of university students, we would expect
most of them to be over 17 years of age. We thus decide to filter out those under this age
and perform their classification' separately. We may consider these students to be outliers
because their ages (and more important their height classifications) are not typical of most
university students. Thus, for classification we have only gender and height. Using these
two attributes, a decision tree building algorithm will construct a tree using a sample of
the database with known classification values. This training sample forms the basis ofhow
the tree is constructed. One possible resulting tree after training is shown in Figure 3.5.
NEURAL NETWORKS
The first proposal to use an artificial neuron appeared in 1943, but computer usage of
neural networks did not actually begin until the 1980s. Neural networks (NN), often
referred to as artificial neural networks (ANN) to distinguish them from biological neural
networks, are modeled after the workings of the human brain. The NN is actually an
information processing system that consists of a graph representing the processing system
as well as various algorithms that access that graph. As with the human brain, the
NN consists of many connected processing elements. The NN, then, is structured as
a directed graph with many nodes (processing elements) and arcs (interconnections)
between them. The nodes in the graph are like individual neurons, while the arcs are
their interconnections. Each of these processing elements functions independently from
the others and uses only local data (input and output to the node) to direct its processing.
This feature facilitates the use of NNs in a distributed and/or parallel environment.
The NN approach, like decision trees, requires that a graphical structure be built
to represent the model and then that the structure be applied to the data. The NN can be
viewed as a directed graph with source (input), sink (output), and internal (hidden) nodes.
The input nodes exist in a input layer, while the output nodes exist in an output layer.
The hidden nodes exist over one or more hidden layers. To perform the data mining

task, a tuple is input through the input nodes and the output node determines what the
prediction is. Unlike decision trees, which have only one input node (the root of the
tree), the NN has one input node for each attribute value to be examined to solve the
data mining function. Unlike decision trees, after a tuple is processed, the NN may be .
changed to improve future performance. Although the structure of the graph does not
change, the labeling of the edges may change.
In addition to solving complex problems, NNs can "learn" from prior applications.
That is, if a poor solution to the problem is made, the network is modified to produce a
better solution to this problem the next time. A major drawback to the use of NNs is the
fact that they are difficult to explain to end users (unlike decision trees, which are easy
to understand). Also, unlike decision trees, NNs usually work only with numeric data.
To better illustrate the process, we reexamine Example 3.7 and show a simple NN
for this problem in Figure 3.6. We first must determine the basic structure of the graph.
Since there are two important attributes, we assume that there are two input nodes. Since
we are to classify into three classes, we use three output nodes. The number of hidden
layers in the NN is not, easy to determine. In most cases, one or two is enough. In
this example, we assum� that there is one hidden layer and thus a total of three layers
in the general structure. We arbitrarily assume that there are two nodes in this hidden
layer. Each node is labeled with a function that indicates its effect on the data coming
into that node. At the input layer, functions !1 and h simply take the corresponding
attribute value in and replicate it as output on each of the arcs coming out of the node.
The functions at the hidden layer, h and j4, and those at the output layer, fs, !6. and
h, perform more complicated functions, which are investigated later in this section. The
arcs are all labeled with weights, where WiJ is the weight between nodes i and j. During
processing, the functions at each node are applied to the input data to produce the output.
For example, the output of node 3 is
(3.27)
where h and g are the input height and gender values. Note that to determine the output
of a node we must know: the values input on each arc, the weights on all input arcs,
the technique used to combine the input values (a weighted sum is used here), and the
function h definition.
As with decision trees, we define a neural network in two parts: one for the data
structure and one for the general approach used, including the data structure and the
algorithms needed to use it.
FIGURE 3.6: Neural network for Example 3.7.
Section 3.5 Neural Networks 63
DEFINITION 3.5. A neural network (NN) is a directed graph, F = (V, A) with
vertices V = { 1 , 2, . . . , n} and arcs A = {(i, j} 1 1 ::::: i, j s n}, with the following
restrictions:
1. V is partitioned into a set of input nodes, VI, hidden nodes, VH, and output
nodes, Vo.
2. The vertices are also partitioned into layers { 1 , . . . , k} with all input nodes
in layer 1 and output nodes in layer k. All hidden nodes are in layers 2 to
k - 1 which are called the hidden layers.
3. Any arc (i, j} must have node i in layer h - 1 and node j in layer h.
4. Arc (i, j} is labeled with a numeric value WiJ·
5. Node i is labeled with a function fi .
Definition 3.5 is a very simplistic view of NNs. Although there are many more compli
cated types that do not fit this definition (as we will see later in the text), this defines
the most common type of NN, and the one that is used most often throughout this text.
More general NNs include some with arcs between any two nodes at any layers. Any
approaches that use a more generalized view of a graph for NNs will be adequately
defined before usage.
DEFINITION 3.6. A neural network (NN) model is a computational model con
sisting of three parts:
1. Neural network graph that defines the data structure of the neural network.
2. Learning algorithm that indicates how learning takes place.
3. Recall techniques that determine how information is obtained from the net
work. We discuss propagation in this text.
NNs have been used in pattern recognition, speech recognition and synthesis, medi
cal applications (diagnosis, drug design), fault detection, problem diagnosis, robot control,
and computer vision. In business, NNs have been used to "advise" booking of airline
seats to increase profitability. As a matter of fact, NNs can be used to compute any func
tion. Although NNs can solve problems that seem more elusive to other AI techniques,
they have a long training time (time during which the learning takes place) and thus are
not appropriate for real-time applications. NNs may contain many processing elements
and thus can be used in massively parallel systems.
Artificial NNs can be classified based on the type of connectivity and learning. The
basic type of connectivity discussed in this text is feed
forward, where connections are
only to layers later in the structure. Alternatively, a NN may be feedback where some
links are back to earlier layers. Learning can be either supervised or unsupervised, as is
discussed in section 4.5.2.
Figure 3.7 shows a sample node, i, in a neural network. Here there are k input
arcs coming from nodes 1, 2, . . . , k. with weights of W ! i , • . • , Wki and input values of
xu , . . . , Xki . The values that flow on these arcs are shown on dashed arcs because they do
not really exist as part of the graph itself. There is one output value Yi produced. During
propagation this value is output on all output arcs of the node. The activation function,

FIGURE 3.7: Neural network node.
f;, is applied to the inputs, which are scaled by applying the corresponding weights.
The weight in the NN may be determined in two ways. In simple cases where much is
known about the problem, the weights may be predetermined by a domain expert. The
more common approach is to have them determined via a learning process.
The structure of ttle NN may also be viewed from the perspective of matrices.
Input and weight on the arcs into node i are
(3.28)
There is one output value from node i, y; , which is propagated to all output arcs during
the propagation process. Using summation to combine the inputs, then, the output of a
node is
Yi = f; (tWji Xji)= f; ([wr; · . . Wk;] [�� ])
]=1 Xb
(3.29)
Here f; is the activation function. Because NNs are complicated, domain experts and
data mining experts are often advised to assist in their use. This in tum complicates the
process.
Overfitting occurs when the NN is trained to fit one set of data almost exactly. The
error that occurs with the given training data is quite small; however, when new data
are examined, the error is very large. In effect, the NN has "memorized" the training set
and cannot generalize to more data. Larger and more complicated NNs can be trained
to represent more complex functions. To avoid overfitting, smaller NNs are advisable.
However, this is difficult to determine beforehand. Another approach that can be used
to avoid overfitting is to stop the learning process early. Using a larger training set
also helps.
3.5.1 Activation Functions
The output of each node i in the NN is based on the definition of a function f; , activation
function, associated with it. An activation function is sometimes called a processing
element function or a squashing function. The function is applied to the set of inputs
coming in on the input arcs. Figure 3.7 illustrates the process. There have been many
proposals for activation functions, including threshold, sigmoid, symmetric sigmoid, and
Gaussian.
Section 3.5 Neural Networks 65
o ;...._
_
_
_
...�.........................
(a) Threshold (b) Sigmoid (c) Gaussian
FIGURE 3.8: Sample activation functions.
An activation function may also be called a firing rule, relating it back to the
workings of the human brain. When the input to a neuron is large enough, it fires,
sending an electrical signal out on its axon (output link). Likewise, in an artificial NN
the output may be generated only if the input is above a certain level; thus, the idea of a
firing rule. When dealing only with binary output, the output is either 0 or 1, depending
on whether the neuron should fire. Some activation functions use - 1 and 1 instead,
while still others output a range of values. Based on these ideas, the input and output
values are often considered to be either 0 or 1. The model used in this text is more
general, allowing any numeric values for weights and input/output values. In addition,
the functions associated with each node may be more complicated than a simple threshold
function. Since many learning algorithms use the derivative of the activation function
in these cases the function should have a derivative that is easy to find.
'
An activation function, fi , is applied to the input values {xli , . . . , Xki } and weights
{wu , ... , Wkd· These inputs are usually combined in a sum of products form S =
CI:�=l (wh; xh;)). If a bias input exists, this formula becomes S = wo; +(I:�=l(whiXhi)).
The following are alternative definitions for activation functions, f; (S) at node i. Activa
tion functions may be unipolar, with values in [0, 1], or bipolar, with values in [-1, 1].
The functions are also shown in Figure 3.8.
·
• Linear: A linear activation function produces a linear output value based on the
input. The following is a typical activation function
j; (S) = cS (3.30)
Here c is a constant positive value. With the linear function, the output value has
no limits in terms of maximum or minimum values.
• Threshold or step: The output value is either a 1 or 0, depending on the sum of the
products of the input values and their associated weights. As seen in Figure 3.8(a),
values above a threshold, T, will be 1 or 0:
f; (S) = { 1 if S > _T }
0 otherwise (3.31)
The binary output values may also be 1 or -1. Alternatively, the 1 value may
be replaced by any constant. A variation of this "hard limit" threshold function

is a linear threshold function. With the linear threshold functio�, a!so calle� a
ramp function or a piecewise linear function, the value of the acttvatwn
.
fun�t10n
increases gradually from the low value to the high value. One such functwn ts
f; (S) = {Is - rl
r2 - r1
0
if s > r2 l
if r1 :::: s :::: r2
if s < r1
(3.32)
Here the linear increase is between T1 and T2. As with the regular threshold
function, the value may be between -1 and 1 or 0 and 1 .
• Sigmoid: As seen in Figure 3.8(b), this is an "S"-shap�d cun:e with
.
output valm:s
between -1 and 1 (or 0 and 1), and which is monotomcally t�creasmg. �t�o�g?,
there are several types of sigmoid functions, they all have thts charactenstlc S
shape. A comm�n one is the logisticfunction
f; (S) =
(1 + e-cS)
(3.33)
Here c is a constant positive value that changes the slope of the function. This
function possesses a simple derivative: �1 = /; (1 - f;).
• Hyperbolic tangent: A variation of the sigmoid function is the hyperbolic tangent
function shown here
(3.34)
This function has an output centered at zero, which may help with learning.
• Gaussian: The Gaussian function, Figure 3.8(c), is a bell-shaped curve with output
values in the range [0, 1]. A typical Gaussian function is
(3.35)
Here s is the mean and v is the predefined positive variance of the function.
These are only a representative subset of the possible set of activation functions that
could be and have been used.
. .
Nodes in NNs often have an extra input called a bias. This bias value of 1 ts mput
on an arc with a weight of -8. The summation with bias input thus becomes
k
s; = L wji xji - e
j=l
(3.36)
The effect of the bias input is to move the activation function on the X axis by a value
of e. Thus, a weight of -8 becomes a threshold of 8 .
Section 3.6 Genetic Algorithms 67
3.6 GENETIC ALGORITH MS
Genetic algorithms are examples of evolutionary computing methods and are optimiza
tion-type algorithms. Given a population of potential problem solutions (individuals),
evolutionary computing expands this population with new and potentially better solu
tions. The basis for evolutionary computing algorithms is biological evolution, where
over time evolution produces the best or "fittest" individuals. Chromosomes, which are
DNA strings, provide the abstract model for a living organism. Subsections of the chro
mosomes, which are called genes, are used to define different traits of the individual.
During reproduction, genes from the parents are combined to produce the genes for
the child.
In data mining, genetic algorithms may be used for clustering, prediction, and even
association rules. You can think of these techniques as finding the "fittest" models from
a set of models to represent the data. In this approach a starting model is assumed and
through many iterations, models are combined to create new models. The best of these,
as defined by a fitness function, are then input into the next iteration. Algorithms differ
in how the model is represented, how different individuals in the model are combined,
and how the fitness function is used.
When using genetic algorithms to solve a problem, the first thing, and perhaps
the most difficult task, that must be determined is how to model the problem as a set
of individuals. In the real world, individuals may be identified by a complete encoding
of the DNA structure. An individual typically is viewed as an array or tuple of values.
Based on the recombination (crossover) algorithms; the values are usually numeric and
may be binary strings. These individuals are like a DNA encoding in that the structure
for each individual represents an encoding of the major features needed to model the
problem. Each individual in the population is represented as a string of characters from
the given alphabet.
DEFINITION 3.7. Given an alphabet A, an individual or chromosome is a string
I = ft , h, . . . , In where Ij E A. Each character in the string, Ij , is called a gene.
The values that each character can have are called the :alleles. A population, P, is
a set of individuals.
Although individuals are often represented as bit strings, any encoding is possible.
An array with nonbinary characters could be used, as could more complicated data
structures including trees and arrays. The only real restriction is that the genetic operators
(mutation, crossover) must be defined.
In genetic algorithms, reproduction is defined by precise algorithms that indicate
how to combine the given set of individuals to produce new ones. These are called
crossover algorithms. Given two individuals (parents from the population, the crossover
technique generates new individuals (offspring or children) by switching subsequences
of the strings. Figure 3.9 illustrates the process of crossover. The locations indicating the
crossover points are shown in the figure with the vertical lines. In Figure 3.9(a) crossover
is achieved by interchanging the last three bits of the two strings. In part (b) the center
three bits are interchanged. Figure 3.9 shows single and multiple crossover points. There
are many variations of the crossover approach, including determining crossover points
randomly. A crossover probability is used to determine how many new offspring are cre
ated via crossover. In addition, the actual crossover point may vary within one algorithm.

ooo looo 000 1111 ooo looo loo ooo l111 loo
111 1111 111 1000 111 1111 111 111 loool11
Parents Children Parents Children
(a) Single crossover (b) Multiple crossover
FIGURE 3.9: Crossover.
As in nature, however, mutations sometimes appear, and these also may be present
in genetic algorithms. The mutation operation randomly changes characters in the off
spring. A very small probability of mutation is set to determine whether a character
should change.
Since genetic algorithms attempt to model nature, only the strong survive. When
new individuals are created, a choice must be made about which individuals will survive.
This may be the new individuals, the old ones, or more likely a combination of the two.
The third major comp�nent of genetic algorithms, then, is the part that determines the
best (or fittest) individuals to survive.
One of the most important components of a genetic algorithm is determining how
to select individuals. A fitness function, f, is used to determine the best individuals
in a population. This is then used in the selection process to choose ?are�ts
:
Given an
objective by which the population can be measured, the fitness functwn md1cates how
well the goodness objective is being met by an individual.
DEFINITION 3.8. Given a population, P, a fitness function, f, is a mapping
f : P --+ R.
The simplest selection process is to select individuals based on their fitness:
f(I;)
PI; =
L f(IJ)
lj EP
(3.37)
Here p1- is the probability of selecting individual /; . This type of selection is called
roulette
'
wheel selection. One problem with this approach is that it is still possible to
select individuals with a very low fitness value. In addition, when the distribution is
quite skewed with a small number of extremely fit individuals, these individuals may
be chosen repeatedly. In addition, as the search continues, the population becomes less
diverse so that the selection process has little effect.
DEFINITION 3.9. A genetic algorithm (GA) is a computational model consisting
of five parts:
1. Starting set of individuals, P.
2. Crossover technique.
3. Mutation algorithm.
4. Fitness function.
Section 3.6 Genetic Algorithms 69
5. Algorithm that applies the crossover and mutation techniques to P iteratively
using the fitness function to determine the best individuals in P to keep. The
algorithm replaces a predefined number of individuals from the population
with each iteration and terminates when some threshold is met.
Suppose that each solution to the problem to be solved is represented as one of these
individuals. A complete search of all possible individuals would yield the best individual
or solution to the problem using the predefined fitness function. Since the search space
is quite large (perhaps infinite), what a genetic algorithm does is to prune from the
search space individuals who will not solve the problem. In addition, it only creates new
individuals who probably will be much different from those previously examined. Since
genetic algorithms do not search the entire space, they may not yield the best result.
However, they can provide approximate solutions to difficult problems.
ALGORITHM 3.3
Input :
p // Initial populat ion
Output :
P // Improved population
Genetic algorithm:
//Algorithm to il lustrate genetic algori thm
repeat
N =l P I ;
P = 0 ;
repeat
i1, i2 = select(P) ;
o1 , 02 = cros s(i1, i2) ;
01 = mutate(o1) ;
o2 = mutate(o2 ) ;
p = p u {01 ' 02 } ;
until I P I= N;
P = P ;
until termination criteria satisf ied;
Algorithm 3.3 outlines the steps performed by a genetic algorithm. Initially, a
population of individuals, P, is created. Although different approaches can be used to
perform this step, they typically are generated randomly. From this population, a new
population, P', of the same size is created. The algorithm repeatedly selects individuals
from whom to create new ones. These parents, i 1 , i2, are then used to produce two
offspring, o 1 , o2, using a crossover process. Then mutants may be generated. The process
continues until the new population satisfies the termination condition. We assume here
that the entire population is replaced with each iteration. An alternative would be to
replace the two individuals with the smallest fitness. Although this algorithm is quite
general, it is representative of all genetic algorithms. There are many variations on this
general theme.
Genetic algorithms have been used to solve most data mining problems, including
classification, clustering, and generating association rules. Typical applications of genetic
algorithms include scheduling, robotics, economics, biology, and pattern recognition.

The major advantage to the use of genetic algorithms is that they are easily paral-
lelized. There are, however, many disadvantages to their use:
• Genetic algorithms are difficult to understand and to explain to end users.
• The abstraction of the problem and method to represent individuals is quite difficult.
• Determining the best fitness function is difficult.
• Determining how to do crossover and mutation is difficult.
3.7 EXERCISES
1. Given the following set of values { 1 , 3, 9, 15, 20}, determine thejackknife estimate
for both the mean and standard deviation of the mean.
2. Redo Example 3.1 assuming that a coin is tossed six times with the following
results: {0, 1 , 0, .0, 1 , 0}.
3. Complete Example 3.3 by determining the correct classification for each x; .
4. Use linear regression with one predictor to determine the formula for the output
given the following samples: ( 1 , 3} and (2, 5}. Then predict the output value with
an input of 5.
5. Calculate the correlation coefficient r for the following:
(a) X values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Y values: 5, 7, 8, 9, 10, 12, 13, 15, 1 8, 20
(b) X values: 1 , 2, 3, 4, 5, 6, 7, 8, 9, 10
Y values: 10, 18, 15, 13, 12, 10, 9, 8, 7, 5
(c) X values: 3, 5, 2, 1 , 10
Y values: 10, 5, 8, 7, 2
6. Find the similarity between (0, 1 , 0.5, 0.3, 1) and ( 1 , 0, 0.5, 0, 0} using the Dice,
Jaccard, and cosine similarity measures.
7. Given the decision tree in Figure 3.5, classify each of the following students:
(Mary, 20, F, 2 m, Senior, Math} , (Dave, 19, M, 1 .7 m, Sophomore, Computer
Science}, and (Martha, 18, F, 1 .2 m, Freshman, English} .
8. Using the NN shown in Figure 3.6, classify the same students as those used in
exercise 7. Assume that the input nodes use an identity function, the hidden nodes
use a hyperbolic tangent activation function, the output layer uses a sigmoidal
function, and a weighted sum is used to calculate input to each node in the hidden
and output layers. You may assume any value for the constants in the activation
functions. Assume that the trained weights are defined by the difference between
the node numbers at either end of the arc. For example, the weight w13 = 0.2 and
W47 = 0.3.
9. Given an initial population {(101010} , (001 100} , (01010 1}, (000010}}, apply the
genetic algorithm to find a better population. Suppose the fitness function is defined
as the sum of the bit values for each individual and that mutation always occurs by
negating the second bit. The termination condition is that the average fitness value
for the entire population must be greater than 4. Also, an individual is chosen
Section 3.8 Bibliographic Notes 71
for crossover only if its fitness is greater than 2. (Hint: You can use different
crossover points.)
J?ata mining owes much of its development to previous work in machine learning, statis
tics, and databa�es [Man96]. The contribution of statistics to data mining can be traced
back to t�e s�rrunal work by Bayes in 1763. The EM algorithm dates back to [DLR77].
An e�arrunatwn of the impact statistics has had on the development of data mining
t�chrnques can
.
be
.
fou
.
nd in [IP96], [GMPS96], and [Mit99]. Many data mining tech
rnques find th�rr birth I
.
n the machine learning field. Excellent studies of the relationship
between machine leammg and data mining can be found in two recent books, [MBK99]
and [WFOO].
�e
.
first pro�osal for NNs was [MP43]. A special issue of Computer was devoted
to exarrurnng NNs m March 1988. Excellent overviews of NNs can be found in [Hay99].
Several excellent texts on NNs exist, including [Kas96], [Has95], and [Sim96].
A co�plete d�scussion of similarity measures can be found in [SM83].
Genetic a�gonth�s wer� first
.
proposed in 1975 by John Holland [Hol75]. A great
survey of genetic algonthms IS avrulable in [Gol89].

C H A P T E R 4
Classification
4.1 INTRODUCTION
4.2 STATISTICAL-BASED ALGORITHMS
4.3 DISTANCE-BASED ALGORITHMS
4.4 DECISION TREE-BASED ALGORITHMS
4.5 NEURAL NETWORK-BASED ALGORITHMS
4.6 RULE-BASED ALGORITHMS
4.7 COMBINING TECHNIQUES
4.8 SUMMARY
4.9 EXERCISES
4.1 INTRODUCTION
Classification is perhaps the most familiar and most popular data mining technique.
Examples of classification applications include image and pattern recognition, medical
diagnosis, loan approval, detecting faults in industry applications, and classifying financial
market trends. Estimation and prediction may be viewed as types of classification. Wheri
someone estimates your age or guesses the number of marbles in a jar, these are actually
classification problems. Prediction can be thought of as classifying an attribute value into
one of a set of possible classes. It is often viewed as forecasting a continuous value, while
classification forecasts a discrete value. Example 1 . 1 in Chapter 1 illustrates the use of
classification for credit card purchases. The use of decision trees and neural networks
(NNs) to classify people according to their height was illustrated in Chapter 3. Before
the use of current data mining techniques, classification was frequently performed by
simply applying knowledge of the data. This is illustrated in Example 4. 1.
EXAMPLE 4.1
Teachers classify students as A, B, C, D, or F based on their grades. By using simple
boundaries (60, 70, 80, 90), the following classification is possible:
90 ::5 grade A
80 ::5 grade < 90 B
70 ::5 grade < 80 C
60 ::5 grade < 70 D
grade < 60 F
75

76 Chapter 4 Classification
All approaches to performing classification assume some knowledge of the data.
Often a training set is used to develop the specific parameters required by the technique.
Training data consist of sample input data as well as the classification assignment for
the data. Domain experts may also be used to assist in the process.
The classification problem is stated as shown in Definition 4. 1 :
DEFINITION 4.1. Given a database D = {t1 , tz, . . . , tn} of tuples (items, records)
and a set of classes C = {C1 , . . . , Cm}, the classification problem is to define a
map�ing f : D -+ C where each ti is assigned to one class. A class, c1, contains
precisely those tuples mapped to it; that is, c1 = {t; 1 f(t;) = c1 , 1 s i s
n, and t; E D}.
Our definition views classification as a mapping from the database to the set of classes.
Note that the classes are predefined, are nonoverlapping, and partition the entire database.
Each tuple in the database is assigned to exactly one class. The classes that exist for a
classification problem are indeed equivalence classes. In actuality, the problem usually
is implemented in twotphases:
1. Create a specific model by evaluating the training data. This step has as input
the t�a
.
ining data (including defined classification for each tuple) and as output a
defirutwn of the model developed. The model created classifies the training data
as accurately as possible.
2. Apply the model developed in step 1 by classifying tuples from the target database.
Although the second step actually does the classification (according to the definition in
Definition 4.1), most research has been applied to step 1. Step 2 is often straightforward.
As discussed in [KLR+98], there are three basic methods used to solve the classi
fication problem:
• Specifying bou:"daries. Here classification is performed by dividing the input
space of potential database tuples into regions where each region is associated
with one class.
• Using probability distributions. For any given class, CJ, P(ti 1 CJ) is the PDF for
the class evaluated at one point, ti.1 If a probability of occurrence for each class
!(CJ) is kno
.
wn (perhaps determined by a domain expert), then P(Cj)P(ti 1 C.)
IS used to estimate the probability that t; is in class CJ.
1
• Using posterior probabilities. Given a data value t; , we would like to determine
the probability that t; is in a class CJ. This is denoted by P(CJ 1 ti) and is called
the posterior probability. One classification approach would be to determine the
posterior probability for each class and then assign ti to the class with the highest
probability.
The naive divisio�s used in Example 4. 1 as well as decision tree techniques are examples
of the first modehng approach. Neural networks fall into the third category.
1 In this discussion each tuple in the database is assumed to consist of a single value rather than a set of
values.
10
9
8
7
6
5
4
3
2
1
-
-
-
-
-
-
-
00
- 1 I I I I I
1 2 3 4 5 6 7 8
Class B Class C
(a) Definition of classes
Class A
10
9
8
7
6
5 X
4 X
3
2
1
00 1 2
X
X
X
X
Section 4. 1
10
9
8
7
6
5
4
3
2
1
Introduction 77
f-
X
X
f-
X
X
,_x X
f-X
X
X
f- X X
I- X
f- X X
0o 1 2 3 4 5 6 7 8
3 4 5 6 7 8
f- I I lx lx I I
(b) Sample database to classify (c) Database classified
FIGURE 4. 1 : Classification problem.
Suppose we are given that a database consists of tuples of the form t = (x, y}
where 0 S x S 8 and 0 S y S 10. Figure 4. 1 illustrates the classification problem.
Figure 4. l (a) shows the predefined classes by dividing the reference space, Figure 4. l (b)
provides sample input data, and Figure 4. l (c) shows the classification of the data based
on the defined classes.
A major issue associated with classification is that of overfitting. If the classification
strategy fits the training data exactly, it may not be applicable to a broader population of
data. For example, suppose that the training data has erroneous or noisy data. Certainly
in this case, fitting the data exactly is not desired.
In the following sections, various approaches to perfonning classification are exam
ined. Table 4.1 contains data to be used throughout this chapter to illustrate the various
techniques. This example assumes that the problem is to classify adults as short, medium,
or tall. Table 4. 1 lists height in meters. The last two columns of this table show two clas
sifications that could be made, labeled Outputl and Output2, respectively. The Outputl
classification uses the simple divisions shown below:
2 m s Height
1 .7 m < Height < 2 m
Height S 1.7 m
Tall
Medium
Short
The Output2 results require a much more complicated set of divisions using both height
and gender attributes.
In this chapter we examine classification algorithms based on the categorization
as seen in Figure 4.2. Statistical algorithms are based directly on the use of statistical
information. Distance-based algorithms use similarity or distance measures to perform
the classification. Decision tree and NN approaches use these structures to perform the
classification. Rule-based classification algorithms generate if
-then rules to perform the
classification.
4. 1 . 1 Issues in Classification
Missing Data. Missing data values cause problems during both the training phase and
the classification process itself. Missing values in the training data must be handled

TABLE 4. 1 : Data for Height Classification
Name Gender Height Outputl Output2
Kristina F 1.6 m Short Medium
Jim M 2 m Tall Medium
Maggie F 1.9 m Medium Tall
Martha F 1.88 m Medium Tall
Stephanie F 1.7 m Short Medium
Bob M 1.85 m Medium Medium
Kathy F 1.6 m Short Medium
Dave M 1.7 m Short Medium
Worth M 2.2 m Tall Tall
Steven M 2. 1 m Tall Tall
Debbie F 1.8 m Medium Medium
Todd M 1.95 m Medium Medium
Kim F 1.9 m Medium Tall
Amy F 1.8 m Medium Medium
Wynette F 1.75 m Medium Medium
S . �
l D'
tatistlca !stance DT NN Rules
FIGURE 4.2: Classification algorithm categorization.
and may produce an inaccurate result. Missing data in a tuple to be classified must be
able t? be �an
.
dled by the resulting classification scheme. There are many approaches to
handlmg ffilssmg data:
• Ignore the missing data.
• Assume a value for the missing data. This may be determined by using some
method to predict what the value could be.
• Assu�e a special value for the missing data. This means that the value of missing
data IS taken to be a specific value all of its own.
�otice
.
t?e similarity between missing data in the classification problem and that of nulls
m traditional databases.
.
Meas�ring Performance. Table 4. 1 shows two different classification results
usmg
.
two different classification tools. Determining which is best depends on the inter
preta�on of the prob�em by users. The performance of classification algorithms is usually
exaffilned by evaluatmg the accuracy of the classification. However, since classification is
Section 4.1 Introduction 79
often a fuzzy problem, the correct answer may depend on the user. Traditional algorithm
evaluation approaches such as determining the space and time overhead can be used, but
these approaches are usually secondary.
Classification accuracy is usually calculated by determining the percentage of tuples
placed in the correct class. This ignores the fact that there also may be a cost asso
ciated with an incorrect assignment to the wrong class. This perhaps should also be
determined.
We can examine the performance of classification much as is done with informa
tion retrieval systems. With only two classes, there are four possible outcomes with
the classification, as is shown in Figure 4.3. The upper left and lower right quad
rants [for both Figure 4.3(a) and (b)] represent correct actions. The remaining two
quadrants are incorrect actions. The performance of a classification could be deter
mined by associating costs with each of the quadrants. However, this would be dif
ficult because the total number of costs needed is m2, where m is the number of
classes.
Given a specific class, Cj, and a database tuple, t; , that tuple may or may not be
assigned to that class while its actual membership may or may not be in that class. This
again gives us the four quadrants shown in Figure 4.3(c), which can be described in the
following ways:
• True positive (TP): t; predicted to be in Cj and is actually in it.
• False positive (FP): t; predicted to be in Cj but is not actually in it.
• True negative (TN): t; not predicted to be in Cj and is not actually in it.
• False negative (FN): t; not predicted to be in Cj but is actually in it.
An OC (operating characteristic) curve or ROC (receiver operating characteristic)
curve or ROC (relative operating characteristic) curve shows the relationship between
false positives and true positives. An OC curve was originally used in the communications
area to examine false alarm rates. It has also been used in information retrieval to examine
fallout (percentage of retrieved that are not relevant) versus recall (percentage of retrieved
that are relevant). In the OC curve the horizontal axis has the percentage of false positives
and the vertical axis has the percentage of true positives for a database sample. At the
RET NOTRET
REL REL
RET NOTRET
NOTREL NOTREL
(a) Information retrieval
Assigned Class A Assigned Class B True positive False negative
in Class A in ClassA
Assigned Class A Assigned Class B False positive True negative
in Class B in Class B
(b) Classificationinto ClassA (c) Class prediction
FIGURE 4.3: Comparing classification performance to information retrieval.

75%
�
:�
0 50%
0..
<1)
=
F:
25%
25% 50% 75% 100%
False positives
FIGURE 4.4: Operating characteristic curve.
1
TABLE 4.2: Confusion Matrix
Actual Assignment
Membership
Short Medium Tall
Short 0 4 0
Medium 0 5 3
Tall 0 2
beginning of evaluating a sample, there are none of either category, while at the end
there are 100 percent of each. When evaluating the results for a specific sample, the
curve looks like a jagged stair-step, as seen in Figure 4.4, as each new tuple is either
a false positive or a true positive. A more smoothed version of the OC curve can also
be obtained.
A confusion matrix illustrates the accuracy of the solution to a classification prob
lem. Given m classes, a confusion matrix is an m x m matrix where entry ci,J indicates the
number of tuples from D that were assigned to class C1 but where the correct class is Ci .
Obviously, the best solutions will have only zero values outside the diagonal. Table 4.2
shows a confusion matrix for the height example in Table 4. 1 where the Outputl assign
ment is assumed to be correct and the Output2 assignment is what is actually made.
4.2 STATISTICAL-BASED ALGORITHMS
4.2.1 Regression
Regression problems deal with estimation of an output value based on input values. When
used for classification, the input values are values from the database D and the output
values represent the classes. Regression can be used to solve classification problems, but
Section 4.2 Statistical-Based Algorithms 81
it can also be used for other applications such as forecasting. Although not explicitly
described in this text, regression can be performed using many different types of tech
niques, including NNs. In actuality, regression takes a set of data and fits the data to
a formula.
Looking at Figure 3.3 in Chapter 3, we see that a simple linear regression problem
can be thought of as estimating the formula for a straight line (in a two-dimensional
space). This can be equated to partitioning the data into two classes. With the banking
example, these would be to approve or reject a loan application. The straight line is the
break-even point or the division between the two classes.
In Chapter 2, we briefly introduced linear regression using the formula
Y = CO + ClXl + · · · + CnXn (4. 1)
By determining the regression coefficients co, C[ , . . • , Cn the relationship between the
output parameter, y, and the input parameters, x1 , . . . , Xn can be estimated. All high
school algebra students are familiar with determining the formula for a straight line,
y = mx + b, given two points in the xy plane. They are determining the regression
coefficients m and b. Here the two points represent the training data.
Admittedly, Example 3.5 is an extremely simple problem. However, it illustrates
how we all use the basic classification or prediction techniques frequently. Figure 4.5
illustrates the more general use of linear regression with one input value. Here there
is a sample of data that we wish to model (shown by the scatter dots) using a linear
model. The line generated by the linear regression technique is shown in the figure.
Notice, however, that the actual data points do not fit the linear model exactly. Thus,
this model is an estimate of what the actual input-output relationship is. We can use the
generated linear model to predict an output value given an input value, but unlike that
for Example 3.5, the prediction is an estimate rather than the actual output value. If we
attempt to fit data that are not linear to a linear model, the results will be a poor model
of the data, as illustrated by Figure 4.5.
FIGURE 4.5: Example of poor fit for linear regression.

There are many reasons why the linear regression model may not be used to
estimate output data. One is that the data do not fit a linear model. It is possible, however,
that the data generally do actually represent a linear model, but the linear model generated
is poor because noise or outliers exist in the data. Noise is erroneous data. Outliers are
data values that are exceptions to the usual and expected data. Example 4.2 illustrates
o?tliers: Iri these cases the observable data may actually be described by the following:
y = CO + C1X1 + · · · + CnXn + E (4.2)
Here E is a random error with a mean of 0. As with point estimation, we can estimate the
accuracy of the fit of a linear regression model to the actual data using a mean squared
error function.
EXAMPLE 4.2
Suppose that a graduate level abstract algebra class has 100 students. Kristina consistently
outperforms the other 4;tudents on exams. On the final exam, Kristina gets a grade of 99.
The next highest grade is 75, with the range of grades being between 5 and 99. Kristina
clearly is resented by the other students in the class because she does not perform at
the same level they do. She "ruins the curve." If we were to try to fit a model to the
grades, this one outlier grade would cause problems because any model that attempted
to include it would then not accurately model the remaining data.
We illustrate the process using a simple linear regression formula and assuming k
points in our training sample. We thus have the following k formulas:
Yi = co + C1X1i + Ei , i = 1, . . . , k (4.3)
With a simple linear regression, given an observable value (xii, Yi), Ei is the error, and
thus the squared error technique introduced in Chapter 2 can be used to indicate the error.
To minimize the error, a method of least squares is used to minimize the least squared
error. This approach finds coefficients co, c1 so that the squared error is minimized for
the set of observable values. The sum of the squares of the errors is
k k
L = LEf = .L:<Yi - co - C1X1i)
2
i=1 i=l
(4.4)
Taking the partial derivatives (with respect to the coefficients) and setting equal to zero,
we can obtain the least squares estimates for the coefficients, c
o and c1 .
Regression can be used to perfonn classification using two different approaches:
1. Division: The data are divided into regions based on class.
2. Prediction: Formulas are generated to predict the output class value.
The first case views the data as plotted in an n-dimensional space without any explicit
class values shown. Through regression, the space is divided into regions-one per
class. With the second approach, a value for each class is included in the graph. Using
regression, the formula for a line to predict class values is generated.
Example 4.3 illustrates the division process, while Example 4.4 illustrates the pre
diction process using the data from Table 4. 1. For simplicity, we assume the training data
include only data for short and medium people and that the classification is performed
using the Outputl column values. If you extend this example to all three classes, you
will see that it is a nontrivial task to use linear regression for classification. It also will
become obvious tl�at the result may be quite poor.
EXAMPLE 4.3
By looking at the data in the Qutput1 column from Table 4.1 and the basic understanding
that the class to which a person is assigned is based only on the numeric value of his or
her height, in this example we apply the linear regression concept to determine how to
distinguish between the short and medium classes. Figure 4.6(a) shows the points under
consideration. We thus have the linear regression formula of y = co + E. This implies
that we are actually going to be finding the value for co that best partitions the height
numeric values into those that are short and those that are medium. Looking at the data
in Table 4. 1, we see that only 12 of the 15 entries can be used to differentiate between
short and medium persons. We thus obtain the following values for Yi in our training
data: { 1.6, 1.9, 1.88, 1.7, 1.85, 1.6, 1.7, 1.8, 1 .95, 1.9, 1.8, 1.75}. We wish to minimize
12 12
L = LEf = L(Yi - co)
2
i=1 i=1
Taking the derivative with respect to co and setting equal to zero we get
2.1
2 i-
�
1.9 i- ¢
0
:=
0
·� 1.8 I- ¢
:t
¢
1.7 r- ¢
1.6 - 0
1.5
12 12
-2 L Yi + L2co = 0
i=1 i=1
2.1
- 2 -
- 1.9 -
:=
- ·� 1.8 i-
:t
- 1.7 r-
- 1.6 I-
1.5
-
�
¢ -
0
¢
¢ -
-
-
'
(a) Short and medium heights (b) Division
Medium
y = 1.786
Short
FIGURE 4.6: Classification using division for Example 4.3.

Solving for co we find that
12
I >i
co =
i
=1 = 1.786
12
We thus have the division between short and medium persons as being determined by
y = 1 .786, as seen in Figure 4.6(b).
EXAMPLE 4.4
We now look at predicting the class using the short and medium data as input and looking
at the Output! classification. The data are the same as those in Example 4.3 except that we
now look at the classes as indicated in the training data. Since regression assumes numeric
data, we assume that the value for the short class is 0 and the value for the medium class
is 1. Figure 4.7(a) sho� the data for this example: {(1.6, 0), (1.9, 1), (1 .88, 1), (1.7, 0),
(1 .85, 1), (1.6, 0), (1.7, 0), (1.8, 1), (1 .95, 1), (1.9, 1), (1.8, 1), (1 .75, 1)}. In this case
we are using the regression formula with one variable:
We thus wish to minimize
Y = CO + C!XJ + E
12 12
L = L Ef = L(y; - co - C1X1i)2
i=1 i=l
Taking the partial derivative with respect to co and setting equal to zero we get
1 1- I
"'
6 0.5 r-
0I- i I
1.6 1.8
aL
12 12 12
- = -2 L Yi + 'L: 2co + L 2C!X1i = 0
aco
i=1 i=1 ;,1
I
I I
2 2.2
Height
I
_
- E o.s
u
I
- 0
2.4 1.6 1.8 2
Height
(a) Short and medium heights with classes (b) Prediction
FIGURE 4.7: Classification using prediction for Example 4.4.
2.2 2.4
To simplify the notation, in the rest of the example we drop the range values for the
summation because all are the same. Solving for co, we find that:
L Yi - L CJX!i
co =
12
Now taking the partial of L with respect to c1 , substituting the value for co, and setting
equal to zero we obtain
aL
- = 2 "'(y; - CO - C!X1i)(-X1;) = 0
ac1 �
Solving for c1, we finally have
We can now solve for co and CJ. Using the data from the 12 points in the training data,
we have L::xli = 21 .43, L Yi = 8, L::<xliy;) = 14.83, and L::<xr;) = 38.42. Thus, we
get q = 3.63 and co = -5.816. The prediction for the class value is thus
y = -5.816 + 3.63x1
This line is plotted in Figure 4.7(b).
In Example 4.4 a line that predicts the class value is generated. This was done for
two classes, but it also could have been done for all three classes. Unlike the division
approach where class membership is obvious based on the region within which a point
occurs, with prediction the class to which a point belongs is less obvious. Here we
predict a class value. In Figure 4.7(b) the class value is predicted based on the height
value alone. Since the prediction line is continuous, howe:ver, the class membership is
not always obvious. For example, ifthe prediction for a value is 0.4, what would its class
be? We can determine the class by splitting the line. So a height is in the short class if
its prediction value is less than 0.5 and it is in the medium class if its value is greater
than 0.5. In Example 4.4 the value of x1 where y = 0.5 is 1.74. Thus, this is really the
division between the short class and the medium class.
If the predictors in the linear regression function are modified by some function
(square, square root, etc.), then the model looks like
(4.5)
where fi is the function being used to transform the predictor. In this case the regression
is called nonlinear regression. Linear regression techniques, while easy to understand,
are not applicable to most complex data mining applications. They do not work well
with nonnumeric data. They also make the assumption that the relationship between the
input value and the output value is linear, which of course may not be the case.

Linear regression is not always appropriate because the data may not fit a straight
line, but also because the straight line values can be greater than 1 and less than 0. Thus,
they certainly cannot be used as the probability of occurrence of the target class. Another
commonly used regression technique is called logistic regression. Instead of fitting the
data to a straight line, logistic regression uses a logistic curve such as is illustrated in
Figure 4.8. The formula for a univariate logistic curve is
e<co+CJxJ)
P =
1 + e<co+cixil
(4.6)
The logistic curve gives a value between 0 and 1 so it can be interpreted as the probability
of class membership. As with linear regression, it can be used when classification into
two classes is desired. To perform the regression, the logarithmic function can be applied
to obtain the logistic function
loge (-
p
- ) = co + C[XJ
1 - p
(4.7)
Here p is the probability of being in the class and 1 - p is the probability that it is
not. However, the process chooses values for co and q that maximize the probability of
observing the given values.
4.2.2 Bayesian Classification
Assuming that the contribution by all attributes are independent and that each contributes
equally to the classification problem, a simple classification scheme called naive Bayes
classification has been proposed that is based on Bayes rule of conditional probability as
stated in Definition 3. 1 . This approach was briefly outlined in Chapter 3. By analyzing
� 0.6
8
·15 0.5
·6b
.s 0.4
0.3
0.2
0.1
4
X
FIGURE 4.8: Logistic curve.
the contribution of each "independent" attribute, a conditional probability is determined.
A classification is made by combining the impact that the different attributes have on the
prediction to be made. The approach is called "naive" because it assumes the indepen
dence between the various attribute values. Given a data value Xi the probability that a
related tuple, ti , is in class C1 is described by P(CJ I Xi). Training data can be used to
determine P(xi ), P(xi 1 C1 ), and P(C1 ). From these values, Bayes theorem allows us
to estimate the posterior probability P (CJ I Xi) and then P (C1 I ti).
Given a training set, the naive Bayes algorithm first estimates the prior probability
P(C1) for each class by counting how often each class occurs in the training data. For
each attribute, Xi , the number of occurrences of each attribute value Xi can be counted
to determine P (xi) . Similarly, the probability P(xi I Cj) can be estimated by counting
how often each value occurs in the class in the training data. Note that we are looking
at attribute values here. A tuple in the training data may have many different attributes,
each with many values. This must be done for all attributes and all values of attributes.
We then use these derived probabilities when a new tuple must be classified. This is
why naive Bayes classification can be viewed as both a descriptive and a predictive
type of algorithm. The probabilities are descriptive and are then used to predict the class
membership for a target tuple.
When classifying a target tuple, the conditional and prior probabilities generated
from the training set are used to make the prediction. This is done by combining the
effects of the different attribute values from the tuple. Suppose that tuple ti has p indepen
dent attribute values {xi ! , Xi2· . . . , Xip} From the descriptive phase, we know P(Xik I CJ),
for each class CJ and attribute Xik· We then estimate P(ti I Cj ) by
p
P(ti I Cj ) = IT P (Xik I Cj ) (4.8)
k=l
At this point in the algorithm, we then have the needed prior probabilities P (CJ) for
each class and the conditional probability P(ti I CJ ) . To calculate P(ti), we can estimate
the likelihood that ti is in each class. This can be done by finding the likelihood that
this tuple is in each class and then adding all these values. The probability that ti is
in a class is the product of the conditional probabilities for each attribute value. The
posterior probability P(CJ I ti) is then found for each class. The class with the highest
probability is the one chosen for the tuple. Example 4.5 illustrates the use of naive Bayes
classification.
EXAMPLE 4.5
Using the Outputl classification results for Table 4. 1, there are four tuples classified as
short, eight as medium, and three as tall. To facilitate classification, we divide the height
attribute into six ranges:
(0, 1 .6], ( 1 .6, 1 .7], ( 1 .7, 1 .8], (1.8, 1 .9], ( 1 .9, 2.0], (2.0, oo)
Table 4.3 shows the counts and subsequent probabilities associated with the attributes.
With these training data, we estimate the prior probabilities:
P (short) =
4/ 15 = 0.267, P (medium) = 8/15 = 0.533, and P (tall) = 3/1 5 = 0.2

TABLE 4.3: Probabilities Associated with Attributes
Attribute Value Count Probabilities
Short Medium Tall Short Medium Tall
Gender M 1 2 3 1/4 2/8 3/3
F 3 6 0 3/4 6/8 0/3
Height (0, 1 .6] 2 0 0 2/4 0 0
(1.6, 1 .7] 2 0 0 2/4 0 0
(1.7' 1.8] 0 3 0 0 3/8 0
(1 .8, 1.9] 0 4 0 0 4/8 0
(1 .9, 2] 0 1 1 0 118 1/3
(2, oo) 0 0 2 0 0 2/3
We use these values to
1
classify a new tuple. For example, suppose we wish to classify
t = (Adam , M, 1 .95 m) . By using these values and the associated probabilities of gender
and height, we obtain the following estimates:
P(t I short)
P (t I medium)
P (t I tall)
Combining these, we get
Likelihood of being short
Likelihood of being medium
Likelihood of being tall
1/4 X 0 = 0
2/8 X 1/8 = 0.03 1
3/3 X 1/3 = 0.333
0 X 0.267 = 0
0.03 1 X 0.533 = 0.0166
0.33 X 0.2 = 0.066
(4.9)
(4.10)
(4.1 1)
We estimate P(t) by summing up these individual likelihood values since t will be either
short or medium or tall:
P (t) = 0 + 0.0166 + 0.066 = 0.0826
Finally, we obtain the actual probabilities of each event:
P(short I t)
P (medium I t)
P (tall l t)
0 X 0.0267
= O
0.0826
0.03 1 X 0.533
= 0.2
0.0826
0.333 X 0.2
0.0826
= 0'799
(4. 12)
(4. 13)
(4. 14)
(4.15)
Therefore, based on these probabilities, we classify the new tuple as tall because it has
the highest probability.
Section 4.3 Distance-Based Algorithms 89
The naive Bayes approach has several advantages. First, it is easy to use. Second,
unlike other classification approaches, only one scan of the training data is required.
The naive Bayes approach can easily handle missing values by simply omitting that
probability when calculating the likelihoods of membership :in each class. In cases where
there are simple relationships, the technique often does yield good results.
Although the naive Bayes approach is straightforward to use, it does not always
yield satisfactory results. First, the attributes usually are not independent. We could use a
subset of the attributes by ignoring any that are dependent on others. The technique does
not handle continuous data. Dividing the continuous values into ranges could be used to
solve this problem, but the division of the domain into ranges is not an easy task, and
how this is done can certainly impact the results.
4.3 DISTANCE-BASED ALGORITHMS
Each item that is mapped to the same class may be thought of as more similar to the
other items in that class than it is to the items found in other classes. Therefore, similarity
(or distance) measures may be used to identify the "alikeness" of different items in the
database. The concept of similarity measure was introduced in Chapter 2 with respect to
IR retrieval. Certainly, the concept is well known to anyone who has performed Internet
searches using a search engine. In these cases, the set of Web pages represents the whole
database and these are divided into two classes: those that answer your query and those
that do not. Those that answer your query should be more alike than those that do not
answer your query. The similarity in this case is defined by the query you state, usually
a keyword list. Thus, the retrieved pages are similar because they all contain (to some
degree) the keyword list you have specified.
The idea of similarity measures can be abstracted and applied to more general
classification problems. The difficulty lies in how the similarity measures are defined
and applied to the items in the database. Since most similarity measures assume numeric
(and often discrete) values, they might be difficult to use for more general or abstract
data types. A mapping from the attribute domain to a subset of the integers may
be used.
Using a similarity measure for classification where the classes are predefined is
somewhat simpler than using a similarity measure for clustering where the classes are
not known in advance. Again, think of the IR example. Each IR query provides the
class definition in the form of the IR query itself. So the classification problem then
becomes one of determining similarity not among all tuples in the database but between
each tuple and the query. This makes the problem an O(n) problem rather than an
O (n
2
) problem.
4.3.1 Simple Approach
Using the IR approach, if we have a representative of each class, we can perform
classification by assigning each tuple to the class to which it is most similar. We
assume here that each tuple, ti, in the database is defined as a vector (til, ti2, . . . , fik)
of numeric values. Likewise, we assume that each class Cj is defined by a tuple
(Cjt. Cj2• . . . , Cjk) of numeric values. The classification problem is then restated in
Definition 4.2.

DEFINITION 4.2. Given a database D = {t1 , t2, . . . , tn} of tuples where each
tuple ti = (ti l , ti2• . . . , tik) contains numeric values and a set of classes C =
{C1 , . . . , Cm} where each class Cj = (Cj ! . Cj2• . . . , Cjk) has numeric values, the
classification problem is to assign each ti to the class Cj such that sim(ti , Cj) =:::
sim(ti . Ct)'VCt E C where Ct "1- Cj .
To calculate these similarity measures, the representative vector for each class
must be determined. Referring to the three classes in Figure 4. l(a), we can determine a
representative for each class by calculating the center of each region. Thus class A is
represented by (4, 7.5), class B by (2, 2.5), and class C by (6, 2.5). A simple classifica
tion technique, then, would be to place each item in the class where it is most similar
(closest) to the center of that class. The representative for the class may be found in other
ways. For example, in pattern recognition problems, a predefined pattern can be used
to represent each class. Once a similarity measure is defined, each item to be classified
will be compared to each predefined pattern. The item will be placed in the class with
the largest similarity 'value. Algorithm 4. 1 illustrates a straightforward distance-based
approach assuming that each class, Ci , is represented by its center or centroid. In the
algorithm we use Ci to be the center for its class. Since each tuple must be compared
to the center for a class and there are a fixed (usually small) number of classes, the
complexity to classify one tuple is O(n).
ALGORITHM 4.1
Input :
c1 , . . . , Cm I/Centers for each class
t // Input tuple to clas sify
Output :
c / /Class to which t i s ass igned
Simple dis tance-based algorithm
dist = oo ;
for i := 1 t o m do
i f dis(ci , t) < dis t , then
c = i ;
dist = dist(ci , t) ;
Figure 4.9 illustrates the use of this approach to perform classification using the
data found in Figure 4.1. The three large dark circles are the class representatives for the
three classes. The dashed lines show the distance from each item to the closest center.
4.3.2 K Nearest Neighbors
One common classification scheme based on the use of distance measures is that of
the K nearest neighbors (KNN). The KNN technique assumes that the entire training set
includes not only the data in the set but also the desired classification for each item. In
effect, the training data become the model. When a classification is to be made for a new
item, its distance to each item in the training set must be determined. Only the K closest
entries in the training set are considered further. The new item is then placed in the
class that contains the most items from this set of K closest items. Figure 4.10 illustrates
�
I
' I
' I
', I X
,I '
CB :l': - -x
/ '
/ '
X '
'
Section 4.3
2 3 4 5
Distance-Based Algorithms 91
Class A
6 7 8
FIGURE 4.9: Classification using simple distance algorithm.
10
9 f-- X
X
X
8 �
X
7 - X
6 1-
t / X
x---- �
I
5 1- I
I
X
4 f-- X
X
X
3 f--
2 1-
X
X
1 1- X
0
0 2 3 4 5 6 7 8
FIGURE 4.1 0: Classification using KNN.
the process used by KNN. Here the points in the training set are shown and K = 3. The
three closest items in the training set are shown; t will be placed in the class to which
most of these are members.
Algorithm 4.2 outlines the use of the KNN algorithm. We use T to represent
the training data. Since each tuple to be classified must be compared to each element
in the training data, if there are q elements in the training set, this is O (q). Given n
elements to be classified, this becomes an 0 (nq) problem. Given that the training data
are of a constant size (although perhaps quite large), this can then be viewed as an O(n)
problem.

ALGORITHM 4.2
Input :
T
K
t
Output :
/ /Tra ining data
/ /Number of ne ighbors
/ / Input tuple to c l a s s i fy
c / / Class to which t is ass igned
KNN algorithm:
/ /Algori thm t o c l a s s i fy tuple us ing KNN
N= 0 ;
/ / Find set o f neighbors , N, for t
for each d E T do
i f I N I.:S K, then
N= NU {d} ;
else
i f 3 u E N such that s im(t, u) _::: s im( t, d) , then
begip
i'!= N- {u} ;
N = NU {d} ;
end
/ / Find class for c la s s i f i cat ion
c = c lass to which the mo st u E N are class i f ie d ;
Example 4.6 illustrates this technique using the sample data from Table 4. 1. The
KNN technique is extremely sensitive to the value of K. A rule of thumb is that K <
jnumber of training items [KLR+98] . For this example, that value is 3.46. Commerci;;j
algorithms often use a default value of 10.
EXAMPLE 4.6
Using the sample data from Table 4. 1 and the Outputl classification as the training set
output value, we classify the tuple (Pat, F, 1 .6} . Only the height is used for distance calcu
lation so that both the Euclidean and Manhattan distance measures yield the same results;
that is, the distance is simply the absolute value of the difference between the values.
Suppose that K = 5 is given. We then have that the K nearest neighbors to the input tuple
are { (Kristina, F, 1 .6} , (Kathy, F, 1 .6} , (Stephanie, F, 1 .7} , (Dave, M, 1.7} , (Wynette,
F, 1.75}}. Of these five items, four are classified as short and one as medium. Thus,
the KNN will classify Pat as short.
4.4 DECISION TRE E-BASED ALGORITH MS
The decision tree approach is most useful in classification problems. With this technique,
a tree is constructed to model the classification process. Once the tree is built, it is applied
to each tuple in the database and results in a classification for that tuple. There are two
basic steps in the technique: building the tree and applying the tree to the database.
Most research has focused on how to build effective trees as the application process is
straightforward.
Section 4.4 Decision Tree-Based Algorithms 93
The decision tree approach to classification is to divide the search space into rect
angular regions. A tuple is classified based on the region into which it falls. A definition
for a decision tree used in classification is contained in Definition 4.3. There are alter
native definitions; for example, in a binary DT the nodes could be labeled with the
predicates themselves and each arc would be labeled with yes or no (like in the "Twenty
Questions" game).
DEFINITION 4.3. Given a database D = {t1 , . . . , tn} where t; = (ti l , . . . , t;h} and
the database schema contains the following attributes {A1 , A2, . • . , Ah }. Also given
is a set of classes C = {C1 , . • • , Cm}. A decision treE� (DT) or classification tree
is a tree associated with D that has the following properties:
• Each internal node is labeled with an attribute, A; .
• Each arc is labeled with a predicate that can be applied to the attribute associated
with the parent.
• Each leaf node is labeled with a class, Cj .
Solving the classification problem using decision trees is a two-step process:
1. Decision tree induction: Construct a DT using training data.
2. For each t; E D, apply the DT to determine its class.
Based on our definition of the classification problem, Definition 4. 1 , the constructed
DT represents the logic needed to perform the mapping. Thus, it implicitly defines the
mapping. Using the DT shown in Figure 3.5 from Chapter 3, the classification of the ·
sample data found in Table 4. 1 is that shown in the column labeled Output2. A different
DT could yield a different classification. Since the application of a given tuple to a
DT is relatively straightforward, we do not consider the second part of the problem
further. Instead, we focus on algorithms to construct decision trees. Several algorithms
are surveyed in the following subsections.
There are many advantages to the use of DTs for classification. DTs certainly
are easy to use and efficient. Rules can be generated that are easy to interpret and
understand. They scale well for large databases because the tree size is independent of
the database size. Each tuple in the database must be filtered through the tree. This takes
time proportional to the height of the tree, which is fixed. Trees can be constructed for
data with many attributes.
Disadvantages also exist for DT algorithms. First, they do not easily handle con
tinuous data. These attribute domains must be divided into categories to be handled.
The approach used is that the domain space is divided into rectangular regions [such as
is seen in Figure 4. l (a)]. Not all classification problems are of this type. The division
shown by the simple loan classification problem in Figure 2.4(a) in Chapter 2 cannot be
handled by DTs. Handling missing data is difficult because correct branches in the tree
could not be taken. Since the DT is constructed from the training data, overfitting may
occur. This can be overcome via tree pruning. Finally, correlations among attributes in
the database are ignored by the DT process.

ALGORITHM 4.3
Input :
D / /Training data
Output :
T / /Decision tree
DTBuild algorithm:
T = 0 ;
/ / S implistic a lgorithm t o i l lustrate naive approach
to bui lding DT
Determine best spl i t ting criterion ;
T = Create root node node and label with spl itting attr ibute ;
T = Add arc to root node for each sp lit predicate and l abel ;
for each arc do
D = Database created by applying splitting predicate to D;
if stopping point reached for this path, then
T = Create leaf node and l abel with appropriate class ;
else 1
T = DTBui ld(D) ;
T = Add T to arc ;
There have been many decision tree algorithms. We illustrate the tree-building
phase in the simplistic DTBuild Algorithm 4.3. Attributes in the database schema that will
be used to label nodes in the tree and around which the divisions will take place are called
the splitting attributes. The predicates by which the arcs in the tree are labeled are called
the splitting predicates. In the decision trees shown in Figure 4.1 1 , the splitting attributes
are {gender, height}. The splitting predicates for gender are {= female, = male}, while
those for height include {< 1 .3 m, > 1 .8 m, < 1 .5 m, >2 m}. The splitting predicates for
height differ based on whether the tuple is for a male or a female. This recursive algorithm
builds the tree in a top-down fashion by examining the training data. Using the initial
training data, the "best" splitting attribute is chosen first. Algorithms differ in how they
determine the "best attribute" and its "best predicates" to use for splitting. Once this
has been determined, the node and its arcs are created and added to the created tree.
The algorithm continues recursively by adding new subtrees to each branching arc. The
algorithm terminates when some "stopping criteria" is reached. Again, each algorithm
determines when to stop the tree differently. One simple approach would be to stop when
the tuples in the reduced training set all belong to the same class. This class is then used
to label the leaf node created.
Note that the major factors in the performance of the DT building algorithm are
the size of the training set and how the best splitting attribute is chosen. The following
issues are faced by most DT algorithms:
• Choosing splitting attributes: Which attributes to use for splitting attributes
impacts the performance applying the built DT. Some attributes are better than
others. In the data shown in Table 4.1, the name attribute definitely should not be
used and the gender may or may not be used. The choice of attribute involves not
only an examination of the data in the training set but also the informed input of
domain experts.
Section 4.4
Gender
�
Decision Tree-Based Algorithms 95
Height
<1.3
,
�2 m
Short Gender Tall
Height Height
/�
< 1 .�8m <l.�m
Height
< =1.8 ;/1.8 m
Height
<1.5 �= 1.5 m
Short Medium T
all Short Medium T
all
Short
I
1.3 1.5 1.8 2.0
(a) Balanced tree
>=1.3 m
<1.5 m
I
Height
Gender Medium Gender
=
1M =
1M
Medium Tall Short Medium
1.3 1.5 1.8 2.0
(b) Deep tree
T
all
Medium Short T
all Medium
Height
< =1 .5�=2 m
Short Medium Tall
:, H H I � I
I I
1.3 1.5 1.8 2.0 1.3 1.5 1.8 2.0
(c) Bushy tree (d) No gender attribute
FIGURE 4.1 1 : Comparing decision trees.
I
• Ordering of splitting attributes: The order in which the attributes are chosen
is also important. In Figure 4.1 l(a) the gender attribute is chosen first. Alterna
tively, the height attribute could be chosen first. As seen in Figure 4.l l(b), in this
case the height attribute must be examined a second time, requiring unnecessary
comparisons.
• Splits: Associated with the ordering of the attributes is the number of splits to
take. With some attributes, the domain is small, so the number of splits is obvious
based on the domain (as with the gender attribute). However, if the domain is
continuous or has a large number of values, the number of splits to use is not
easily determined.

• Tree structure: To improve the performance of applying the tree for classification,
a balanced tree with the fewest levels is desirable. However, in this case, more
complicated comparisons with multiway branching [see Figure 4.11(c)] may be
needed. Some algorithms build only binary trees.
• Stopping criteria: The creation of the tree definitely stops when the training data
are perfectly classified. There may be situations when stopping earlier would be
desirable to prevent the creation of larger trees. This is a trade-offbetween accuracy
of classification and performance. In addition, stopping earlier may be performed
to prevent overfitting. It is even conceivable that more levels than needed would
be created in a tree if it is known that there are data distributions not represented
in the training data.
• Training data: The structure of the DT created depends on the training data. If the
training data set is too small, then the generated tree might not be specific enough
to work properly with the more general data. If the training data set is too large,
then the created tree may overfit.
• Pruning: Once a tree is constructed, some modifications to the tree might be
needed to improve the performance of the tree during the classification phase. The
pruning phase might remove redundant comparisons or remove subtrees to achieve
better performance.
To illustrate some of these design decisions, Figure 4. 1 1 shows four different deci
sion trees that can be used to classify persons according to height. The first tree is a
duplicate of that from Chapter 3. The first three trees of this figure all perform the same
classification. However, they all perform it differently. Underneath each tree is a table
showing the logical divisions used by the associated tree for classification. A nice feature
of Figure 4.11(a) is that it is balanced. The tree is of the same depth for any path from
root to leaf. Figures 4.11(b) and (c), however, are not balanced. In addition, the height of
the tree in (b) is greater than that of any of the others, implying a slightly worse behavior
when used for classification. However, all of these factors impact the time required to do
the actual classification. These may not be crucial performance issues unless the database
is extremely large. In that case a balanced shorter tree would be desirable. The tree shown
in Figure 4. l l(d) does not represent the same classification logic as the others.
The training data and the tree induction algorithm determine the tree shape. Thus,
the best-shaped tree that performs perfectly on the training set is desirable. Some algo
rithms create only binary trees. Binary trees are easily created, but they tend to be
deeper. The performance results when applying these types of trees for classification
may be worse because more comparisons usually are needed. However, since these com
parisons are simpler than those that require multiway branches, the ultimate performance
may be comparable.
The DT building algorithms may initially build the tree and then prune it for more
effective classification. With pruning techniques, portions of the tree may be removed
or combined to reduce the overall size of the tree. Portions of the tree relating to clas
sification using an unimportant attribute may be removed. This sort of change with a
node close to the root could ripple down to create major changes in the lower parts of
the tree. For example, with the data in Figure 4. 1, if a tree were constructed by looking
4.4.1
at values of the name attribute, all nodes labeled with that attribute would be removed
L�wer-level nodes w�
uld move up or be combined in some way. The approach to doin�
this could become qmte com�licated. In the case of overfitting, lower-level subtrees may
be removed completely. Prumng may be performed while the tree is being created th
.
, us
�rev�ntmg a tree from becoming too large. A second approach prunes the tree after it
Is bmlt.
The time and space �omplexity of DT algorithms depends on the size of the training
data, q ; the number of attnbutes, h; and the shape of the resulting tree. In the worst case
the DT that is built may be quite deep and not bushy. As the tree is built, for each of
the
.
se nodes, ea�h attrib�te will be examined to determine if it is the best. This gives
a time complexity to build the tree of 0(h q log q). The time to classify a database of
size n is based on the height of the tree. Assuming a height of 0(log q), this is then
O(n log q).
In the following subsections we examine several popular DT approaches.
103
The �3 �echnique to building a decision tree is based on information theory and attempts
t� rmmrmze the exp�cted number of comparisons. The bask idea of the induction algo
nthm IS to ask questions whose answers provide the most information. This is similar to
the intuitive approach taken by adults when playing the "1venty Questions" game. The
�rst question an adult might ask could be "Is the thing alive?" while a child might ask "Is
1t II_lY Daddy?" The first question divides the search space into two large search domains,
:Vhile the second performs little division of the space. The basic strategy used by ID3
1s to choose splitting attributes with the highest information gain first. The amount of
information associated with an attribute value is related to the probability of occurrence.
�ooking at the "Twenty Questions" example, the child's question divides the search space
mto two sets. One set (Daddy) has an infinitesimal probability associated with it and the
?ther set is almost
.
certain, while the question the adult makes divides the search space
mto two subsets with almost equal probability of occurring.
The concept used to quantify information is called entropy. Entropy is used to
measure the amount of uncertainty or surprise or randomness in a set of data. Certainly,
when all data in a set belong to a single class, there is no uncertainty. In this case the
entropy is zero. The objective of decision tree classification is to iteratively partition
the given data set into subsets where all elements in each final subset belong to the
same class. In Figure 4. 12(a, b, and c) will help to explain the concept. Figure 4.12(a)
shows log(l/p) as the probability p ranges from 0 to 1. This intuitively shows the
amount of surprise based on the probability. When p = J., there is no surprise. This
means that if an event has a probability of 1 and you are told that the event occurred,
you would not be surprised. As p � 0, the surprise increases. When we deal with a
divide and conquer approach such as that used with decision trees, the division results in
multiple probabilities whose sum is 1 . In the "1venty Questions" game, the P(Daddy) <
P(-,Daddy) and P(Daddy) + P(-,Daddy) = 1. To measure the information associated
with this division, we must be able to combine the inforrnation associated with both
events. That is, we must be able to calculate the average information associated with
the division. This can be performed by adding the two values together and taking into
account the probability that each occurs. Figure 4.12(b) shows the function p log(ljp),

4
3
2
0.5 .------,..---,---r---r--,
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8
0
0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
(c) H(p, 1 - p)
(a) log(l/p) (b) p log(lip)
FIGURE 4. 12: Entropy.
which is the expected information based on probability of an event. To determine the
expected informatiori associated with two events, we add the individual values together.
This function p log(l /p) + (1 - p) log(1/(1 - p)) is plotted in Figure 4.12(c). Note that
the maximum occurs when the two probabilities are equal. This supports our intuitive
idea that the more sophisticated questions posed by the adult are better than those posed
by the child.
The formal definition of entropy is shown in Definition 4.4. The value for entropy
is between 0 and 1 and reaches a maximum when the probabilities are all the same.
DEFINITION 4.4. Given probabilities PI, P2, . . . , Ps where l:f=I Pi = 1, entropy
is defined as
H(p! , P2, . . . , Ps) = L(p; log(1/p;)) (4.16)
i = !
Given a database state, D, H (D) finds the amount of order (or lack thereof) in that
state. When that state is split into s new states S = {DJ , D2, . . . , Ds }, we can again look
at the entropy of those states. Each step in ID3 chooses the state that orders splitting
the most. A database state is completely ordered if all tuples in it are in the same class.
ID3 chooses the splitting attribute with the highest gain in information, where gain is
defined as the difference between how much information is needed to make a correct
classification before the split versus how much information is needed after the split.
Certainly, the split should reduce the information needed by the largest amount. This is
calculated by determining the differences between the entropies of the original dataset and
the weighted sum of the entropies from each of the subdivide� datasets. �he entr�p�e� of
the split datasets are weighted by the fraction of the dataset bemg placed
�
n that dlVlswn.
The ID3 algorithm calculates the gain of a particular split by the followmg formula:
Gain(D, S) = H(D) - L P(D;)H(D;) (4. 17)
i=l
Example 4.7 and associated Figure 4. 13 illustrate this process using the heig�t
example. In this example, six divisions of the possible ranges of heights are used. T
hts
Short
Height
<=1.7 m
>1.7 m
<=1.95 m
Short Medium
>1.95 m
T
all
Medium Tall
(a) Original tree (b) Optimized tree
FIGURE 4. 1 3: Classification problem.
division into ranges is needed when the domain of an attribute is continuous or (as in this
case) consists of many possible values. While the choice of these divisions is somewhat
arbitrary, a domain expert shoulQ be able to perform the task.
EXAMPLE 4.7
The beginning state of the training data in Table 4.1 (with the Outputl classification) is
that (4/15) are short, (8/15) are medium, and (3/15) are tall. Thus, the entropy of the
starting set is
4/15 log(15/4) + 8/15 log(15j8) + 3/15 log(15j3) = 0.4384
Choosing the gender as the splitting attribute, there are nine tuples that are F and six
that are M. The entropy of the subset that are F is
3/9 log(9/3) + 6/9 log(9/6) = 0.2764 (4. 1 8)
whereas that for the M subset is
1/6 log(6/1) + 2/6 log(6/2) + 3/6 log(6/3) = 0.4392 (4. 19)
The ID3 algorithm must determine what the gain in information is by using this split.
To do this, we calculate the weighted sum of these iast two entropies to get
((9/15) 0.2764) + ((6/15) 0.4392) = 0.34152 (4.20)
The gain in entropy by using the gender attribute is thus
0.4384 - 0.34152 = 0.09688 (4.21)
Looking at the height attribute, we have two tuples that are 1.6, two are 1 .7, one is
1.75, two are 1 .8, one is 1.85, one is 1 .88, two are 1.9, one js 1 .95, one is 2, one is
2.1 , and one is 2.2. Determining the split values for height is hot easy. Even though the
training dataset has these 1 1 values, we know that there will be many more. Just as with
continuous data, we divide into ranges:
(0, 1.6], (1 .6, 1 .7], (1.7, 1 .8], (1 .8, 1 .9], (1.9, 2.0], (2.0, oo)

There are 2 tuples in the first division with entropy (2/2(0) + 0 + 0) = 0, 2 in (1.6, 1.7]
with entropy (2/2(0) + 0 + 0) = 0, 3 in (1.7, 1.8] with entropy (0 + 3/3(0) + 0) = 0, 4 in
(1.8, 1.9] with entropy (0+4/4(0) +0) = 0, 2 in (1.9, 2.0] with entropy (0+ 1/2(0.301)+
1 /2(0.301)) = 0.301, and two in the last with entropy (0 + 0 + 2/2(0)) = 0. All of these
states are completely ordered and thus an entropy of 0 except for the (1.9, 2.0] state. The
gain in entropy by using the height attribute is thus
0.4384 - 2/15(0.301) = 0.3983 (4.22)
Thus, this has the greater gain, and we choose this over gender as the first splitting
attribute. Within this division there are two males, one medium and one tall. This has
occurred because this grouping was too large. A further subdivision on height is needed,
and this generates the DT seen in Figure 4.13(a).
Figure 4.13(a) illustrates a problem in that the tree has multiple splits with identical
results. In addition, there is a subdivision of range (1 .9, 2.0]. Figure 4.13(b) shows an
optimized version ofthe tree.
4.4.2 C4.5 and CS.O
The decision tree algorithm C4.5 iwproves ID3 in the following ways:
• Missing data: When the decision tree is built, missing data are simply ignored.
That is, the gain ratio is calculated by looking only at the other records that have
a value for that attribute. To classify a record with a missing attribute value, the
value for thai item can be predicted based on what is known about the attribute
values for the other r�cords.
• Continuous d;:lta: The basic idea is to divide the data into ranges based on the
attribute values for that item that are found irt the training sample.
� Pruning: There are two primary pruning strategies proposed in C4.5:
- With subtree replacement, a subtree is replaced by a leaf node if this replace
ment results in an error rate close to that of the original tree. Subtree replace
ment works from the bottom of the tree up to the root.
- Another pruning strategy, called subtree raising, replaces a subtree by its
most used subtree. Here a subtree is raised from its current location to a node
higher up in the tree. Again, we must determine the increase in error rate for
this replacement.
• Rules: C4.5 allows classification via either decision trees or rules generated from
them. In addition, some techniques to simplify complex rules are proposed. One
approach is to replace the left-hand side of a rule by a simpler version if all records
iri the training set are treated identically. An "otherwise" type of rule can be used
to indicate what should be done if no other rules apply.
• Splitting: The ID3 approach favors attributes with many divisions and thus may
lead to overfitting. In the extreme, an attribute that has a unique value for each
Section 4.4 Decision Tnee-Based Algorith ms 101
tuple in the training set would be the best because there would be only one tuple
(and thus one class) for each division. An improvement can be made by taking
into account the cardinality of each division. This approach uses the GainRatio as
opposed to Gain. The GainRatio is defined as
. . Gain(D, S)
GamRatw(D, S) = (lE!J �)
H I D l , . . . , I D l
(4.23)
For splitting purposes, C4.5 uses the largest GainRatio that ensures a larger than average
information gain. This is to compensate for the fact that the GainRatio value is skewed
toward splits where the size of one subset is close to that of the starting one. Example 4.8
shows the calculation of GainRatio for the first split in Example 4.7.
EXAMPLE 4.8
To calculate the GainRatio for the gender split, we first find the entropy associated with
the split ignoring classes
(9 6 ) 9 (15 ) 6 (15 )
H
- - = - log - + - log -- = 0.292
15
,
15 15 9 15 6
This gives the GainRatio value for the gender attribute as
0.09688
= 0.332
0.292
The entropy for the split on height (ignoring classes) is
( 2 2 3 4 2 )
H 15
'
15
'
15
'
lS' 15
(4.24)
(4.25)
(4.26)
C5.0 (called See 5 on Windows) is a commercial version of C4.5 now widely used
in many data mining packages such as Clementine and RuleQuest. It is targeted toward
use with large datasets. The DT induction is close to that of C4.5, but the rule generation
is different. Unlike C4.5, the precise algorithms used for C5.0 have not been divulged.
C5.0 does include improvements to generate rules. Results show that C5.0 improves on
memory usage by about 90 percent, runs between 5.7 and 240 times faster than C4.5,
and produces more accurate rules [ResO1].
One major improvement to the accuracy of C5.0 is based on boosting. Boosting
is an approach to combining different classifiers. While boosting normally increases the
time that it takes to run a specific classifier, it does improve the accuracy. The error
rate has been shown to be less than half of that found with C4.5 on some datasets
[ResOl]. Boosting does not always help when the training data contains a lot of noise.
Boosting works by creating multiple training sets from one training set. Each item in
the training set is assigned a weight. The weight indicates the importance of this item to
the classification. A classifier is constructed for each combination of weights used. Thus,
multiple classifiers are actually constructed. When C5.0 performs a classification, each
classifier is assigned a vote, voting is performed, and the target tuple is assigned to the
class with the most number of votes.

4.4.3 CART
Classification and regression trees (CART) is a technique that generates a binary decision
tree. As with ID3, entropy is used as a measure to choose the best splitting attribute and
criterion. Unlike ID3, however, where a child is created for each subcategory, only two
children are created. The splitting is performed around what is determined to be the best
split point. At each step, an exhaustive search is used to determine the best split, where
"best" is defined by
m
¢> (sjt) = 2h PR L I P(CJ I tL) - P (CJ I tR) I
}=I
(4.27)
This formula is evaluated at the current node, t, and for each possible splitting attribute
and criterion, s. Here L and R are used to indicate the left and right subtrees of the
current node in the tree. P[, PR are the probability that a tuple in the training set will
. . . . !tuples in subtree! nr
be on the left or nght side of the tree. Thts IS defined as !tuples in training setl "
v
ve assume
that the right branch is taken on equality. P(CJ I tL) or P(CJ I tR) is the probability
that a tuple is in this class, C1, and in the left or right subtree. This is defined as the
!tu
ples of class 1 in subtree!
. At each step only one criterion is chosen as the best over all
!tuples at the target nodel .
'
. . .
possible criteria. Example 4.9 shows Its use with the height example With Output1 results.
EXAMPLE 4.9
The first step is to determine the split attribute and criterion for the first split. We again
assume that there are six subranges to consider with the height attribute. Using these
ranges, we have the potential split values of 1.6, 1 .7, 1.8, 1 .9, 2.0. We thus have a choice
of six split points, which yield the following goodness measures:
¢> (Gender) 2(6/ 15) (9/ 15)(2/ 15 + 4/15 + 3/15) = 0.224 (4.28)
¢> (1 .6) 0 . (4.29)
¢> (1 .7) 2(2/15) ( 13/15)(0 + 8/15 + 3/ 15) = 0. 169 (4.30)
¢> (1 .8) 2(5/ 15) ( 10/15) (4/ 15 + 6/15 + 3/15) = 0.385 (4.3 1)
¢> (1 .9) 2(9/ 15)(6/15)(4/ 15 + 2/15 + 3/ 15) = 0.256 (4.32)
¢> (2.0) 2(12/15)(3/15)(4/ 15 + 8/15 + 3/ 15) = 0.32 (4.33)
The largest of these is the split at 1. 8. The remainder of this example is left as an exercise.
Since gender is really unordered, we assume M < F.
As illustrated with the gender attribute, CART forces that an ordering of the
attributes be used. CART handles missing data by simply ignoring that record in calcu
lating the goodness of a split on that attribute. The tree stops growing when no split will
improve the performance. Note that even though it is the best for the training data, it
may not be the best for all possible data to be added in the future. The CART algorithm
also contains a pruning strategy, which we will not discuss here but which can be found
in [KLR+98] .
Section 4.5 Neural Network-Based Algorithms 103
4.4.4 Scalable DT Techniques
We briefly examine some DT techniques that address creation of DTs for large datasets.
The SPRINT (Scalable PaRallelizable INduction of decision Trees) algorithm
addresses the scalability issue by ensuring that the CART technique can be applied
regardless of availability of main memory. In addition, it can be easily parallelized. With
SPRINT, a gini index is used to find the best split. Here gini for a database D is defined as
gini(D) = 1 - L PJ (4.34)
where p1 is the frequency of class C1 in D. The goodness of a split of D into subsets
Dr and D2 is defined by
ginisplit (D) = � (gini(Dr)) +
n2
(gini(D2))
n n
(4.35)
The split with the best gini value is chosen. Unlike the earlier approaches, SPRINT
does not need to sort the data by goodness value at each node during the DT induction
process. With continuous data, the split point is chosen to be the midpoint of every pair
of consecutive values from the training set.
By maintaining aggregate metadata concerning database attributes, the RainForest
approach allows a choice of split attribute without needing a training set. For each node
of a DT, a table called the attribute-value class (AVC) label group is used. The table
summarizes for an attribute the count of entries per class or attribute value grouping.
Thus, the AVC table summarizes the information needed to determine splitting attributes.
The size of the table is not proportional to the size of the database or training set, but
rather to the product of the number of classes, unique attribute values, and potential
splitting attributes. This reduction in size (for large training sets) facilitates the scaling of
DT induction algorithms to extremely large training sets. During the tree-building phase,
the training data are scanned, the AVC is built, and the best splitting attribute is chosen.
The algorithm continues by splitting the training data and constructing the AVC for the
next node.
4.5 NEURAL NETWORK-BASED ALGORITHMS
With neural networks (NNs), just as with decision trees, a model representing how to
classify any given database tuple is constructed. The activation functions typically are
sigmoidal. When a tuple must be classified, certain attribute values from that tuple are
input into the directed graph at the corresponding source nodes. There often is one sink
node for each class. The output value that is generated indicates the probability that
the corresponding input tuple belongs to that class. The tuple will then be assigned to
the class with the highest probability of membership. The learning process modifies the
labeling of the arcs to better classify tuples. Given a starting structure and value for all
the labels in the graph, as each tuple in the training set is sent through the network, the
projected classification made by the graph can be compared with the actual classification.
Based on the accuracy of the prediction, various labelings in the graph can change. This

learning process continues with all the training data or until the classification accuracy
is adequate.
Solving a classification problem using NNs involves several steps:
1. Determine the number of output nodes as well as what attributes should be used as
input. The number of hidden layers (between the source and the sink nodes) also
must be decided. This step is performed by a domain expert.
2. Determine weights (labels) and functions to be used for the graph.
3. For each tuple in the training set, propagate it through the network and evaluate
the output prediction to the actual result. If the prediction is accurate, adjust labels
to ensure that this prediction has a higher output weight the next time. If the
prediction is not correct, adjust the weights to provide a lower output value for this
class.
4. For each tuple ti E D, propagate t; through the network and make the appropriate
classification.
1
There are many issues to be examined:
• Attributes (number of source nodes): This is the same issue as determining which
attributes to use as splitting attributes.
• Number of hidden layers: In the simplest case, there is only one hidden layer.
• Number of hidden nodes: Choosing the best number of hidden nodes per hid
den layer is one of the most difficult problems when using NNs. There have been
many empirical and theoretical studies attempting to answer this question. The
answer depends on the structure of the NN, types of activation functions, training
algorithm, and problem being solved. If too few hidden nodes are used, the target
function may not be learned (underfitting). If too many nodes are used, overfit
ting may occur. Rules of thumb are often given that are based on the size of the
training set.
• Training data: As with DTs, with too much training data the NN may suffer from
overfitting, while too little and it may not be able to classify accurately enough.
• Number of sinks: Although it is usually assumed that the number of output nodes
is the same as the number of classes, this is not always the case. For example, with
two classes there could only be one output node, with the resulting value being the
probability of being in the associated class. Subtracting this value from one would
give the probability of being in the second class.
• Interconnections: In the simplest case, each node is connected to all nodes in the
next level.
• Weights: The weight assigned to an arc indicates the relative weight between those
two nodes. Initial weights are usually assumed to be small positive numbers and
are assigned randomly.
• Activation functions: Many different types of activation functions can be used.
Section 4.5 Neural Network-Based Al gorithms 105
• Learning technique: The technique for adjusting the weights is called the le�ing
technique. Although many approaches can be used, the most common approach is
some form of backpropagation, which is discussed in a subsequent subsection.
• Stop: The learning may stop when all the training tuples have propagated through
the network or may be based on time or error rate.
There are many advantages to the use of NNs for classification:
• NNs are more robust than DTs because of the weights.
• The NN improves its performance by learning. This may continue even after the
training set has been applied.
• The use of NNs can be parallelized for better perfom1ance.
• There is a low error rate and thus a high degree of accuracy once the appropriate
training has been performed.
• NNs are more robust than DTs in noisy environments.
Conversely, NNs have many disadvantages:
• NNs are difficult to understand. Nontechnical users may have difficulty understand
ing how NNs work. While it is easy to explain decision trees, NNs are much more
difficult to understand.
• Generating rules from NNs is not straightforward.
• Input attribute values must be numeric.
• Testing
• Verification
• As with DTs, overfitting may result.
• The learning phase may fail to converge.
• NNs may be quite expensive to use.
4.5.1 Propagation
The normal approach used for processing is called propagation. Given a tuple of values
input to the NN, X = (XJ , . . . , Xh ), one value is input at each node in the input layer.
Then the summation and activation functions are applied at each node, with an output
value created for each output arc from that node. These values are in tum sent to the
subsequent nodes. This process continues until a tuple of output values, Y = (y1 , . . . , Ym),
i s produced from the nodes in the output layer. The process of propagation i s shown in
Algorithm 4.4 using a neural network with one hidden layer. Here a hyperbolic tangent
activation function is used for the nodes in the hidden layer, while a sigmoid function
is used for nodes in the output layer. We assume that the constant c in the activation
function has been provided. We also use k to be the number of edges corning into
a node.

1Q6 Chapter 4 Classification
ALGORITHM 4.4
Input :
N I/neural network
,, , .
X = (x1;. . . , Xh) // Input tuple cori� isting of values for
input attributes only
Output :
Y = (y1 , . . . , Ym) I/Tuple consisting of output values from NN
Propagation algori thm :
/ /Algorithm il lustrates propagat ion of a tuple
through a NN
for each node i in the input layer do
Output Xi on each output arc from i ;
for each hidden layer do
for· each node i do
si = (I:�=1(wjiXji));
for each output arc f rom i do
(1-e-5i)
Output
(1+e csi) ;
for each node i in the output layer do
Si = (I:�=1(WjiXji))i
1
Output Yi = (1+e c si) ;
A simple application of propagation is shown in Example 4. 10 for the height data.
Here the classification performed is the same as that seen with the decision tree in
Figure 4. ll(d).
EXAMPLE 4.10
Figure 4.14 shows a very simple NN used to classify university students as short, medium,
or tall. There are two input nodes, one for the gender data and one for the height
data. There are three output nodes, each associated with one class and using a simple
threshold activation function. Activation function h is associated with the short class,
!4 is associated with the medium class, and fs is associated with the tall class. In this
case, the weights of each arc from the height node is 1. The weights on the gender arcs
is 0. This implies that in this case the gender values are ignored. The plots for the graphs
of the three activation functions are shown.
4.5.2 NN Supervised Learning
The NN starting state is modified based on feedback of its petformance with the data in
the training set. This type of learning is referred to as supervised because it is known a
priori what the desired output should be. Unsupervised learning can also be performed if
the output is not known. With unsupervised approaches, no external teacher set is used. A
training set may be provided, but no labeling of the desired outcome is included. In this
case, similarities and differences between different tuples in the training set are uncovered.
in this chapter, we examine supervised learning. We briefly explore unsupervised learning
in Chapter 5.
Section 4.5 Neural NetwCJrk-Based Algorithms 107
:ll -
�
Small
Tall
lM-
0 -
1 2
FIGURE 4. 14: Example propagation for tall data.
Supervised learning in an NN is the process of adjusting the arc weights based
on its performance with a tuple from the training set. The behavior of the training data
is known a priori and thus can be used to fine-tune the network for better behavior
in future similar situations. Thus, the training set can be used as a "teacher" during
the training process. The output from the network is compared to this known desired
behavior. Algorithm 4.5 outlines the steps required. One potential problem with super
vised learning is that the error may not be continually reduced. It would, of course,
be hoped that each iteration in the learning process reduces the error so that it is ulti
mately below an acceptable level. However, this is not always the case. This may be
due to the error calculation technique or to the approach used for modifying the weights.
This is actually the general problem of NNs. They do not guarante� convergence or
optimality.
ALGORITHM 4.5
Input :
N
X
D
Output :
/ /Starting neural network
/ / Input tuple from training set
//Output tuple des ired
N // Improved neural network
SupLearn algorithm :
//Simplis t ic algori thm to il lustrate approach
to NN learning
Propagate X through N producing output Y;
Calculate error by comparing v' to Y;
Upcj.ate weights on arcs in N to reduce error ;
Notice that this algorithm must be associated with a means to calculate the error
as well as some technique to adjust the weights. Many techniques have been proposed
to calculate the error. Assuming that the output from node i is Yi but should be di, the

error produced from a node in any layer can be found by
The mean squared error (MSE) is found by
(Yi - di)2
2
(4.36)
(4.37)
This MSE can then be used to find a total error over all nodes in the network or over
only the output nodes. In the following discussion, the assumption is made that only the
final output of the NN is known for a tuple in the training data. Thus, the total MSE
error over all m output nodes in the NN is
t (Yi - di)2
i=l
m
(4.38)
This formula couffl be expanded over all tuples in the training set to see the total error
over all of them. Thus, an error can be calculated for a specific test tuple or for the total
set of all entries.
Tlie Hebb and delta rules are approaches to change the weight on an input arc to a
node based on the knowledge that the output value from that node is incorrect. With both
techniques, a learning rule is used to modify the input weights. Suppose for a given node,
j, the input weights are represented as a tuple (wlj , . . . , Wkj), while the input and output
values are (xu. . . . , Xkj) and YJ· respectively. The objective of a learning technique is to
change the weights based on the output obtained for a specific input tuple. The change
in weights using the Hebb rule is represented by the following rule
(4.39)
Here c is a constant often called the learning rate. A rule of thumb is that c =
1
I# entries in training setl ·
A variation of this approach, called the delta rule, examines not only the output
value YJ but also the desired value dj for output. In this case the change in weight is
found by the rule
(4.40)
The nice feature of the delta rule is that it minimizes the error d1 - YJ at each node.
Backpropagation is a learning technique that adjusts weights in the NN by prop
agating weight changes backwa.td from the sink to the source nodes. Backpropagation
is the most well known form of learning because it is easy to understand and generally
applicable. Backpropagation can be thought of as a generalized delta rule approach.
Figure 4.15 shows the structure and use of one tiode, j, in a neural network graph.
The basic node structure is shown in part (a). Here the representative input arc has
a weight cif W?j. where ? is used to show that the input to node j is corning from
another node shown here as ?. Of course, there probably are multiple input arcs to
a node. The output weight is similarly labeled wJ?· During propagation, data values
input at the input layer flow through the network, with final values corning out of the
network at the output layer. The propagation technique is shown in part (b) Figure 4.15.
�
X?j Yj? �
!!.w?j dw
j?
(a) Node j in NN (b) Propagation at Nodej (c) Back-propagation at Node j
FIGURE 4. 1 5: Neural network usage.
Here the smaller dashed arrow underneath the regular graph arc shows the input value
X?j flowing into node j. The activation function fJ is applied to all the input values
and weights, with output values resulting. There is an associated input function that is
applied to the input values and weights before applying the activation function. This
input function is typically a weighted sum of the input values. Here YJ? shows the output
value flowing (propagating) to the next node from node j. Thus, propagation occurs
by applying the activation function at each node, which then places the output value
on the arc to be sent as input to the next nodes. In most cases, the activation function
produces only one output value that is propagated to the set of connected nodes. The
NN can be used for classification and/or learning. During the classification process, only
propagation occurs. However, when learning is used after the output of the classification
occurs, a comparison to the known classification is used to determine how to change
the weights in the graph. In the simplest types of learning, learning progresses from the
output layer backward to the input layer. Weights are changed based on the changes
that were made in weights in subsequent arcs. This backward learning process is called
backpropagation and is illustrated in Figure 4. 15(c). Weight wJ? is modified to become
wj? + 6wj?. A learning rule is applied to this 6wj? to determine the change at the next
higher level 6W?j.
ALGORITHM 4.6
Input :
N
X = (xl , . . . , Xh)
D = (d1, . . . , dm)
Output :
/ / Start ing neural network
/ / Input tuple f rom tra ining set
/ /Output tuple des ired
N / / Improved neural network
Backpropagation algorithm:
/ / I l lustrate backpropagation
Propagat i on (N, X) ;
E = 1/2 L7=l (di - Yi)2 ;
Gradient (N, E) ;
A simple version of the backpropagation algorithm is shown in Algorithm 4.6. The
MSE is used to calculate the error. Each tuple in the training set is input to this algorithm.
The last step ofthe algorithm uses gradientdescent as the technique to modify the weights
in the graph. The basic idea of gradient descent is to find the set of weights that minimizes
the MSE. a�i gives the slope (or gradient) of the error function for one weight. We thus
wish to find the weight where this slope is zero. Figure 4.16 and Algorithm 4.7 illustrate
the concept. The stated algorithm assumes only one hidden layer. More hidden layers
would be handled in the same manner with the error propagated backward.

1 1 0 Chapter 4 Classification
E
--
--�------------r-------------- w
Desired weight
FIGURE 4. 1 6: Gradient descent.
.ALGORITHM 4.7
Input :
N / /Start ing neural network
E / /Error found from back algorithm
Output :
N / / Improved neural network
Gradient algorithm:
/ / I llustrates incremental gradient des cent
for each node i in output layer do
for each node j input to i do
ll.wji = TJ(di - Yi)Yj(l - Yi)Yi ;
Wji = Wji + fl.Wji ;
layer = previous layer ;
for each node j in thi s l ayer do
for each node k input to j do
l-(y ·)2
fl.Wkj =
T/Yk--
-
2_
J_ Lm(dm - Ym)WjmYm(l - Ym) ;
Wkj = Wkj + fl.Wkj ;
This algorithm changes weights by working backward from the output layer to the
input layer. There are two basic versions of this algorithm. With the batch or offline
approach, the weights are changed once after all tuples in the training set are applied
and a total MSE is found. With the incremental or online approach, the weights are
changed after each tuple in the training set is applied. The incremental technique is
usually preferred because it requires less space and may actually examine more potential
solutions (weights), thus leading to a better solution. In this equation, 17 is referred to
as the learning parameter. It typically is found in the range (0, 1), although it may be
larger. This value determines how fast the algorithm learns.
Applying a learning rule back through multiple layers in the network may be
difficult. Doing this for the hidden layers is not as easy as doing it with the output layer.
Overall, however, we are trying to minimize the error at the output nodes, not at each
node in the network. Thus, the approach that is used is to propagate the output errors
backward through the network.
Section 4.5 Neural Network-Based Algorithms 1 1 1
Output
wkj
I wji I Y;
0
. '0------�>
- - - - - - - - - - � - - - - - - - - - - - - - - - - - - �
Yk Yi
FIGURE 4. 1 7: Nodes for gradient descent.
Figure 4.17 shows the structure we use to discuss the gradient descent algorithm.
Here node i is at the output layer and node j is at the hidden layer just before it; y; is
the output of i and yJ is the output of j .
The learning function i n the gradient descent technique is based o n using the
following value for delta at the output layer:
aE aE ay; as;
/).Wji = -1] -- = -1] - - -- (4.41)
awji ay; as; awji
Here the weight wJi is that at one arc coming into i from j. Assuming a sigmoidal
activation function in the output layer, for the output layer we have
:� = a�i ((1 +�-S;)) = (1 - (1 +�-S;)) (1 +�-S;) = (1 -y;)y; (4.42)
Also,
as;
-- = YJ
awji
For nodes in the output layer, the third partial is
::, = a:, (�pdm -Ym)2) = -(d, -y;)
We thus have the formula in Algorithm 4.7:
f).w.. = 17(d· -y-)y·
(1 - 1
)
1 = 17(d; -y;)y1·(1 - y;)y;
Jl r r J
1 + e-S; 1 + e-S;
(4.43)
(4.44)
(4.45)
For a node j in the hidden layer, calculating the change in the weight for arcs
coming into it is more difficult. Using Figure 4.17, we derive this change as
where
aE
/).Wkj = -1] -- (4.46)
awkj
(4.47)
Here the variable m ranges over all output nodes with arcs from j . We then derive
aE -(dm -Ym) (4.48)
(4.49)
Wjm (4.50)

1 12 Chapter 4 Classification
For hidden layers, where a hyperbolic tangent activation function is assumed, we have
Also,
asJ
= Yk
awkj
This gives us the formula in Algorithm 4.7:
1 - (yj)2
/).Wkj = 11 Yk
2
L(dm - Ym)WjmYm O - Ym)
m
Another common formula for the change in weight is
aE
/).Wji (t + 1) = - 1'}
BWji
+ af).Wji (t)
1 - y2
__
}
2
(4.5 1)
(4.52)
(4.53)
(4.54)
Here the change in weight at time t + 1 is based not only on the same partial derivative
as earlier, but also on the last change in weight. Here a is called the momentum and is
used to prevent oscillation problems that may occur without it.
4.5.3 Radial Basis Function Networks
A radial function or a radial basisfunction (REF) is a class of functions whose value
decreases (or increases) with the distance from a central point. The Gaussian activation
function shown in Equation 3.35 is an RBF with a central point of 0. An RBF has a
Gaussian shape, and an RBF network is typically an NN with three layers. The input layer
is used to simply input the data. A Gaussian activation function is used at the hidden
layer, while a linear activation function is used at the output layer. The objective is to
have the hidden nodes learn to respond only to a subset of the input, namely, that where
the Gaussian function is centered. This is usually accomplished via supervised learning.
When RBF functions are used as the activation functions on the hidden layer, the nodes
can be sensitive to a subset of the input values. Figure 4.18 shows the basic structure of
an RBF unit with one output node.
4.5.4 Perceptrons
The simplest NN is called a perceptron. A perceptron is a single neuron with multiple
inputs and one output. The original perceptron proposed the use of a step activation
function, but it is more common to see another type of function such as a sigmoidal
function. A simple perceptron can be used to classify into two classes. Using a unipolar
activation function, an output of 1 would be used to classify into one class, while an
output of 0 would be used to pass in the other class. Example 4. 1 1 illustrates this.
EXAMPLE 4.1 1
Figure 4.19(a) shows a perceptron with two inputs and a bias input. The three weights
are 3, 2, and - 6, respectively. The activation function !4 is thus applied to the value
FIGURE 4. 1 8: Radial basis function network.
Xz
y
- - �
(a) Classification perceptron (b) Classification problem
FIGURE 4. 1 9: Petceptron classification example.
S = 3x1 + 2x2 - 6. Using a simple unipolar step activation function, we get
{ 1 if S > O }
!4 =
0 otherwise
(4.55)
An alternative way to view this classification problem is shown in Figure 4. 1 9(b). Here
x1 is shown on the horizontal axis and x2 is shown on the vertical axis. The area of the
plane to the right of the line x2 =
3 '_ 3j2xi represents one class and the rest of the
plane represents the other class.
The simple feedforward NN that was introduced in Chapter 3 is actually called a
multilayerperceptron (MLP). An MLP Is a network of perceptrons. Figure 3.6 showed
an MLP used for classifying the height example given in Table 4. 1 . The neurons are
placed in iayers with outputs always flowing toward the output layer. If only one layer
exists, it is called a perceptron. If multiple layers exist, it is an MLP.
In the 1950s a Russian mathematician, Andrey Kolmogorov, proved that an MLP
needs no more than two hidden layers. Kolmogorov's theorem states that a mapping

between two sets of numbers can be performed using an NN with only one hidden
layer. In this case, the NN is to have one input node for each attribute input, and
given n input attributes the hidden layer should have (2n + 1) nodes, each with input
from each of the input nodes. The output layer has one node for each desired out
put value.
4.6 RULE-BASED ALGORITHMS
One straightforward way to perform classification is to generate if
-then rules that cover
all cases. For example, we could have the following rules to determine classification
of grades:
If 90 ::S grade, then class
If 80 ::S grade and grade < 90, then class
If 70 ::S grade and grade < 80, then class
A
B
c
i 60 ::S grade and grade < 70, then class D
If grade < 60, then class F
A classification rule, r = (a, c), consists of the if or antecedent, a, part and the then or
consequent portion, c. The antecedent contains a predicate that can be evaluated as true
or false against each tuple in the database (and obviously in the training data). These
rules relate directly to the corresponding DT that could be created. A DT can always be
used to generate rules, but they are not equivalent. There are differences between rules
and trees:
• The tree has an implied order in which the splitting is performed. Rules have no
order.
. ·
• A tree is created based on looking at all classes. When generating rules, only one
class must be examined at a time.
There are algorithms that generate rules from trees as well as algorithms that generate
rules without first creating DTs.
4.6. 1 Generating R�les from a DT
The process to generate a rule from a DT is straightforward and is outlined in Algo
rithm 4.8. This algorithm will generate a rule for each leaf node in the decision tree. All
rules with the same consequent could be combined together by ORing the antecedents
of the simpler rules.
ALGORlTHM 4.8
Input :
T / /De cis ion tree
Output :
R / /Rules
Section 4.6 ��ule-Based Algorithms 1 1 5
Gen algorithm :
/ / I l lustrate s imple approach to generating
clas s i f i cat ion rules f rom a DT
R = 0
for each path from root to a leaf in T do
a = True
for each non - l eaf node do
a = a/ ( l abel of node combined with label of incident
outgoing arc )
c = label of leaf node
R = R U r = (a, c)
Using this algorithm, the following rules are generated for the DT in Figure 4.13(a):
{((Height ::S 1 .6 m), Short)
(((Height > 1 .6 m) 1 (Height ::S 1 .7 m)), Short)
(((Height > 1 .7 m) 1 (Height ::S 1.8 m)), Medium)
(((Height > 1.8 m) 1 (Height ::s 1 .9 m)), Medium)
(((Height > 1 .9 m) 1 (Height ::: 2 m) 1 (Height ::: 1.95 m)), Medium)
(((Height > 1.9 m) 1 (Height ::S 2 m) 1 (Height > 1.95 m)), Tall)
((Height > 2 m), Tall)}
An optimized version of these rules is then:
{((Height ::S 1 .7 m), Short)
(((Height > 1 .7 m) 1 (Height ::: 1 .95 m)), Medium)
((Height > 1.95 m), Tall)}
4.6.2 Generating Rules from a Neural Net
To increase the understanding of an NN, classification rules may be derived from it.
While the source NN may still be used for classification, the derived rules can be used to
verify or interpret the network. The problem is that the rules do not explicitly exist. They
are buried in the structure of �p.e graph itself. In addition, if learning is still occurring,
the rules themselves are dynaffiic. The rules generated tend both to be more concise
and to have a lower error rate than rules used with DTs. The basic idea of the RX
algorithm is to cluster output values with the associated hidden nodes and input. A major
problem with rule extraction is the potential size that these rules should be. For example,
if you have a node with n inputs each having 5 values, there are 5n different input
combinat�ons to this one node alone. These patterns would all have to be accounted for
when constructing rules. To overcome this problem and that of having continuous ranges
of output values from nodes, the output values for both the hidden and output layers are
first discretized. This is accomplished by clustering the values and dividing continuous
values into disjoint ranges. The rule extraction algorithm, RX, shown in Algorithm 4.9
is derived from [LSL95].

1 1 6 Chapter 4 Classification
ALGORITHM 4.9
Input :
D //Training data
N / / Init ial neural network
Output :
R //Derived rules
RX algorithm:
/ /Rule extract i on algorithm to extract rules from NN
cluster output node act ivation values ;
c luster hidden node act ivation values ;
generate rules that describe the output values in terms of
the hidden act ivation values ;
generate rules that des cribe hidden output values in
terms of inputs ;
combine the two sets of rules .
4.6.3 Generating Rules iNithout a DT or N N
These techniques are sometimes called covering algorithms because they attempt to
generate rules exactly cover a specific class [WFOO]. Tree algorithms work in a top
down divide and conquer approach, but this need not be the case for covering algorithms.
They generate the best rule possible by optimizing the desired classification probability.
Usually the "best" attribute-value pair is chosen, as opposed to the best attribute with
the tree-based algorithms. Suppose that we wished to generate a rule to classify persons
as tall. The basic format for the rule is then
If ? then class = tall
The objective for the covering algorithms is to replace the "?" in this statement with
predicates that can be used to obtain the "best" probability of being tall.
One simple approach is called 1R because it generates a simple set of rules that are
equivalent to a DT with only one level. The basic idea is to choose the best attribute to
perform the classification based on the training data. "Best" is defined here by counting
the number of errors. In Table 4.4 this approach is illustrated using the height example,
TABLE 4.4: 1R Classification
Option Attribute Rules Errors Total Errors
Gender F � Medium 3/9 6/15
M � Tall 3/6
2 Height (0, 1 .6] � Short 0/2 1/15
(1 .6, 1 .7] � Short 0/2
(1 .7, 1.8] � Medium 0/3
(1 .8, 1.9] � Medium 0/4
(1 .9, 2.0] � Medium 1/2
(2.0, oo) � Tall 0/2
Section 4.6 f{ule-Based Algorithms 1 1 7
Outputl . If we only use the gender attribute, there are a total of 6/15 errors whereas
if we use the height attribute, there are only 1/15. Thus, the height would be chosen
and the six rules stated in the table would be used. As with ID3, 1R tends to choose
attributes w�th a large
.
n�mber o� values leading to overfitting. l R can handle missing
dat� b� addmg an additional attnbute value for the value of missing. Algorithm 4.10,
which 1s adapted from [WFOO], shows the outline for this algorithm.
ALGORITHM 4.10
Input :
D
R
c
Output :
R
/ /Training data
/ /Att ributes to consider for rules
/ / Classes
/ / Rules
lR algorithm:
/ / lR algori thm generates rules based on one attribute
R = 0 ;
for each A E R do
RA = 0 ;
for each possible value , v, of A do
/ / v may be a range rather than a speci f i c value
for each Cj E c f ind count (Cj) ;
I I Here count i s the number of occurrences o f this
class for thi s att ribute
let Cm be the class with the largest count ;
RA = RA U ((A = v) ---r (class = Cm)) ;
ERRA = number of tuples incorrectly clas s i f ied by RA i
R = RA where ERRA i s minimum ;
Another approach to generating rules without first having a DT is called PRISM.
PRISM generates rules for each class by looking at the training data and adding rules that
completely describe all tuples in that class. Its accuracy is 100 percent. Example 4.12
illustrates the use of PRISM. Algorithm 4.1 1, which is adapted from [WFOO], shows the
process. Note that the algorithm refers to attribute-value pairs. Note that the values will
include an operator so that in Example 4.12 the first attribute-value pair chosen is with
attribute height and value 72.0. As with earlier classification techniques, this must be
modified to handle continuous attributes. In the example, we have again used the ranges
of height values used in earlier examples.
EXAMPLE 4.1 2
Using the data in Table 4.1 and the Outputl classification, the following shows the basic
probability of putting a tuple in the tall class based on the given attribute-value pair:
Gender = F 0/9
Gender = M 3/6
Height <= 1.6 0/2
1 .6 < Height <= 1.7 0/2

1.7 < Height <= 1.8
1.8 < Height <= 1.9
1 .9 < Height <= 2.0
2.0 < Height
0/3
0/4
1/2
2/2
Based on this analysis, we would generate the rule
If 2.0 < height, then class = tall
Since all tuples that satisfy this predicate are tall, we do not add any additional predicates
to this rule. We now need to generate additional rules for the tall class. We thus look at the
remaining 13 tuples in the training set and recalculate the accuracy of the corresponding
predicates:
Gender = F 0/9
Gender = M 1/4
Height <= 1.6 0/2
1.6 < Height <= 1.7 0/2
1.7 < Height <= 1.8 0/3
1.8 < Height <= 1.9 0/4
1 .9 < Height <= 2.0 1/2
Based on the analysis, we see that the last height range is the most accurate and thus
generate the rule:
However, only one of the tuples that satisfies this is actually tall, so we need to add
another predicate to it. We then look only at the other predicates affecting
.
these two
tuples. We now see a problem in that both of these are males. The problem 1s actually
caused by our "arbitrary" range divisions. We now divide the range into two subranges:
1.9 < Height <= 1.95 0/1
1.95 < Height <= 2.0 1/1
We thus add this second predicate to the rule to obtain
If 2.0 < height and 1.95 < height <= 2.0, then class = tall
or
This problem does not exist if we look at tuples individually using the attrib�te:-value
pairs. However, in that case we would not generate the needed ranges f�r class1fymg the
actual data. At this point, we have classified all tall tuples. The algonthm would then
proceed by classifying the short and medium classes. This is left as an exercise.
Section 4.7 Combining Techniques 1 19
ALGORITHM 4.11
Input :
D
c
Output :
//Training data
/ /Clas ses
R / /Rul e s
PRISM algorithm :
R = 0 ;
//PRISM algori thm generates rules based on best
attribut e - value pairs
for each Cj E C do
repeat
T = D; I/Al l ins tances of class Cj wi ll be systematically
removed from T
·
p = true ; I /Create new rule with empty left - hand s i de
r = ( I f p then Cj) ;
repeat
for each attribute A value v pair found in T do
l l (l(tupleSET with A=v)ApA(ECj)l )
ca cu ate
l(tuples ET with A=v)Apl ;
find A = v that maximi zes this value ;
p = p 1 (A = v
) ;
T = {tuples in T that sat i s fy A = v} ;
until a l l tuples in T belong to Cj ;
D = D - T;
R = R U r;
until there are no tuples in D that belong to Cj ;
4.7 COMBINING TECHNIQUES
Given a classification problem, no one classification technique always yields the best
results. Therefore, there have been some proposals that look at combining techniques.
While discussing C5.0, we briefly introduced one technique for combining classifiers
called boosting. 1vo basic techniques can be used to accomplish this:
• A synthesis of approaches takes multiple techniques and blends them into a new
approach. An example of this would be using a prediction technique, such as linear
regression, to predict a future value for an attribute that is then used as input to a
classification NN. In this way the NN is used to predict a future classification value.
• Multiple independent approaches can be applied to a classification problem, each
yielding its own class prediction. The results of these individual techniques can then
be combined in some manner. This approach has been referred to as combination
o
fmultiple classifiers (CMC).
One approach to combine independent classifiers assumes that there are n inde
pendent classifiers and that each generates the posterior probability Pk(CJ 1 ti) for each
class. The values are combined with a weighted linear combination
n
L WkPk(Cj I ti)
k=l
(4.56)

0 0
� A � �
A Thple in Class 1
and correctly classified
0 • 0 0
0
X 0
� Tuple in Class 1
andincorrectly classified
A �
• •
A 0
A 0
o Thple in Class 2
and correctly classified
(a) Classifier 1 (b) Classifier2
• Thple in Class 2
and incorrectly classified
FIGURE 4.20: Combination of multiple classifiers.
Here the weights, Wk. can be assigned by a user or learned based on the past accuracy
of each classifier. Another technique is to choose the classifier that has the best accuracy
in a database sample. This is referred to as a dynamic classifier selection (DCS). �x
�
m
ple 4. 13, which is modified from [LJ98], illustrate�
the use
.
of
.
DCS. Anothe�
vanat10n
is simple voting: assign the tuple to the class to which a maJonty of the classifiers have
assigned it. This may have to be modified slightly in case there are many classes and no
majority is found.
EXAMPLE 4. 13
Two classifiers exist to classify tuples into two classes. A target tuple, X, needs to be
classified. Using a nearest neighbor approach, the 10 tuples closest to X are identified.
Figure 4.20 shows the 10 tuples closest to X. In Figure 4.20(a) the res�
lts for the first
classifier are shown, while in Figure 4.20(b) those for the second classifier are shown.
The tuples designated with triangles should be in c�as
�
1, whil�
those shown
.
as s
�
uares
should be in class 2. Any shapes that are darkened mdicate an mcorrect classificauon by
that classifier. To combine the classifiers using DCS, look at the general accuracy of each
classifier. With classifier 1 , 7 tuples in the neighborhood of X are correctly classified,
while with the second classifier, only 6 are correctly classified. Thus, X will be classified
according to how it is classified with the first classifier.
Recently, a new CMC technique, adaptive classifier combination (ACC), has been
proposed [LJ98]. Given a tuple to classify, the neighborhood around it is first determined,
then the tuples in that neighborhood are classified by each classifier, and finall
�
the
accuracy for each class is measured. By examining the accuracy across all classifiers
for each class, the tuple is placed in the class that has the highest local accuracy. In
effect, the class chosen is that to which most of its neighbors are accurately classified
independent of classifier. Example 4.14 illustrates the use of ACC.
EXAMPLE 4.14
Using the same data as in Example 4.13, the ACC technique examines how accurat�
all
classifiers are for each class. With the tuples in class 1, classifier 1 accurately classifies
3 tuples, while classifier 2 accurately classifies only 1 tuple. A measure of the accuracy
SE!Ction 4.9 Exercises 121
for both classifiers with respect to class 1 is then: 3/4 + 1/4. When looking at
·
class 2,
the measure is: 4/6 + 5/6. Thus, X is placed in class 2.
4.8 SUMMARY
No one classification technique is always superior to the others in terms of classification
accuracy. However, there are advantages and disadvantages to the use of each. The
regression approaches force the data to fit a predefined model. If a linear model is
chosen, then the data are fit into that model even though it might not be linear. It
requires that linear data be used. The KNN technique requires only that the data be
such that distances can be calculated. This can then be applied even to nonnumeric data.
Outliers are handled by looking only at the K nearest neighbors. Bayesian classification
assumes that the data attributes are independent with discrete values. Thus, although it is
easy to use and understand, results may not be satisfactory. Decision tree techniques are
easy to understand, but they may lead to overfitting. To avoid this, pruning techniques
may be needed. ID3 is applicable only to categorical data. Improvements on it, C4.5
and CS, allow the use of continuous data and improved techniques for splitting. CART
creates binary trees and thus may result in very deep trees.
When looking at the approaches based on complexity analysis, we see that they
are all very efficient. This is due to the fact that once the model is created, applying
it for classification is relatively straightforward. The statistical techniques, regression
and naive Bayes, require constant time to classify a tuple once the models are built.
The distance-based approaches, simple and KNN, are also constant but require that each
tuple be compared either to a representative for each class or to all items in the training
set. Assuming there are q of these, the KNN then requires O(q) time per tuple. DT
classification techniques, ID3, C4.5, and CART require a number of comparisons that
are (in the worst case) equal to the longest path from a root to a leaf node. Thus, they
require 0 (log q) time per tuple. Since q is a constant, we qm view these as being
performed in constant time as well. The NN approaches again require that a tuple be
propagated through the graph. Since the size of the graph is constant, this can be viewed
as being performed in constant time. Thus, all algorithms are 0(n) to classify the n items
in the database.
4.9 EXERCISES
1. Explain the differences between the definition of the classification problem found
in Definition 4. 1 and an alternative one with the mapping from C to D.
2. Using the data in Table 4.1, draw OC curves assuming that the Output2 column is
the correct classification and Outputl is what is seen. You will need to draw three
curves, one for each class.
3. Using the data in Table 4. 1 , construct a confusion matrix assuming Output is the
correct assignment and Output1 is what is actually made.
4. Apply the method of least squares technique to determine the division between
medium and tall persons using the training data in Table 4. 1 and the classification
shown in the Output1 column (see Example 4.3). Y
ou may use either the division
technique or the prediction technique.

5. Apply the method of least squares technique to determine the divisio
1
n ?
fi
etw�en
medium and tall persons using the training data in Table 4.1 and the c assi catwn
shown in the Output2 column. This uses both the height data and the gender data
to do the classification. Use the division technique.
6. Redo Exercise 5 using the prediction technique.
7. Use KNN to classify (Jim, M, 2.0) with K = 5 using the height data and assuming
that Output2 is correct.
8. Explain the difference between P Cti I Cj) and P (Cj I t;) ·
9. Redo Example 4.5 using Output2 data.
10. Determine the expected number of comparisons for each tree shown in Figure 4. 1 1 .
11. Generate a DT for the height example in Table 4.1 using the ID 3 algorithm and
the training classifications shown in the Output2 column of that table.
12. Repeat Exercise 1 1 using the GainRatio instead of the Gain.
13. Construct the confusion matrix for the results of Exercises 1 1 and 12.
14. Using 1R, gerf
erate rules for the height example using the Output2 column in
Table 4.1 .
15. Complete Example 4.9 by generating the D� for the height example (using the
Outputl classification) using the CART algonthm.
16. Suppose that the output for Mary in Exercise 8 in Chapter 3 should �ave been O
d
��r
small, 0 for medium, and 1 for tall. Use the gradient descent algonthm to mo .I Y
the weights in the NN.
17. Complete Example 4.12 by generating rules for the short and medium
.
classes.
.
18. (Implementation) Various classification algorithms can be faun� onhne. Ob
1
ta�n
code for both CART and C4.5. Apply these programs to the height examp e m
Table 4.1 using the training classifications shown in the Output2 column. Compare
these results to those found in Exercises 1 1 and 12.
19. Generate rules from each of the trees found in Exercise 18.
20. (Implementation/research) Various datasets that
.
have b�en u�ed for classifi
d
cation
benchmarks can be found online. Obtain a real-hfe classificatiOn dataset an gen
erate decision trees using the programs you found in Exercise 18. Compare these
two trees. Which is better? Why?
21. (Research) Compare at least three different guideline that have been proposed for
determining the optimal number of hidden nodes in an NN.
4.1 0 BIBLIOGRAPHIC NOTES
Classification is perhaps the oldest data mining technique. Plant and anim�l cl�ssifica
tion dates back to the 1700s or earlier. A historical investigation of class1ficatwn and
clustering can be found in one of the first clustering texts [Har75]. The�e are many
classification texts, including [BFOS98]. A recent study has compared 33 different clas-
sification algorithms.2
2Tjen-Sien Lim, Wei-Yin Loh, and Yu-Shan-Shih, "A Comparison of Prediction Accuracy, Complexity,
and Training Time of Thirty-three Old and New Classification Algorithms," Machine Learning, vol. 40, 2000,
pp. 203-229.
Section 4. i 0 Bibliographic Notes 123
There are detailed discussions of linear regression for classification in [HTF01].
It was reported that the accuracy for multiclass problems was very poor. A better lin
ear approach is to use linear discriminant analysis (LDA), which is also discussed in
that book.
Chi squared automatic interaction detection (CHAID) was one of the earliest deci
sion tree algorithms, proposed by Hartigan in 1975 [Har75]. CHAID uses the chi squared
statistic to determine splitting. It can use only categorical values. The chi squared test
is used to prevent the tree from growing too big. A modified version of CHAID was
proposed in 1980 by Kass; it reduces the overall time, but does not guarantee the best
result. Kass proposes that pairs of categories (for the predictor variable) be merged
if there is no statistically significant difference between them. The resulting set of
categories defines the split [Kas80]. With exhaustive CHA1D, which was proposed in
1991, the split point is determined by merging pairs of categories until only one pair
remains [BdVS91]. The predictor with the best overall prediction is then chosen for
the split.
ID3 was first proposed in the mid 1970s by Quinlan [Qui86] . A thorough investi
gation of C4.5 can be found in the seminal text on the subject [Qui93].
CART was developed by Breimen in 1984 [BFOS84]. In 1997, another binary
decision tree technique was proposed [LS97]. QUEST (quick unbiased efficient statistical
tree) addresses the problems in CART that it tends to select variables with many values,
which creates a bias iri the model. QUEST handles variable selection and split point
differently. Also, unlike CART, QUEST does not perform an exhaustive search, so it is
more efficient. The approach used by QUEST is to determine the association between
each predictor variable and target variable. The variable with the largest association is
chosen. The split point (for that variable) is then determined.
Many different techniques have been proposed to pnme decision trees. A survey
by Breslow and Aha in 19973 looked at techniques to simplify trees. These included
techniques to control the tree site (pruning), m<;>dify the test space, change the way the
searching of the test space was conducted, reduce the size of the input data set (e.g.,
feature selection), and use alternative data structures. Pruning approaches include pre
pruning algorithms that affect the size of the tree as it is created.4 Post-pruning algorithms
change the tree after it is created.5
Pattern recognition is a type of classification used in many diverse applications,
including medical diagnosis, assembly line parts inspection, speech recognition, printed
character recognition, military target recognition, robot navigation, and fingerprint analy
sis. While these applications can use the general strategies outlined in this chapter, there
has been much work in the development of algorithms specifically targeted to individual
applications. The interested reader is referred to some of the many texts available on
pattern recognition, including [Bis95], [DHSOO], [Fuk90], [GJJ96], and [TK98].
3L. Breslow and D.W. Aha, "Comparing Tree-Simplification Procedures," Proceedings o
f the Sixth Inter
national W
orkshop on Artificial Intelligence and Statistics, 1997, pp. 67-74.
4
Floriana Esposito, Donato Malerba, and Giovanni Semeraro, "A Comparative Analysis of Methods for
Pruning Decision Trees, "IEEE T
ransactions on Pattern Analysis and Machine Intelligence, vol. 19, no. 5,
May 1997, pp. 476-491.
5
Tapio Elomaa and Matti Kaariainen, "An Analysis of Reduced Error Pruning," Journal o
f Intelligence
Research, vol. 15, 2001, pp. 163-187.

Classification of large datasets has been proposed to be handled in several ways.
Sampling has been proposed in [Cat91]. Partitioning the data and creating classifiers
for each was proposed in [CS93]. Parallelization of classification algorithms has been
examined [Fif92]. The SLIQ approa�h addresses scalability by a presorting step instead
of sorting at each node, which some decision tree techniques require [MAR96]. The
second improvement with SLIQ is that the tree is grown in a breadth-first fashion as
opposed to a depth-first manner. In [SAM96], the authors not only propose SPRINT,
but also design and compare several parallel implementations of SPRINT and SLIQ.
Rainforest was proposed in 1 998 [GRG98].
Although an RBF network could conceivably be of any shape, a seminal paper
by Broomhead and Lowe proposed the structure discussed in this chapter [BL88]. An
excellent and complete introduction to RBF networks is found in [Orr96]. The idea of the
perceptron was proposed by Rosenblatt in [Ros58]. The MLP is due to Rumelhart and
McClelland[RM86]. Discussions of choice for number of hidden layers in an MLP can
be found in [Bis95] and [Hay99]. Kolmogorov's famous theorem is reported in [Kol57].
The. lR algorithrri was studied in [Hol93] and [WFOO]. PRISM was proposed in
[Cen87]. Extracti�g rules from neural networks has been investigated since the early
1990s [Liu95], [TS93], [LSL95], and [LSL96].
5.1
C H A P T E R 5
Cl ustering
5.1 INTRODUCTION
5.2 SIMILARITY AND DISTANCE MEASURES
5.3 OUTLIERS
5.4 HIERARCHICAL ALGORITHMS
5.5 PARTITIONAL ALGORITHMS
5.6 CLUSTERING LARGE DATABASES
5.7 CLUSTERING WITH CATEGORICAL ATTRIBUTES
5.8 COMPARISON
5.9 EXERCISES
INTRODUCTION
Clustering is similar to classification in that data are grouped. However, unlike classifi
cation, the groups are not predefined. Instead, the grouping is accomplished by finding
similarities between data according to characteristics found in the actual data. The groups
are called clusters. Some authors view clustering as a special type of classification. In
this text, however, we follow a more conventional view in that the two are different.
Many definitions for clusters have been proposed:
• Set of like elements. Elements from different clusters are not alike.
• The distance between points in a cluster is less than the distance between a point
in the cluster and any point outside it.
A term similar to clustering is database segmentation, where like tuples (records) in a
database are grouped together. This is done to partition or segment the database into
components that then give the user a more general view of the data. In this text, we do
not differentiate between segmentation and clustering. A simple example of clustering is
found in Example 5 . 1 . This example illustrates the fact that determining how to do the
clustering is not straightforward.
EXAMPLE 5.1
An international online catalog company wishes to group its customers based on common
features. Company management does not have any predefined labels for these groups.
Based on the outcome of the grouping, they will target marketing and advertising cam
paigns to the different groups. The information they have about the customers includes
125

1 26 Chapter 5 Clustering
TAB LE 5.1 : Sample Data for Example 5.1
Income Age Children Marital Status Education
$25,000 35 3 Single High school
$15,000 25 Married High school
$20,000 40 0 Single High school
$30,000 20 0 Divorced High school
$20,000 25 3 Divorced College
$70,000 60 0 Married College
$90,000 30 0 Married Graduate school
$200,000 45 5 Married Graduate school
$100,000 so 2 Divorced College
• • • •
•
•
•
•
• •
0
•
•
. . • •
• • • •
�
•
•
•
•
• •
•
•
•
• •
• •
• •
•
•
• •
(a) Group of homes (b) Geographic distance-based (c) Size-based
FIGURE 5.1 : Different clustering attributes.
income, age, number of children, marital status, and education. Ta�le 5 shows some
tuples from this database for customers in the United States. Dependmg on �h� ty?e of
advertising not all attributes are important. For example, suppose the advertlsmg 1s for
a special s�le on children's clothes. We could target the ad:�r�ising only to the persons
with children. One possible clustering is that shown by the dmswn� of the table. The fir�t
group of people have young children and a high school d�gree, whlle the second group 1s
similar but have no children. The third group has both chlldren and a college degree. The
last two groups have higher incomes and at least a college degr�e
.
' The very last
.
group has
children. Different clusterings would have been found by exanumng age or mantal status.
As illustrated in Figure 5.1, a given set of data may be clustered on different
attributes. Here a group of homes in a geographic area is shown. The
_
first type of
clustering is based on the location of the home. Homes that are geographically close to
each other are clustered together. In the second clustering, homes are grouped based on
the size of the house.
Clustering has been used in many application domains, including biology, medi�ine,
anthropology, marketing, and economics. Clustering applications include plant and ammal
Section 5.1 Introduction 1 27
classification, disease classification, image processing, pattern recognition, and document
retrieval. One of the first domains in which clustering was used was biological taxonomy.
Recent uses include examining Web log data to detect usage patterns.
When clustering is appliedto a real-world database, manyinteresting problems occur:
• Outlier handling is difficult. Here the elements do not naturally fall into any cluster.
They can be viewed as solitary clusters. However, if a clustering algorithm attempts
to find larger clusters, these outliers will be forced to be placed in some cluster.
This process may result in the creation of poor clusters by combining two existing
clusters and leaving the outlier in its own cluster.
• Dynamic datain thedatabase implies that cluster membershipmay change over time.
• Interpreting the semantic meaning of each cluster may be difficult. With classifica
tion, the labeling of the classes is known ahead of time. However, with clustering,
this may not be the case. Thus, when the clustering process finishes creating a set
of clusters, the exact meaning of each cluster may not be obvious. Here is where
a domain expert is needed to assign a label or interpretation for each cluster.
• There is no one correct answer to a clustering problem. In fact, many answers may
be found. The exact number of clusters required is not easy to determine. Again, a
domain expert may be required. For example, suppose we have a set of data about
plants that have been collected during a field trip. Without any prior knowledge of
plant classification, if we attempt to divide this set of data into similar groupings,
it would not be clear how many groups should be created.
• Another related issue is what data should be used for clustering. Unlike learning
during a classification process, where there is some a priori knowledge concern
ing what the attributes of each classification should be, in clustering we have no
supervised learning to aid the process. Indeed, clustering can be viewed as similar
to unsupervised learning.
We can ilien summarize some basic features of clustering (as opposed to classification):
• The (best) number of clusters is not known.
• There may not be any a priori knowledge concerning the clusters.
• Cluster results are dynamic.
The clustering problem is stated as shown in Definition 5.1. Here we assume
that the number of clusters to be created is an input value, k. The actual content (and
interpretation) of each cluster, KJ , 1 ::: j ::: k, is determined as a result of the function
definition. Without loss of generality, we will view that the result of solving a clustering
problem is that a set of clusters is created: K = {K1 , K2 , . . . , Kk}.
DEFINITION 5.1. Given a database D = {tJ , tz, . . . , tn} of tuples and an integer
value k, the clustering problem is to define a mapping f : D -+ {1, . . . , k} where
each t; is assigned to one cluster KJ , 1 ::: j ::: k. A cluster, KJ. contains precisely
those tuples mapped to it; that is, KJ = {t; I f(t;) = KJ , 1 ::: i ::: n, and t; E D}.
A classification ofthe different types of clustering algorithms is shown in Figure 5.2.
Clustering algorithms themselves may be viewed as hierarchical or partitional. With

128 Chapter 5 Clustering
FIGU RE 5.2: Classification of clustering algorithms.
hierarchical clustering, a nested set of clusters is created. Each level in the hierarchy has
a separate set of clusters. At the lowest level, each item is in its own unique cluster. At the
highest level, all items belong to the same cluster. With hierarchical clustering, the desired
number of clusters i$ not input. With partitional clustering, the algorithm creates only
one set of clusters. These approaches use the desired number of clusters to drive how the
final set is created. Traditional clustering algorithms tend to be targeted to small numeric
databases that fit into memory. There are, however, more recent clustering algorithms that
look at categorical data and are targeted to larger, perhaps dynamic, databases. Algorithms
targeted to larger databases may adapt to memory constraints by either �ampling the
database or using data structures, which can be compressed or pruned to fit mto memory
regardless of the size of the database. Clustering algorithms may also differ base� on
whether they produce overlapping or nonoverlapping clusters. Even though we consider
only nonoverlapping clusters, it is possible to place an item in multiple clusters. In turn,
nonoverlapping clusters can be viewed as extrinsic or intrinsic. Extrinsic techniques use
labeling of the items to assist in the classification process. These algorithms are the
traditional classification supervised learning algorithms in which a special input training
set is used. Intrinsic algorithms do not use any a priori category labels, but depend
only on the adjacency matrix containing the distance between objects. All algorithms we
examine in this chapter fall into the intrinsic class.
The types of clustering algorithms can be furthered classified based on the_im�le
mentation technique used. Hierarchical algorithms can be categorized as agglomerative
or divisive. "Agglomerative" implies that the clusters are created in a bottom-up fashion,
while divisive algorithms work in a top-down fashion. Although both hierarchical and
partitional algorithms could be described using the agglomerative vs. d�vi�ive lab�l, �t
typically is more associated with hierarchical algorithms. Another descnptlve tag mdi
cates whether each individual element is handled one by one, serial (sometimes called
incrementa[), or whether all items are examined together, simultaneous. If a specific tuple
is viewed as having attribute values for all attributes in the schema, then clustering al�o
rithms could differ as to how the attribute values are examined. As is usually done with
decision tree classification techniques, some algorithms examine attribute values one at a
time, monothetic. Polythetic algorithms consider all attribute values at one time. Finally,
clustering algorithms can be labeled based on the mathematical formulation given to
the algorithm: graph theoretic or matrix algebra. In this chapter we generally use t�e
graph approach and describe the input to the clustering algorithm as an adjacency matnx
labeled with distance measures.
5.2
Section 5.2 Similarity and Distance Measures 1 29
We discuss many clustering algorithms in the following sections. This is only a
representative subset of the many algorithms that have been proposed in the literature.
Before looking at these algorithms, we first examine possible similarity measures and
examine the impact of outliers.
SIMILARITY AND DISTANCE MEASURES
There are many desirable properties for the clusters created by a solution to a specific
clustering problem. The most important one is that a tuple within one cluster is more
like tuples within that cluster than it is similar to tuples outside it. As with classification,
then, we assume the definition of a similarity measure, sim(t;, t1), defined between any
two tuples, t; , tt E D. This provides a more strict and alternative clustering definition, as
found in Definition 5.2. Unless otherwise stated, we use the first definition rather than the
second. Keep in mind that the similarity relationship stated within the second definition
is a desirable, although not always obtainable, property.
DEFINITIO
� 5.2. Given a database D = {tr , ,t2• . . . , tn} of tuples, a simjlarity
measure, sim(t; , t1), defined between any two tuples, t; , t1 E D, and an integer
value k, the clustering problem is to define a mapping f : D � {1, . . . , k} where
each t; is assigned to one cluster K1·, I < j < k Given a cluster K · Vt ·z t · E K .
. - - . ' J • J ' ;m J
and t; � Kj, sirn(tjl . fjm) > sim(tjl. t;).
A dis
�ance measure, dis(t; , tj), as opposed to similarity, is often used in clustering.
The clustenng problem then has the desirable property that given a cluster, Kj. Vtj1,
fjm E Kj and t; � Kj, dis(tjl , fjm) .::: dis(tjl, t;).
Some clustering algorithms look only at numeric data, usually assuming metric
data points. Metric attributes satisfy the triangular inequality. The clusters can then
be described by using several characteristic yalues. Given a cluster, Km of N points
{tml , tm2• . . . , tmN}, we make the following definitions [ZRL96]:
centroid
radius
N
_L)tm;)
c - i=l
m - -
-
N
-
-
N
LCtmi - Cm)
2
i=l
N
N N
LL(tmi - lmj)2
i=l j=l
diameter = Dm =
(N)(N - 1)
(5. 1)
(5.2)
(5.3)
Here the centroid is the "middle" of the cluster; it need not be an actual point in the
cluster. Some clustering algorithms alternatively assume that the cluster is represented by
one centrally located object in the cluster called a medoid. The radius is the square root
of the average mean squared distance from any point in the cluster to the centroid, and
the d�ame
.
ter is the square root of the average mean squared distance between all pairs
of pomts m the cluster. We use the notation Mm to indicate the medoid for cluster Km .

Many clustering algorithms require that the distance between clusters (rather than
elements) be determined. This is not an easy task given that there are many interpretations
for distance between clusters. Given clusters Ki and KJ, there are several standard
alternatives to calculate the distance between clusters. A representative list is:
• Single link: Smallest distance between an element in one cluster and an eiement
in the other. We thus have dis(Ki , KJ) = min(dis(tu , tJm))Vtu E Ki fj KJ and
Vtjm E KJ rj Ki .
• Complete link: Largest distance between an element in one cluster and an element
in the other. We thus have dis(Ki , KJ) = max(dis(tu, tjm))Vtu E Ki rj KJ and
VtJm E KJ rj Ki .
• Average: Average distance between an element in one cluster and an element in
the other. We thus have dis(Ki , Kj) = mean(dis(tu , tJm))Vtu E Ki fj KJ and
Vtjm E KJ fj Ki .
• Centroid: If �lusters have a representative centroid, then the centroid distance
is defined as the distance between the centroids. We thus have dis(Ki , K1) =
dis(Ci , CJ), where Ci is the centroid for Ki and similarly for Cj .
• Medoid: Using a medoid to represent each cluster, the distance between the clusters
can be defined by the distance between the medoids: dis(Ki , Kj) = dis(Mi , MJ).
5.3 OUTLIERS
As mentioned earlier, outliers are sample points with values much different from those
of the remaining set of data. Outliers may represent errors in the data (perhaps a mal
functioning sensor re.corded an incorrect data value) or could be correct data values that
are simply much different from the remaining data. A person who is 2.5 meters tall is
much taller than most people. In analyzing the height of individuals; this value probably
would be viewed as an outlier.
Some clustering techniques do ri�t perform well with the presence of outliers. This
problem is illustrated in Figure 5.3. Here if three clusters are found (solid line), the outlier
will occur in a cluster by itself. However, if two clusters are found (dashed line), the two
(obviously) different sets of data will be placed in one cluster because they are closer
together than the outlier. This problem is complicated by the fact that many clustering
algorithms actually have as input the number of desired clusters to be found.
Clustering algorithms may actually find and remove outliers to ensure that they
perform better. However, care must be taken in actually removing outliers. For example,
suppose that the data mining problem is to predict flooding. Extremely high water level
values occur very infrequently, and when compared with the normal water level values
may seem to be outliers. However, removing these values may not allow the data mining
algorithms to work effectively because there would be no data that showed that floods
ever actually occurred.
Outlier detection, or outlier mining, is the process of identifying outliers in a set
of data. Clustering, or other data mining, algorithms may then choose to remove or
treat these values differently. Some outlier detection techniques ate based on statistical
techniques. These usually assume that the set of data follows a known distribution and that
outliers can be detected by well-known tests such as discordancy tests. However, these
Section 5.4 H ierarchical Algorithms 131
0 Three clusters
�) 1vo clusters
FIGU RE 5.3: Outlier clustering problem.
B c D E F
FIGURE 5.4: Dendrogram for Example 5.2.
tests are not very realistic for real-world data because real..world data values may not
follow well-defined data distributions. Also, most of these tests assume a single attribute
value, and many attributes are- involved in real-world datasets. Alternative detection
techniques may be based on distance measures.
5.4 HIERARCHICAL ALGORITHMS
As mentioned earlier, hierarchical clustering algorithms actuaily cre�tes sets of cluste�s.
Example 5.2 illustrates the concept. Hierarchical algorithms differ in how the sets are
created. A tree data structure, called a dendrogram, can be used to illustrate the hierar
chical clu�tering technique and the sets of different clusters. The root in a dendrogram
tree conta�ns one �luster where all elements are together. The leaves in the dendrogram
each consist of a smgle element cluster. Internal nodes in the dendrogram represent new
clusters formed by merging the clusters that appear as its children in the tree. Each level
in the tree is associated with the distance measure that was used to merge the clusters.
All clusters created at a particular level were combined because the children clusters had
a distance between them less than the distance value associated with this level in the
tree. A dendrogram for Example 5.2 is seen in Figure 5.4.
EXAMPLE 5.2
Figure 5.5 shows six elements, {A, :S, C, D, E, F}, to be clustered. Parts (a) to (e) of the
figure show five different sets of clusters. In part (a) each cluster is viewed to consist of

1 32
10
9
8
7
6
5
4
3
2
1
0
Chapter 5 Clustering
10
9
8
X 7
Ex 6
�D 5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6
(a) Six clusters (b) Four clusters
1 2 3 4 5 6 7 8
(d) Two clusters
10
9
8
7
6
5
4
3
2
1
0
7 8 0 1 2 3 4 5 6 7
(c) Three clusters
(e} One cluster
FIGURE 5.5: Five levels of clustering for Example 5.2.
8
a single element. Part (b) illustrates four clusters. Here there are two sets of two-element
dusters. These clusters are formed at this level because these two elements are closer
to each other than any of the other elements. part (c) shows a new cluster formed by
adding a close element to one of the rwo-f!lement clusters. In part (d) the two-element
and three-element clusters are merged to give a five-element cluster. This is done because
these two clusters are closer to each other than to the remote element cluster, {F}. At the
last stage, part (e), all six elements are merged.
·
The space complexity for hierarchical algorithms is O(n2) because this is the space
required for the adjacency matrix. The space required for the dendrogram is O (kn),
which is much less than O (n2). The time complexity for hierarchical algorithms is
0 (kn2
) because there is one iteration for each level in the dendrogram. Depending on
the specific algorithm, however, this could actually be O(maxd n2) where maxd is the
maximum distance between points. Different algorithms may actually merge the closest
clusters from the next lowest level or simply create new clusters at each level with
progressively larger distances.
Hierarchical techniques are well suited for many clustering applications that natu
rally exhibit a nesting relationship between clusters. For example, in biology, plant and
animal taxonomies could easily be viewed as a hierarchy of clusters.
5.4.1 Agglomerative Algorithms
Agglomerative algorithms start with each individual item in its own cluster and iteratively
merge clusters until all items belong in one cluster. Different agglomerative algorithms
differ in how the clusters are merged at each level. Algorithm 5.1 illustrates the typi
cal agglomerative clustering algorithm. It assumes that a set of elements and distances
between them is given as input. We use an n x n vertex adjacency matrix, A, as input.
Section 5.4 Hierarchical Algorithms 1 33
Here the adjacency matrix, A, contains a distance value rather than a simple boolean
value: A[i, j] = dis(t; , tj) . The output of the algorithm is a dendrogram, DE, which we
represent as a set of ordered triples (d, k, K) where d is the threshold distance, k is the
number of clusters, and K is the set of clusters. The dendrogram in Figure 5.7(a) would
be represented by the following:
{(0, 5, {{A}, {B}, {C}, {D}, {£}}), (1 , 3, {{A , B}, {C, D}, {£}})
(2, 2, {{A , B, C, D}, {£}}), (3, 1, {{A, B, C, D, £}})}
Outputting the dendrogram produces a set of clusters rather than just one clustering. The
user can determine which of the clusters (based on distance threshold) he or she wishes
to use.
ALGORITHM 5.1
Input :
D = { t 1 , t2 , . . . , tn} / /Set of element s
A / /Adj acency matrix showing dis tance between elements
Output :
DE // Dendrogram represented as a set of ordered triples
Agglomerative algorithm :
d = 0 ;
k = n ;
K = {{ tl}, . . . , { tn}} ;
DE = { (d, k, K) } ; / / Init ially dendrogram contains each element
in its own cluster .
repeat
oldk = k ;
d = d + l ;
Ad = Vertex adj acency matrix for graph with thre shold
di stance of d;
(k, K) = NewCl usters(Ad, D) ;
if oldk =I= k then
DE = DE U (d, k, K) ; I/ New set of clusters added to dendrogram .
unti l k = 1
This algorithm uses a procedure called NewClusters to determine how to create
the next level of clusters from the previous level. This is where the different types
of agglomerative algorithms differ. It is possible that only two clusters from the prior
level are merged or that multiple clusters are merged. Algorithms also differ in terms
of which clusters are merged when there are several clusters with identical distances.
In addition, the technique used to determine the distance between clusters may vary.
Single link, complete link, and average link techniques are perhaps the most well known
agglomerative techniques based on well-known graph theory concepts.
All agglomerative approaches experience excessive time and space constraints.
The space required for the adjacency matrix is O (n2) where there are n items to cluster.
Because of the iterative nature of the algorithm, the matrix (or a subset of it) must be
accessed multiple times. The simplistic algorithm provided in Algorithm 5.1 performs
at most maxd examinations of this matrix, where maxd is the largest distance between
any two points. In addition, the complexity of the NewClusters procedure could be
expensive. This is a potentially severe problem in large databases. Another issue with

the agglomerative approach is that it is not incremental. Thus, when new elements are
added or old ones are removed or changed, the entire algorithm must be rerun. More
recent incremental variations, as discussed later in this text, address this problem.
.
Single Link Technique. !he single link technique is based on the idea of finding
maxunal connected components m a graph. A connected component is a graph in which
there exists a path between any two vertices. With the single link approach, two clusters
are merged if there is at least one edge that connects the two clusters; that is, if the
minimum distance between any two points is less than or equal to the threshold dis
tance
_
being considered. For this reason, it is often called the nearest neighbor clustering
techmque. Example 5.3 illustrates this process.
EXAMPLE 5.3
Table 5.2 contains five sample data items with the distance between the elements indicated
in the t
�
ble entries. When viewed as a graph problem, Figure 5.6(a) shows the general
graph With all eqges labeled with the respective distances. To understand the idea behind
the hierarchical approach, we show several graph variations in Figures 5.6(b), (c), (d),
and (e). Figure 5.6(b) shows only those edges with a distance of 1 or less. There are
only two edges. The first level of single link clustering then will combine the connected
clusters (single elements from the first phase), giving three clusters: {A,B}, {C,D}, and
{E}. During the next level of clustering, we look at edges with a length of 2 or less. The
graph representing this threshold distance is shown in Figure 5.6(c). Note that we now
have an edge (actually three) between the two clusters {A,B} and {C,D}. Thus, at this
level of the single link clustering algorithm, we merge these two clusters to obtain a total
of two clusters: {A,B,C,D} and {E} . The graph that is created with a threshold distance
of 3 is shown in Figure 5.6(d). Here the graph is connected, so the two clusters from the
last le
_
vel
_
are m�rged into one large cluster that contains all elements. The dendrogram
f
?
r this smgle link example is shown in Figure 5.7(a). The labeling on the right-hand
side shows the threshold distance used to merge the clusters at each level.
The single link algorithm is obtained by replacing the NewClusters procedure in
the agglomerative a_Jgorithm with a procedure to find connected components of a graph.
We assume that this connected components procedure has as input a graph (actually
represented by a vertex adjacency matrix and set of vertices) and as outputs a set of
TABLE 5.2: Sample Data for Example 5.3
Item A B c D E
A 0 2 2 3
B 0 2 4 3
c 2 2 0 1 5
D 2 4 0 3
E 3 3 5 3 0
Section 5.4 Hierarchica l Algorithms 135
�
E�c E·
A 1 B
�c E· �c
D D D
(a) Graph with all distances (b) Graph with threshold of 1 (c) Graph with threshold of 2
A
A 1 B
E �
�c
D
(d) Graph with threshold of 3
A 1 B
E �
<;�c
D
(e) Graph with threshold of 4
FIGURE 5.6: Graphs for Example 5.3.
B C D E
(a) Single link
3
2
0
A B E C D
(b) Complete link
4
3
2
0
A B c D
(c) Average link
FIG URE 5.7: Dendrograms for Example 5.3.
5
4.5
4
3.5
3
2.5
2
1.5
10.5
0
E
connected components defined by a number (indicating the number of components) and
an array containing the membership of each component. Note that this is exactly what
the last two entries in the ordered triple are used for by the dendrogram data structure.
The single link approach is quite simple, but it suffers from several problems. This
algorithm is not very efficient because the connected components procedure, which is an
O(n2) space and time algorithm, is called at each iteration. A more efficient algorithm
could be developed by looking at which clusters from an earlier level can be merged at
each step. Another problem is that the clustering creates clusters with long chains.
An alternative view to merging clusters in the single link approach is that two
clusters are merged at a stage where the threshold distance is d if the minimum distance
between any vertex in one cluster and any vertex in the other cluster is at most d.
There have been other variations of the single link algorithm. One variation, based
on the use of a minimum spanning tree (MST), is shown in Algorithm 5.2. Here we
assume that a procedure, MST, produces a minimum spanning tree given an adjacency
matrix as input. The clusters are merged in increasing order of the distance found in the
MST. In the algorithm we show that once two clusters are merged, the distance between

them in the tree becomes oo. Alternatively, we could have replaced the two nodes and
edge with one node.
ALGORITHM 5.2
Input :
D = { t1 , t2, . . . , tn) I/Set of elements
A / /Adj acency matrix showing distance between elements
Output :
DE II Dendrogram repres ented as a set of ordered triples
MST single link algorithm:
d = O
k = n
K = {{t1}, . . . , { tn))
DE = (d, k, K) ; I I Ini ti ally dendrogram contains each element in
its own cluster .
M = MST(A) ;
repeat
oldk = k;f
Ki , Kj = two clusters closest t ogether in MST ;
K = K - {Ki) - {Kj ) U {Ki U Kj ) i
k = oldk - 1 ;
d = dis(Ki , Kj ) i
DE = DEU (d, k, K) ; / / New set of c lusters added to dendrogram .
dis (Ki , Kj) = oo
until k = 1
We illustrate this algorithm using the data in Example 5.3. Figure 5.8 shows one
MST for the example. The algorithm will merge A and B and then C and D (or the
reverse). These two clusters will then be merged at a threshold of 2. Finally, E will
be merged at a threshold of 3. Note that we get exactly the same dendrogram as in
Figure 5.7(a).
The time complexity of this algorithm is 0 (n2) because the procedure to create
the minimum spanning tree is O (n2) and it dominates the time of the algorithm. Once
it is created having n - 1 edges, the repeat loop will be repeated only n - 1 times.
The single linkage approach is infamous for its chain effect; that is, two clusters
are merged if only two of their points are close to each other. There may be points in
the respective clusters to be merged that are far apart, but this has no impact on the
algorithm. Thus, resulting clusters may have points that are not related to each other at
all, but simply happen to be near (perhaps via a transitive relationship) points that are
close to each other.
E
3
A 1 B
D
2
c
FIG URE 5.8: MST for Example 5.3.
Section 5.4 Hi1�rarch ical Algorithms 1 37
Complete Link Algorithm. Although the complete link algorithm is similar to
the single link algorithm, it looks for cliques rather than connected components. A clique
is a maximal graph in which there is an edge between any two vertices. Here a procedure
is used to find the maximum distance between any clusters so that two clusters are merged
if the maximum distance is less than or equal to the distance threshold. In this algorithm,
we assume the existence of a procedure, clique, which finds all cliques in a graph. As
with the single link algorithm, this is expensive because it i:; an 0 (n2) algorithm.
Clusters found with the complete link method tend to be more compact than those
found using the single link technique. Using the data found in Example 5.3, Figure 5.7(b)
shows the dendrogram created. A variation of the complete link algorithm is called
the f
arthest neighbor algorithm. Here the closest clusters are merged where the dis
tance is the smallest measured by looking at the maximum distance between any two
points.
Average Link. The average link technique merges two clusters if the average
distance between any two points in the two target clusters is below the distance threshold.
The algorithm used here is slightly different from that found in single and complete link
algorithms because we must examine the complete graph (not just the threshold graph)
at each stage. Thus, we restate this algorithm in Algorithm :5.3.
ALGORITH� 5.3
Input :
D = {tl , t2 · · · · · tn} // Set of el ements
A / /Adj acency matrix showing distance between elements
Outpu t :
DE / / Dendrogram represented as a set of ordered triples
Average link algorithm:
d = 0 ;
k = n ;
K = {{ tl}, . . . , { tn}} ;
DE = (d, k, K) ; II Init ial ly dendrogram contains each e lement
in its own cluster .
repeat
oldk = k;
d = d + 0 . 5 ;
for each pair of Ki , Kj E K do
ave = average distance between all ti E Ki and tj E Kj ;
if ave :S d , then
K = K - {Ki ) - {Kj } U {Ki U Kj } ;
k = oldk - 1 ;
DE = DE U (d, k, I() ; I I New set of clusters added
to dendrogram .
unti l k = 1
Note that in this algorithm we increment d by 0.5 rather than by 1. This is a rather
arbitrary decision based on understanding of the data. Cettainly, we could have used
an increment of 1 , but we would have had a dendrogram different from that seen in
Figure 5.7(c).

5.4.2
5.5
Divisive Clustering
With divisive clustering, all items are initially placed in one cluster and clusters are
repeatedly split in two until all items are in their own cluster. The idea is to split up
clusters where some elements are not sufficiently close to other elements.
One simple example of a divisive algorithm is based on the MST version of the
single link algorithm. Here, however, we cut out edges from the MST from the largest
to the smallest. Looking at Figure 5.8, we would start with a cluster containing all
items: {A, B, C, D, E}. Looking at the MST, we see that the largest edge is between
D and E. Cutting this out of the MST, we then split the one cluster into two: {E} and
{A, B, C, D}. Next we remove the edge between B and C. This splits the one large
cluster into two: {A , B} and {C, D}. These will then be split at the next step. The order
depends on how a specific implementation would treat identical values. Looking at the
dendrogram in Figure 5.7(a), we see that we have created the same set of clusters as
with the agglomerative approach, but in reverse order.
PARTITIONAL ALGORITHMS
Nonhierarchical or partitional clustering creates the clusters in one step as opposed to
several steps. Only one set of clusters is created, although several different sets of clusters
may be created internally within the various algorithms. Since only one set of clusters is
output, the user must input the desired number, k, of clusters. In addition, some metric
or criterion function is used to determine the goodness of any proposed solution. This
measure of quality could be the average distance between clusters or some other metric.
The solution with the best value for the criterion function is the clustering solution used.
One common measure is a squared error metric, which measures the squared distance
from each point to the centroid for the associated cluster:
k
L L dis(Cm , tmi)2 (5.4)
A problem with partitional algorithms is that they suffer from a combinatorial
explosion due to the number of possible solutions. Clearly, searching all possible cluster
ing alternatives usually would not be feasible. For example, given a measurement criteria,
a naive approach could look at all possible sets of k clusters. There are S(n, k) possible
combinations to examine. Here
k
S(n, k) = �L(-l)k�i ( 7 )(i)n
I = l
(5.5)
There are 1 1,259,666,000 different ways to cluster 19 items into 4 clusters. Thus, most
algorithms look only at a small subset of all the clusters psing some strategy to identify
sensible clusters. Because of the plethora of partitional algorithms, we will look at only
a representative few. We have chosen some of the fncist well known algorithms as well
as some others that have appeared recently in the literature.
5.5.1 Minimum Spanning Tree
Since we have agglomerative and divisive algorithms based on the use of an MST,
we also present a partitional MST algorithm. This is a very simplistic approach, but it
Section 5.5 Partitional Algorithms 139
illustrates how partitional algorithms work. The algorithm is shown in Algorithm 5.4.
Since the clustering problem is to define a mapping, the output of this algorithm shows
the clusters as a set of ordered pairs (ti , j) where fCti ) = KJ .
ALGORITHM 5.4
Input :
D = { t l , t2 , · · · · tn} //Set of elements
A //Adjacency matrix showing distance between elements
k //Number of desired clusters
Output :
f //Mapping represented as a set of ordered pairs
Partitional MST algorithm:
M = MST(A)
identify inconsistent edges in M;
remove k - 1 inconsistent edges ;
create output representation;
The problem is how to define "inconsistent." It could be defined as in the earlier
division MST algorithm based on distance. This would remove the largest k - 1 edges from
the starting completely connected graph and yield the same results as this corresponding
level in thedendrogram. Zahn proposesmorereasonable inconsistent measures based on the
weight (distance) of an edge as compared to those close to it. For example, an inconsistent
edge would be one whose weight is much larger than the average of the adjacent edges.
The time complexity of this algorithm is again dominated by the MST procedure,
which is O(n2). At most, k - 1 edges will be removed, so the last three steps of the
algorithm, assuming each step takes a constant time, is only O (k - 1). Although deter
mining the inconsistent edges in M may be quite complicated, it will not require a time
greater than the number of edges in M. When looking at edges adjacent to one edge,
there are at most k - 2 of these edges. In this case, then, the last three steps are O (k2),
and the total algorithm is still 0 (n
2
).
5.5.2 Squared Error Clustering Algorithm
The squared error clustering algorithm minimizes the squared error. The squared error
for a cluster is the sum of the squared Euclidean distances between each element in the
cluster and the cluster centroid, Ck. Given a cluster Ki , let the set of items mapped to
that cluster be {ti l , ti2, . . . , tim}. The squared error is defined as
m
seK; = L llti} - ckf
J=l
(5.6)
Given a set of clusters K = {K, , K2, . . . , Kk}, the squared error for K is defined as
k
seK = LseKj
J=l
(5.7)
In actuality, there are many different examples of squared error clustering algo
rithms. They all follow the basic algorithm structure shown in Algorithm 5.5.

140 Chapter 5
ALGORiTHM 5.5
Input :
ciustering
D = { t1, t2, . . . , tn) I /Set of elements
Output :
K //Set of clusters
Squared error algorithm:
assign each item ti to a cluster;
calculate center for each cluster;
repeat
assign each item ti to the cluster which has the closest center;
calculate new center for each cluster;
calculate squared error;
unt il the difference between successive squared errors
is below a threshold;
For each iteration in the squared error algorithm, each tuple is assigned to the
cluster with the cl�sest center. Since there are k clusters and n items, this is an 0 (kn)
operation. Assuming t iterations, this becomes an O(tkn) algorithm. The amount of space
may be only O(n) because an adjacency matrix is not needed, as the distance between
all items is not used.
5.5.3 K-Means Clustering
K-means is an iterative clustering algorithm in which items are moved among sets ofclus-
. ters until the desired set is reached. As such, it may be viewed as a type of squared error
algorithm, although the convergence criteria need not be defined based on the squared
error. A high degree of similarity among elements in clusters is obtained, while a high
degree of dissimilarity among elements in different clusters is achieved simultaneously.
The cluster mean of Ki = {tn , ti2• . . . , tim} is defined as
1 m
mi = -
LtiJ (5.8)
m
J=l
This definition assumes that each tuple has only one numeric value as opposed to a
tuple with many attribute values. The K-means algorithm requires that some definition
of cluster mean exists, but it does not have to be this particular one. Here the mean
is defined identically to our earlier definition of centroid. This algorithm assumes that
the desired number of clusters, k, is an input parameter. Algorithm 5.6 shows the K
means algorithm. Note that the initial values for the means are arbitrarily assigned. These
could be assigned randomly or perhaps could use the values from the first k input items
themselves. The convergence criteria could be based on the squared error, but they need
not be. For example, the algorithm could stop when no (or a very small) number of tuples
are assigned to different clusters. Other termination techniques have simply looked at a
fixed number of iterations. A maximum number of iterations may be included to ensure
stopping even without convergence.
ALGORITHM 5.6
Input :
D = { tl, t2 , · · · • tn) //Set of elements
Output :
K //Set of clusters
K-means algorithm:
assign initial values for means m1, m2 , . . . , mk ;
repeat
assign each item ti to the cluster which has the closest mean;
calculate new mean for each cluster;
until convergence criteria is met ;
The K-means algorithm is illustrated in Example 5.4.
EXAMPLE 5.4
Suppose that we are given the following items to cluster:
{2, 4, 10, 12, 3, 20, 30, 1 1, 25} (5.9)
and suppose that k = 2. We initially assign the means to the first two values: m 1 = 2
and m2 = 4. Using Euclidean distance, we find that initially Kt = {2, 3} and
.
K2 . =
{4, 10, 12, 20, 30, 1 1, 25}. The value 3 i s equally close to both means, so we arb1trarlly
choose K1 • Any desired assignment could be used in the case of ties. We then recalculate
the means to get m 1 = 2.5 and m2 = 16. We again make assignments to clusters to g�t
K1 = {2, 3, 4} and K2 = { 10, 12, 20, 30, 1 1, 25}. Continuing in this fashion, we obtam
the following:
m 1 m2 K! K2
3 18 {2, 3, 4, 10} {12, 20, 30, 11, 25}
4.75 19.6 {2, 3, 4, 10, 1 1 , 12} {20, 30, 25}
7 25 {2, 3, 4, 10, 1 1 ' 12} {20, 30, 25}
Note that the clusters in the last two steps are identical. This will yield identical means,
and thus the means have converged. Our answer is thus Kt = {2, 3, 4, 10, 1 1 , 12} and
K2 - {20, 30, 25}.
The time complexity of K-means is O (tkn) where t is the number of iterations.
K-means finds a local optimum and may actually miss the global optimum. K-�eans
does not work on categorical data because the mean must be defined on the attnbute
type. Only convex-shaped clusters are found. It also does not handle outli�rs well. On
.
e
variation of K-means, K-modes, does handle categorical data. Instead of usmg means, 1t
uses modes. A typical value for k is 2 to 10.
. . . .
Although the K-means algorithm often produces good results, 1t 1s not t1me-effic1ent
and does not scale well. By saving distance information from one iteration to the next,
the actual number of distance calculations that must be made can be reduced.
Some K-means variations examine ways to improve the chances of finding the global
optimum. This often involves careful selection of the initial clu�ters an� �eans. Anoth�r
variation is to allow clusters to be split and merged. The vanance w1thin a cluster 1s
examined, and if it is too large, a cluster is split. Similarly, if the distance between two
cluster centroids is less than a predefined threshold, they will be combined.

5.5.4 Nearest Neighbor Algorithm
An algorithm similar to the single link technique is called the nearest neighbor algorithm.
With this serial algorithm, items are iteratively merged into the existing clusters that are
closest. In this algorithm a threshold, t, is used to determine if items will be added to
existing clusters or if a new cluster is created.
ALGORITHM 5.7
Input :
D = {t1 , t2 , . . . , tn} / /Set of elements
A / /Adj acency matrix showing distance between el ements
Output :
K / /Set of clus ters
Nearest neighbor algorithm:
K1 = ft1 } ;
K = {Kl} i
k = 1 ;
for i = 2 to.; n do
f ind the tm in some cluster Km in K such that di s(ti, tm) i s
the smallest ;
if dis (ti , tm), :::; t then
Km = Km U ti
else
k = k + l ;
Kk = {ti} ;
Example 5.5 shows the application of the nearest neighbor algorithm to the data
shown in Table 5.2 assuming a threshold of 2. Notice that the results are the same as
those seen in Figure 5.7(a) at the level of 2.
EXAMPLE 5.5
Initially, A is placed in a cluster by itself, so we have K1 = {A}. We then look at B to
decide if it should be added to K1 or be placed in a new cluster. Since dis(A , B) = 1,
which is less than the threshold of 2, we place B in K1 to get K1 = {A , B }. When
looking at C, we see that its distance to both A and B is 2, so we add it to the cluster ·
to get K1 = {A , B, C}. The dis(D, C) = 1 < 2, so we get K1 = {A, B , C, D }. Finally,
looking at E, we see that the closest item in K1 has a distance of 3, which is greater
than 2, so we place it in its own cluster: K2 = {E}.
The complexity of the nearest neighbor algorithm actually depends on the number
of items. For each loop, each item must be compared to each item already in a cluster.
Obviously, this is n in the worst case. Thus, the time complexity is O(n2). Since we do
need to examine the distances between items often, we assume that the space requirement
is also O(n2).
5.5.5 PAM Algorithm
The P
AM (partitioning around medoids) algorithm, also called the K-medoids algorithm,
represents a cluster by a medoid. Using a medoid is an approach that handles outliers
well. The PAM algorithm is shown in Algorithm 5.8. Initially, a random set of k items
is taken to be the set of medoids. Then at each step, all items from the input dataset that
are not currently medoids are examined one by one to see if they should be medoids.
That is, the algorithm determines whether there is an item that should replace one of the
existing medoids. By looking at all pairs of medoid, non-medoid objects, the algorithm
chooses the pair that improves the overall quality of the clustering the best and exchanges
them. Quality here is measured by the sum of all distances from a non-medoid object
to the medoid for the cluster it is in. An item is assigned to the cluster represented by
the medoid to which it is closest (minimum distance). We assume that K; is the cluster
represented by medoid t; . Suppose t; is a current medoid and we wish to determine
whether it should be exchanged with a non-medoid th . We wish to do this swap only
if the overall impact to the cost (sum of the distances to cluster medoids) represents an
improvement.
Following the lead in [NH94], we use Cjih to be the cost change for an item fj
associated with swapping medoid t; with non-medoid fh. The cost is the change to the
sum of all distances from items to their cluster medoids. There are four cases that must
be examined when calculating this cost:
1. tj E K; , but 3 another medoid tm where dis(tj , tm) S dis(tj , fh);
2. fj E K; , but dis(tj . th) S dis(tj , tm)V other medoids tm ;
3. lj E Km , tf. K; , and dis(fj , tm) S dis(tj , fh); and
4. fj E Km , tf. K; , but dis(fj , fh) S dis(fj , tm) .
We leave it as an exercise to determine the cost of each of these cases. The total impact
to quality by a medoid change TCh then is given by
n
TCth = L Cjth
j=i
ALGORITHM 5.8
Input :
D = {tl, t2, . . . , tn} //Set of elements
A //Adj acency matrix showing distance between element s
k //Number of de s ired clusters
Output :
K //Set of c lusters
PAM algorithm :
arbi trarily select k medoids from D;
repeat
for each th not a medoid do
for each medoid ti do
calculate TCih ;
find i, h where TCih is the smallest ;
i f TCih < 0 , then
replace medo id ti with th ;
until TCih :::: 0 ;
(5. 10)
for each ti E D do
ass ign ti to Kj , where dis(ti , tj) is the smallest over al l medoids ;

Example 5.6 shows the application of the PAM algorithm to the data shown in
Table 5.2 assuming a threshold of 2.
EXAMPLE 5.6
Suppose that the two medoids that are initially chosen are A and B. Based on the
distances shown in Table 5.2 and randomly placing items when distances are identical to
the two medoids, we obtain the clusters {A, C, D} and {B, E} The three non-medoids,
{C, D, E}, are then examined to see which (if any) should be used to replace A or B.
We thus have six costs to determine: TCAc, TCAD· TCAE, TCBc, TCBD, and TCBE·
Here we use the name of the item instead of a numeric subscript value. We obtain the
following:
TCAc = CAAc + CBAC + CcAC + CDAC + CEAC = 1 + 0 - 2 - 1 + 0 = -2 (5. 1 1)
Here A is no longer a medoid, and since it is closer to B, it will be placed in the cluster
with B as medoid, and thus its cost is CAAC = 1. The cost for B is 0 because it stays
a cluster medoid.
1
C is now a medoid, so it has a negative cost based on its distance to
the old medoid; that is, CcAB = -2. D is closer to C than it was to A by a distance
of 1, so its cost is CDAC = - 1 . Finally, E stays in the same cluster with the same
distance, so its cost change is 0. Thus, we have that the overall cost is a reduction of 2.
Figure 5.9 illustrates the calculation of these six costs. Looking at these, we see that
the minimum cost is 2 and that there are several ways to reduce this cost. Arbitrarily
choosing the first swap, we get C and B as the new medoids with the clusters being
{C, D} and {B, A, E}. This concludes the first iteration of PAM. At the next iteration,
we examine changing medoids again and pick the choice that best reduces the cost. The
iterations stop when no changes will reduce the cost. We leave the rest of this problem
to the reader as an exercise.
P
AM does not scale well to large datasets because of its computational complexity.
For each iteration, we have k(n -k) pairs of objects i, h for which a cost, TCih, should be
determined. Calculating the cost during each iteration requires that the cost be calculated
for all other non-medoids tj. There are n - k of these. Thus, the total complexity per
iteration is n(n - k)2. The total number of iterations can be quite large, so PAM is not
an alternative for large databases. However, there are some clustering algorithms based
on PAM that are targeted to large datasets.
CLARA (Clustering LARge Applications) improves on the time complexity of PAM
by using samples of the dataset. The basic idea is that it applies PAM to a sample of the
underlying database and then uses the medoids found as the medoids for the complete
clustering. Each item from the complete database is then assigned to the cluster with the
medoid to which it is closest. To improve the CLARA accuracy, several samples can be
drawn with PAM applied to each. The sample chosen as the final clustering is the one
that performs the best. Because of the sampling, CLARA is more efficient than PAM for
large databases. However, it may not be as effective, depending on the sample size. Five
samples of size 40 + 2k seem to give good results [KR90].
CLARANS (clustering large applications based upon randomized search) improves
on CLARA by using multiple different samples. In addition to the normal input to PAM,
5.5.6
<c
A
D
·�.
(a) Start: Medoids A, B
A
B
3
E
(d) TC8c: Change -2
Section 5.5
I�
A
1
B
3
E
(b) TCAC: Change -2;
TCAv: Change -2
c
D
3
E
(e) TC8v: Change -2
Partitional Algorithms
E
(c) TCAE: Change -1
• E
(f) TC8E: Change -2
FIGURE 5.9: Cost calculations for Example 5.6.
145
CLARANS requires two additional parameters: m�neighbor an� numlocal. Maxneigh
bor is the number of neighbors of a node to which any spec1fic n?de can be com
pared. As maxneighbor increases, CLARANS looks more and more hke PAM because
all nodes will be examined. Numlocal indicates the numb�r of sa�p�es to be taken.
Since a new clustering is performed on each sample, tlus also mdicates the num
ber of clusterings to be made. Performance studies indicate that numlocal = 2 a�d
maxneighbor = max((0.0125 x k(n - k)), 250) are good choices
.
[NH94]. CLARANS 1s
shown to be more efficient than either PAM or CLARA for any s1ze d�taset. CL�NS
assumes that all data are in main memory. This certainly is not a valid assumpt10n for
large databases.
Bond Energy Algorithm
The bond energy algorithm (BEA) was developed and has been used in the datab
.
ase
design area to determine how to group data and how to physically place
.
data on a d�sk.
It can be used to cluster attributes based on usage and then perform log1�al or �hystcal
design accordingly. With BEA, the affinity (bond) between database �tU:bu
.
tes lS based
on common usage. This bond is used by the clustering algorithm as a su
rulanty meas�re.
The actual measure counts the number of times the two attributes are used together m a
given time. To find this, all common queries must be identified.
The idea is that attributes that are used together form a cluster and should be stored
together. m a distributed database, each resulting cluster is called a verticalfragment and

FIGURE 5.10: Clustered affinity matrix for BEA (modified from [OV99]).
may be stored at different sites from other fragments. The basic steps of this clustering
algorithm are:
·
1. Create an attribute affinity matrix in which each entry indicates the affinity between
the two associate attributes. The entries in the similarity matrix are based on the
freqJlency of common usage of attribute pairs.
2. The BEA then converts this similarity matrix to a BOND matrix in which the
entries represent a type of nearest neighbor bonding based on probability of co
access. The BEA algorithm rearranges rows or columns so that similar attributes
appear dose together in the matrix.
3. Finally, the designer draws boxes around regions �n the matrix with high similarity.
The resulting matrix, modified from [OV99], is illustrated in Figure 5.10 The two shaded
boxes represent the attributes that have been grouped together into two clusters.
Two attributes Ai and AJ have a high affinity if they are frequently used together
in database applications. At the heart of the BEA algorithm is the global affinity measure.
Suppose that a database schema consists of n attributes {A1 , A2, . . . , An}. The global
affinity measure, AM, is defined as
n
A M = l)bond(Ai , Ai-l ) + bond(Ai , Ai+I))
i =l
5.5.7 Clustering with Genetic Algorithms
(5.12)
There have been clustering techniques based on the use of genetic algorithms. To deter
mine how to perform clustering with genetic algorithms, we first must determine how
to. represent each cluster. One simple approach would be to use a bit-map representation
for each possible clu�ter. So, given a database with four items, {A, B, C, D}, we would
represent one solution to creating two clusters as 1001 and 01 10. This represents the two
clusters {A, D } and { B , C}.
Section 5.5 Partitional Algorithms 1 47
Algorithm 5.9 shows one possible iterative refinement technique for clustering that
uses a genetic algorithm. The approach is similar to that in the squared error approach iu
that an initial random solution is given and successive changes to this converge on a local
optimum. A new solution is generated from the previous solution using crossover and
mutation operations. Our algorithm shows only crossover. The use of crossover to create
a new solution from a previous solution is shown in Example 5.7. The new "solution"
must be created in such a way that it represents a valid k clustering. A fitness function
must be used and may be defined based on an inverse of the squared error. Because
of the manner in which crossover works, genetic clustering algorithms perform a global
search rath�r than a local search of potential solutions.
ALGORITHM 5.9
Input :
D = { t 1 , t2 , . . . , tn} //Set of el ements
Output :
K //Set of clusters
GA clustering algorithm:
randomly create an initial solut ion ;
repeat
use crossover to create a new solut ion ;
unt il termination criteria is met ;
EXAMPLE 5.7
Suppose a database contains the following eight items {A, B, C, D, E, F, G, H}, which
are to be placed into three clusters. We could �mtially place the items into the three clusters
{A, C, E}, {B, F}, and {D, G, H}, which are represented by 10101000, 01000100, and
0001001 1, respectively. Suppose we choose the first and third individuals as parents and
do a simple crossover at point 4. This yields the new solution: 00011000, 01000100, and
101000 1 1 .
5.5.8 Clustering with Neural Networks
Neural networks (NNs) that use unsupervised learning attempt to find features in the
data that characterize the desired output. They look for clusters of like data. These types
of NNs are often called self-organizing neural networks. There are two basic types of
unsupervised learning: noncompetitive and competitive.
With the noncompetitive or · Hebbian learning, the weight between two nodes is
changed to be proportional to both output values. That is
b.Wji = T/YJYi (5. 13)
With competitive learning, nodes are allowed to compete and the winner takes all.
This approach usually assumes a two-layer NN in which all nodes from one layer are
connected to all nodes in the other layer. As training occurs, nodes in the output layer
become associated with certain tuples in the input dataset. Thus, this provides a grouping

148 Chapter 5 Cl ustering
of these tuples together into a cluster. Imagine every input tuple having each attribute
value input to a specific input node in the NN. The number of input nodes is the same
as the number of attributes. We can thus associate each weight to each output node with
one of the attributes from the input tuple. When a tuple is input to the NN, all output
nodes produce an output value. The node with the weights more similar to the input
tuple is declared the winner. Its weights are then adjusted. This process' continues with
each tuple input from the training set. With a large and varied enough training set, over
time each output node should become associated with a set of tuples. The input weights
to the node are then close to an average of the tuples in this cluster.
Self-Organizing Feature Maps. A self-organizingfeature map (SOFM) or self
organizing map (SOM) is an NN approach that uses competitive unsupervised learning.
Learning is based on the concept that the behavior of a node should impact only those
nodes and arcs near it. Weights are initially assigned randomly and adjusted during the
learning process to produce better results. During this learning process, hidden features or
patterns in the data ,are uncovered and the weights are adjusted accordingly. SOFMs were
developed by obsefving how neurons work in the brain and in ANNs. That is [BS97] :
• The firing of neurons impact the firing of other neurons that are near it.
• Neurons that are far apart seem to inhibit each other.
• Neurons seem to have specific nonoverlapping tasks.
The term self-organizing indicates the ability of these NNs to organize the nodes into
clusters based on the similarity between them. Those nodes that are closer together are
more similar than those that are far apart. This hints at how the actual clustering is
performed. Over time, nodes in the output layer become matched to input nodes, and
patterns of nodes in the output layer emerge.
Perhaps the most common example of a SOFM is the Kohonen self-organizing
map, which is used extensively in commercial data mining products to perform clustering.
There is one input layer and one special layer, which produces output values that compete.
In effect, multiple outputs are created and the best one is chosen. This extra layer is
not technically either a hidden layer or an output layer, so we refer to it here as the
competitive layer. Nodes in this layer are viewed as a two-dimensional grid of nodes as
seen in Figure 5. 1 1 . Each input node is connected to each node in this grid. Propagation
occurs by sending the input value for each input node to each node in the competitive
layer. As with regular NNs, each arc has an associated weight and each node in the
competitive layer has an activation function. Thus, each node in the competitive layer
produces an output value, and the node with the best output wins the competition and is
determined to be the output for that input. An attractive feature of Kohonen nets is that
the data can be fed into the multiple competitive nodes in parallel. Training occurs by
adjusting weights so that the best output is even better the next time this input is used.
"Best" is determined by computing a distance measure.
A common approach is to initialize the weights on the input arcs to the com
petitive layer with normalized values. The similarity between output nodes and input
vectors is then determined by the dot product of the two vectors. Given an input tuple
X = (x1 , . . . , xh) and weights on arcs input to a competitive node i as WJi, . . . , Whi, the
5.6
Section 5.6 Cl ustering Large Databases 149
FIGURE 5. 1 1 : Kohonen network.
similarity between X and i can be calculated by
h
sim(X, i) = I>j Wji
j=l
(5.14)
The competitive node most similar to the input node wins the competitive. Based on this,
the weights corning into i as well as those for the nodes immediately surrounding it in
the matrix are increased. This is the learning phase. Given a node i, we use the notation
Ni to represent the union of i and the nodes near it in the matrix. Thus, the learning
process uses
D.w . _ { C(Xk - Wkj) if j E Ni }
kJ - 0 otherwise
(5. 15)
In this formula, c indicates the learning rate and may actually vary based on the node rather
than being a constant. The basic idea of SOM learning is that after each input tuple in the
training set, the winner and its neighbors have their weights changed to be closer to that of
the tuple. Over time, a pattern on the output nodes emerges, which is close to that of the
training data. At the beginning of the training process, the neighborhood of a node may be
defined to be large. However, the neighborhood may decrease during the processing.
CLUSTERING LARGE DATABASES
The clustering algorithms presented in the preceding sections are some of the classic clus
tering techniques. When clustering is used with dynamic databases, these algorithms may
not be appropriate. First, they all assume that [because most are O(n2)] sufficient main
memory exists to hold the data to be clustered and the data structures needed to support
them. With large databases containing thousands of items (or more), these assumptions
are not realistic. In addition, performing 1/0s continuously through the multiple itera
tions of an algorithm is too expensive. Because of these main memory restrictions, the
algorithms do not scale up to large databases. Another issue is that some assume that the

1 50 Chapter 5 Clusteri ng
data are present all at once. These techniques are not appropriate for dynamic databases.
Clustering techniques should be able to adapt as the database changes.
The algorithms discussed in the following subsections each examine an issue asso
ciated with performing clustering in a database environment. It has been argued that to
perform effectively on large databases, a clustering algorithm should [BFR98]:
1. require no more (preferably less) than one scan of the database.
2. have the ability to provide status and "best" answer so far during the algorithm
execution. This is sometimes referred to as the ability to be online.
3. be suspendable, stoppable, and resumable.
4. be able to update the results incrementally as data are added or removed from the
database.
5. work with limited main memory.
6. be capable of performing different techniques for scanning the database. This may
include sampling.
7. process each tJple only once.
Recent research at Microsoft has examined how to efficiently perform the clustering
algorithms with large databases [BFR98]. The basic idea of this scaling approach is as
follows:
1. Read a subset of the database into main memory.
2. Apply clustering technique to data in memory.
3. Combine results with those from prior samples.
4. The in-memory data are then divided into three different types: those items that
will always be needed even when the next sample is brought in, those that can
be discarded with appropriate updates to data being kept in order to answer the
problem, and those that will be saved in a compressed format. Based on the type,
each data item is then kept, deleted, or compressed in memory.
5. If termination criteria are not met, then repeat from step 1.
This approach has been applied to the K-means algorithm and has been shown to be
effective.
5.6.1 BIRCH
BIRCH (balanced iterative reducing and clustering using hierarchies) is designed for
clustering a large amount of metric data. It assumes that there may be a limited amount
of main memory and achieves a linear 1/0 time requiring only one database scan. It is
incremental and hierarchical, and it uses an outlier handling technique. Here points that
are found in sparsely populated areas are removed. The basic idea of the algorithm is that
a tree is built that captures needed information to perform clustering. The clustering is
then performed on the tree itself, where labelings of nodes in the tree contain the needed
information to calculate distance values. A major characteristic of the BIRCH algorithm
is the use of the clusteringfeature, which is a triple that contains information about a
cluster (see Definition 5.3). The clustering feature provides a summary of the information
about one cluster. By this definition it is clear that BIRCH applies only to numeric data.
Section 5.6 Clustering Large Databases 151
This algorithm uses a tree called a CF tree as defined in Definition 5.4. The size of the
tree is determined by a threshold value, T, associated with each leaf node. This is the
maximum diameter allowed for any leaf. Here diameter is the average of the pairwise
distance between all points in the cluster. Each internal node corresponds to a cluster
that is composed of the subclusters represented by its children.
DEFINITION 5.3. A clustering feature (CF) is a triple (N,
{
s, SS), where the
number of the points in the cluster is N,
{
s is the sum of the points in the cluster,
and SS is the sum of the squares of the points in the cluster.
DEFINITION 5.4. A CF tree is a balanced tree with a branching factor (maximum
number of children a node may have) B. Each internal node contains a CF triple
for each of its children. Each leaf node also represents a cluster and contains a CF
entry for each subcluster in it. A subcluster in a leaf node must have a diameter
no greater than a given threshold value T.
Unlike a dendrogram, a CF tree is searched in a top-down fashion. Each node in
the CF tree contains clustering feature information about its subclusters. As points are
added to the clustering problem, the CF tree is built. A point is inserted into the cluster
(represented by a leaf node) to which it is closest. If the diameter for the leaf node is
greater than T, then a splitting and balancing of the tree is performed (similar to that
used in a B-tree). The algorithm adapts to main memory size by changing the threshold
value. A larger threshold, T, yields a smaller CF tree. This process can be performed
without rereading the data. The clustering feature data provides enough information to
perform this condensation. The complexity of the algorithm is O (n).
ALGORITHM 5.10
Input :
D = {t1, t2 , . . . , tn} / / Set of elements
T II Threshold for CF tree construction
Output :
K // Set of cl usters
BIRCH clustering algorithm :
for each ti E D do
determine correct leaf node for ti insertion ;
i f threshold condition i s not violated , then
add ti to cluster and update CF triples ;
else
if room to insert ti , then
insert ti as s ingle cluster and update CF triples ;
else
spl i t leaf node and redist ribute CF features ;
Algorithm 5.10 outlines the steps performed in BIRCH. Not shown in this algorithm
are the parameters needed for the CF tree construction, such as its branching factor, the
page block size, and memory size. Based on size, each node has room for a fixed number,
B, of clusters (i.e., CF triples). The first step creates the CF tree in memory. The threshold
value can be modified if necessary to ensure that the tree fits into the available memory
space. Insertion into the CF tree requires scanning the tree from the root down, choosing
the node closest to the new point at each level. The distance here is calculated by looking

at the distance between the new point and the centroid of the cluster. This can be easily
calculated with most distance measures (e.g., Euclidean or Manhattan) using the CF
triple. When the new item is inserted, the CF triple is appropriately updated, as is each
triple on the path from the root down to the leaf. It is then added to the closest leaf node
found by adjusting the CF value for that node. When an item is inserted into a cluster at
the leaf node of the tree, the cluster must satisfy the threshold value. If it does, then the
CF entry for that cluster is modified. If it does not, then that item is added to that node
as a single-item cluster.
Node splits occur if no space exists in a given node. This is based on the size of
the physical page because each node size is determined by the page size. An attractive
feature of the CF values is that they are additive; that is, if two clusters are merged, the
resulting CF is the addition of the CF values for the starting clusters. Once the tree is
· built, the leaf nodes of the CF tree represent the current clusters.
In reality, this algorithm, Algorithm 5. 10, is only the first of several steps proposed
for the use of BIRCH with large databases. The complete outline of steps is:
1. Create initiil. CF tree using a modified version of Algorithm 5.10. This in effect
"loads" the database into memory. If there is insufficient memory to construct the
CF tree with a given threshold, the threshold value is increased and a new smaller
CF tree is constructed. This can be done by inserting the leaf nodes of the previous
tree into the new small tree.
2. The clustering represented by the CF tree may not be natural because each entry
has a limited size. In addition, the input order can negatively impact the results.
These problems can be overcome by another global clustering approach applied
to the leaf nodes in the CF tree. Here each leaf node is treated as a single point
for clustering. Although the original work proposes a centroid-based agglomer
ative hierarchical clustering algorithm to cluster the subclusters, other clustering
algorithms could be used.
3. The last phase (which is optional) reclusters all points by placing them in the clus
ter that has the closest centroid. Outliers, points that are too far from any centroid,
can be removed during this phase.
BIRCH is linear in both space and I/0 time. The choice of threshold value is
imperative to an efficient execution of the algorithm. Otherwise, the tree may have to be
rebuilt many times to ensure that it can be memory-resident. This gives the worst-case
time complexity of O(n2).
5.6.2 DBSCAN
The approach used by DBSCAN (density-based spatial clustering o
f applications with
noise) is to create clusters with a minimum size and density. Density is defined as a
minimum number of points within a certain distance of each other. This handles the
outlier problem by ensuring that an outlier (or a small set of outliers) will not create a
cluster. One input parameter, MinPts, indicates the minimum number of points in any
cluster. In addition, for each point in a cluster there must be another point in the cluster
whose distance from it is less than a threshold input value, Eps. The Eps-neighborhood
or neighborhood of a point is the set of points within a distance of Eps. The desired
number of clusters, k, is not input but rather is determined by the algorithm itself.
Eps Minpts = 4
..... - - -...
Section 5.6 Clustering Large Databases 1 53
I /
/ '
I ' I I
(
p I ' I
I
I
I .--------! I
•
•
I
•
I
/I
I
'
' /
...... _ _ _
/
I q
I
I
I I
(a) Bps-neighborhood (b) Core points (c) Density reachable
FIGURE 5. 1 2: DBSCAN example.
DBSCAN uses a new concept of density. We fitst must look at some definitions
froin [EKSX96]. Definition 5.5 defines directly density-reachable. The first part of the
definition ensures that the second point is "close enough" to the first point. The second
portion of the definition ensures that there are enough core points close enough to each
other. These core points form the main portion of a cluster in that they are all close to
each other. A directly density-reachable point must be dose to one of these core points,
but it need not be a core point itself. In that case, it is called a border point. A point is
said to be density-reachable from another point if there is a chain from one to the other
that contains only points that are directly density-reachable from the previous point. This
guarantees that any cluster will have a core set of points very close to a large number of
other points (core points) and then some other points (border points) that are sufficiently
close to at least one core point.
DEFINITION 5.5. Given values Eps and MinPts, a point p is directly density
reachable from q if
• dis(p, q) ::; Eps
and
• I {r I dis(r, q) ::; Eps} I� MinPts
Figure 5.12 illustrates the concepts used by DBSCAN. This figure shows 12 points.
The assumed Eps value is illustrated by the straight line. In part (a) it is shown that there
are 4 points within the neighborhood of point p. As is seen, p is a core point because
it has 4 (MinPts value) points within its neighborhood. Part (b) shows the 5 core points
in the figure. Note that of the 4 points that are in the neighborhood of p, only 3 are
themselves core points. These 4 points are said to be directly density-reachable from p.
Point q is not a core point and is thus called a border point. We have partitioned the
points into a core set of points that are all close to each other; then border points, which
are close to at least one of the core points; and finally the remaining points, which are
not close to any core point. Part (C) shows that even though point r is not a core point,
it is density-reachable from q.
Algorithm 5.11 outlines the DBSCAN algorithm. Because of the restrictions on
what constitutes a cluster when the algorithm finishes, there will be points not assigned
to a cluster. These are defined as noise.

ALGORITHM 5.11
Input :
D = { t1 , t2 , . . . , tn} //Set of elements
MinPts // Number of points in cluster
Eps II Maximum distance for density measure
Output :
K = {K1 , K2, . . . , Kk} I /Set of clusters
DBSCAN algorithm:
k = 0 ; II Initially there are no clusters .
for i = 1 to n do
i f ti is not in a cluster, then
X = { tj I tj is density-reachable from ti } i
i f X is a valid cluster, then
k = k + l ;
Kk = X;
The expected time complexity of DBSCAN is 0 (n lg n). It is possible that a border
point could belong
'
to two clusters. The stated algorithm will place this point in whichever
cluster is generated first. DBSCAN was compared with CLARANS and found to be more
efficient by a factor of 250 to 1900 [EKSX96]. In addition, it successfully found all
clusters and noise from the test dataset, whereas CLARANS did not.
5.6.3 CURE Algorithm
One objective for the CURE (Clustering Using REpresentatives) clustering algorithm is to
handle outliers well. It has both a hierarchical component and a partitioning component.
First, a constant number of points, c, are chosen from each cluster. These well-scattered
points are then shrunk toward the cluster's centroid by applying a shrinkage factor, a .
When a is 1, all points are shrunk to just one-the centroid. These points represent the
cluster better than a single point (such as a medoid or centroid) could. With multiple rep
resentative points, clusters of unusual shapes (not just a sphere) can be better represented.
CURE then uses a hierarchical clustering algorithm. At each step in the agglomerative
algorithm, clusters with the closest pair of representative points are chosen to be merged.
The distance between them is defined as the minimum distance between any pair of
points in the repr��entative sets from the two clusters.
The basic approach used by CURE is shown in Figure 5.13. The first step shows
a sample of the data. A set of clusters with its representative points exists at each step
in the processing. In Figure 5.1 3(b) there are three clusters, each with two representative
points. The representative points are shown as darkened circles. As discussed in the
following paragraphs, these representative points are chosen to be far from each other as
well as from the mean of the cluster. In part (c), two of the clusters are merged and two
new representative points are chosen. Finally, in part (d), these points are shrunk toward
the mean of the cluster. Notice that if one representative centroid had been chosen for
the clusters, the smaller cluster would have been merged with the bottom cluster instead
of with the top cluster.
CURE handles limited main memory by obtaining a random sample to find the
initial clusters. The random sample is partitioned, and each partition is then partially
clustered. These resulting clusters are then completely clustered in a second pass. The
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
0 0
(a) Sample of data
(c) Merge clusters with closest points
0
Section 5.6 Clustering Large Databases 1 55
(b) Three clusters with representative points
'
"' I
/ 0 . /
/ I
/
/ 0 0 /
/ /
/ 0 0 /
I 0 I
/� 0 (
--"'l ·l} I 0 I
(Q_�... ./ /
/ 0 0 I
"'D t o
( 0
0
. _ _ _ _ _ _ _ 0 0
_
_ _ _ ,'
(d) Shrink representative points
FIGURE 5. 1 3: CURE exampie.
sampling and partitioning are done solely to ensure that the data (regardless of database
size) can fit into available main memory. When the clustering of the sample is complete,
the labeling of data on disk is performed, A data item is assigned to the cluster with the
closest representative points. The basic steps of CURE for large databases are:
1. Obtain a sample of the database.
2. Partition the sample into p partitions of size �. This is done to speed up the
algorithm because clustering is first performed on each partition.
3. Partially cluster the points in each partition using the hierarchical algorithm (see
Algorithm 5. 12). This provides a first guess at what the clusters should be. The
number of clusters is
P
� for some constant q.
4. Remove outliers. Outliers are eliminated by the use of two different techniques.
The first technique eliminates clusters that grow very slowly. When the number of
clusters is below a threshold, those clusters with only one or two items are deleted.
It is possible that close outliers are part of the sample and would not be identified
by the first outlier elimination technique. The second technique removes very small
clusters toward the end of the clustering phase.

5. Completely cluster all data in the sample using Algorithm 5.12. Here, to ensure
processing in main memory, the input includes only the cluster representatives from
the clusters found for each partition during the partial clustering step (3).
6. Cluster the entire database on disk using c points to represent each cluster. An item
in the database is placed in the cluster that has the closest representative point to
it. These sets of representative points are small enough to fit into main memory,
so each of the n points must be compared to ck representative points.
The time complexity of CURE is O (n2 lg n), while space is O (n). This is worst
case behavior. The improvements proposed for main memory processing certainly
improve on this time complexity because the entire clustering algorithm is performed
against only the sample. When clustering is performed on the complete database, a time
complexity of only O(n) is required. A heap and k-D tree data structure are used to
ensure this performance. One entry in the heap exists for each cluster. Each cluster has
not only its representative points, but also the cluster that is closest to it. Entries in the
heap are stored in jncreasing order of the distances between clusters. We assume that
each entry u in the heap contains the set of representative points, u.rep; the mean of
the points in the cluster, u.mean; and the cluster closest to it, u.closest. We use the heap
operations: heapify to create the heap, min to extract the minimum entry in the heap,
insert to add a new entry, and delete to delete an entry. A merge procedure is used to
merge two clusters. It determines the new representative points for the new cluster. The
basic idea of this process is to first find the point that is farthest from the mean. Sub
sequent points are then chosen based on being the farthest from those points that were
previously chosen. A predefined number of points is picked. A k-D tree is a balanced
binary tree that can be thought of as a generalization of a binary search tree. It is used to
index data of k dimensions where the i1h level of the tree indexes the i1h dimension. In
CURE, a k-D tree is used to assist in the merging of clusters. It stores the representative
points for each cluster. Initially, there is only one representative point for each cluster,
the sole item in it. Operations performed on the tree are: delete to delete an entry form
the tree, insert to insert an entry into it, and build to initially create it. The hierarchical
clustering algorithm itself, which is from [GRS98], is shown in Algorithm 5.12. We do
not include here either the sampling algorithms or the merging algorithm.
ALGORITHM 5.12
Input :
D = {t1, t2 , . . . , tn} //Set of elements
k // Desired number of clusters
Output :
Q //Heap containing one entry for each cluster
CURE algorithm:
T = buil d(D) ;
Q = heapi fy(D) ; // Initially build heap with one entry per item;
repeat
u = min(Q) ;
delete (Q, u . close) ;
w = merge(u, v) ;
del ete(T, u) ;
del ete(T, v) ;
Section 5.7 Cl ustering with Categorical Attributes 1 57
insert(T, w) ;
for each x E Q do
x . close = find closest cluster to x;
if x is closest to w, then
w. close = x;
insert(Q, w) ;
until number of nodes in Q is k;
Performance experiments compared CURE to BIRCH and the MST approach
[GRS98]. The quality of the clusters found by CURE is better. While the value of
the shrinking factor a does impact results, with a value between 0.2 and 0.7, the correct
clusters are still found. When the number of representative points per cluster is greater
than five, the correct clusters are still always found. A random sample size of about 2.5%
and the number of partitions is greater than one or two times k seem to work well. The
results with large datasets indicate that CURE scales well and outperforms BIRCH.
5.7 CLUSTERING WITH CATEGORICAL ATTRI BUTES
Traditional algorithms do not always work with categorical data. Example 5.8 illus
trates some problems that exist when clustering categorical data. This example uses a
hierarchical-based centroid algorithm to illustrate the problems. The problem illustrated
here is that the centroid tends to weaken the relationship between the associated cluster
and other clusters. The problems worsens as more and more clusters are merged. The
number of attributes appearing in the mean increases, while the individual values actually
decreases. This makes the centroid representations very similar and makes distinguishing
between clusters difficult.
EXAMPLE 5.8
Consider an information retrieval system where documents may contain keywords {book,
water, sun, sand, swim, read}. Suppose there are four documents, where the first contains
the word {book}, the second contains {water, sun, sand, swim}, the third contains {water,
sun, swim, read}, and the fourth contains {read, sand}. We can represent the four books
using the following boolean points: (1, 0, 0, 0, 0, 0), (0, 1, 1 , 1 , 1, 0), (0, 1 , 1, 0, 1 , 1),
(0, 0, 0, 1 , 0, 1). We can use the Euclidean distance to develop the following adjacency
matrix of distances:
2 3 4
1 0 2.24 2.24 1 .73
2 2.24 0 1 .41 2
3 2.24 1.41 0 2
4 1 .73 2 2 0
The distance between points 2 and 3 is the smallest (1 .41), and thus they are merged.
When they are merged, we get a cluster containing {(0, 1 , 1 , 1 , 1 , 0), (0, 1 , 1 , 0, 1 , 1)}
with a centroid of (0, 1, 1 , 0.5, 1 , 0.5). At this point we have a distance from this new
cluster centroid to the original points 1 and 4 being 2.24 and 1.73, respectively, while the
distance between original points 1 and 4 is 1 .73. Thus, we could next merge these points

even though they have no keywords in common. So with k = 2 we have the following
clusters: {{1, 4}, {2, 3}}.
The ROCK (RObust Clustering using linKs) clustering algoritlun is targeted to both
boolean data and categorical data. A novel approach to identifying similarity is based
on the number of links between items. A pair of items are said to be neighbors if their
similarity exceeds some threshold. This need not be defined based on a precise metric,
but rather a more intuitive approach using domain experts could be used. The number of
links between two items is defined as the number of common neighbors they have. The
objective of the clustering algoritlun is to group together points that have more links.
The algorithm is a hierarchical agglomerative algorithm using the number of links as the
similarity measure rather than a measure based on distance.
Instead of using a Euclidean distance, a different distance, such as the Jaccard
coefficient, has been proposed. One proposed similarity measure based on the Jaccard
coefficient is defin�d as
1
I t· n t · I
sim(t; , tJ) = ' 1
I t; u t1 I
(5. 16)
If the tuples are viewed to be sets of items purchased (i.e., market basket data), then we
look at the number of items they have in common divided by the total number in both.
The denominator is used to normalize the value to be between 0 and 1 .
The number of links between a pair of points can be viewed as the number of
unique paths of length 2 between them. The authors argue that the use of links rather
than similarity (distance) measures provides a more global approach because the similarity
between points is impacted by other points as well. Example 5.9 illustrates the use of links
by the ROCK algorithm using the data from Example 5.8 using the Jaccard coefficient.
Note that different threshold values for neighbors could be used to get different results.
Also note that a hierarchical approach could be used with different threshold values for
each level in the dendrogram.
EXAMPLE 5.9
Using the data from Example 5.8, we have the following table of similarities (as opposed
to the distances given in the example):
2 3 4
1 0 0 0
2 0 1 0.6 0.2
3 0 0.6 1 0.2
4 0 0.2 0.2
Suppose we say that the threshold for a neighbor is 0.6, then we have the following are the
neighbors: { (2, 3), (2, 4) , (3, 4)}. Note that in the following we add to these that a point is
a neighbor of itself so that we have the additional neighbors: {(1 , 1), (2, 2) , (3, 3) , (4, 4)}.
The following table shows the number of links (common neighbors between points)
assuming that the threshold for a neighbor is 0.6:
Section 5.8 Comparison 1 59
2 3 4
1 1 0 0 0
2 0 3 3 3
3 0 3 3 3
4 0 3 3 3
In this case, then, we have the following clusters: {{1}, {2, 3, 4}}. Comparing this to the
set of clustering found with a traditional Euclidean distance, we see that a "better" set
of clusters has been created.
The ROCK algorithm is divided into three general parts:
1. Obtaining a random sample of the data.
2. Performing clustering on the data using the link agglomerative approach. A good
ness measure is used to determine which pair of points is merged at each step.
3. Using these clusters the remaining data on disk are assigned to them.
The goodness measure used to merge clusters is:
link(Ki , KJ)
g(Ki , K1· ) = -
-
-
-
-
-
-;-:-o;.
=
·--:-
�
;;:::-
1+2/(E>)-n1+2/(8) -n 1
+2/(8) •
(ni + nJ) ' 1
(5. 17)
Here link(K; , KJ) is the number of links between the two clusters. Also, ni and nJ are
the number of points in each cluster. The denominator is used to normalize the number
of links because larger clusters would be expected to have more links simply because
they are larger. nJ+2f(E>) is an estimate for the number of links between pairs of points in
Ki when the threshold used for the similarity measure is e. The function /(8) depends
on the data, but it is found to satisfy the property that each item in K; has approximately
n{(E>) neighbors in the cluster. Obviously, if all points in the cluster are connected,
/(8) = 1. Then n[ is the number of links between points in K; .
The first step in the algorithm converts the adjacency matrix into a boolean matrix
where an entry is 1 if the two corresponding points are neighbors. As the adjacency matrix
is of size n
2, this is an O(n2) step. The next step converts this into a matrix indicating the
links. This can be found by calculating S x S, which can be done in 0 (n237) [GRS99].
The hierarchical clustering portion of the algorithm then starts by placing each point in
the sample in a separate cluster. It then successively merges clusters until k clusters are
found. To facilitate this processing, both local and global heaps are used. A local heap,
q, is created to represent each cluster. Here q contains every cluster that has a nonzero
link to the cluster that corresponds to this cluster. Initially, a cluster is created for each
point, t; . The heap for t;, q [t;], contains every cluster that has a nonzero link to {td
The global heap contains information about each cluster. PJl information in the heap is
ordered based on the goodness measure, which is shown in Equation 5.17.
5.8 COMPARISON
The different clustering algoritluns discussed in this chapter are compared in Table 5.3.
Here we include a classification of the type of algorithm, space and time complexity,
and general notes concerning applicability.

TABLE 5.3: Comparison of Clustering Algorithms
Algorithm Type Space Time Notes
Single link Hierarchical O(n2) O (kn2) Not incremental
Average link Hierarchical O(n2) O (kn2) Not incremental
Complete link Hierarchical O (n2) O (kn2) Not incremental
MST Hierarchical/ O (n2) O (n2) Not incremental
partitional
Squared error Partitional O (n) O(tkn) Iterative
K-means Partitional O (n) O (tkn) Iterative; No categorical
Nearest neighbor Partitional O (n2) O(n2) Iterative
PAM Partitional O (n2) O(tk(n - k)2) Iterative; Adapted agglo-
merative; Outliers
BIRCH Partitional O (n) O(n) CF-tree; Incremental;
(no rebuild) Outliers
CURE Mixed O (n) O (n) Heap; k-D tree; Incre-
mental; Outliers;
Sampling
ROCK Agglomerative O (n2) O(n2 lgn) Sampling; Categorical;
Links
DBSCAN Mixed O (n2) O(n2) Sampling; Outliers
The single link, complete link, and average link techniques are all hierarchical tech
niques with O(n2) time and space complexity. While we discussed the agglomerative
versions of these are also divisive versions, which create the clusters in a top-down man
ner. They all assume the data are present and thus are not incremental. There are several
clustering algorithms based on the construction of an MST. There are both hierarchical
and partitional versions. Their complexity is identical to that for the other hierarchical
techniques, and since they depend on the construction of the MST, they are not incre
mental. Both K-means and the squared error techniques are iterative, requiring O (tkn)
time. The nearest neighbor is not iterative, but the number of clusters is not prede
termined. Thus, the worst-case complexity can be O(n2). BIRCH appears to be quite
efficient, but remember that the CF-tree may need to be rebuilt. The time complexity
in the table assumes that the tree is not rebuilt. CURE is an improvement on these by
using sampling and partitioning to handle scalability well and uses multiple points rather
than just one point to represent each cluster. Using multiple points allows the approach
to detect nonspherical clusters. With sampling, CURE obtains an O(n) time complexity.
However, CURE does not handle categorical data well. This also allows it to be more
resistant to the negative impact of outliers. K-means and PAM work by iteratively reas
signing items to clusters, which may not find a global optimal assignment. The results
of the K-means algorithm is quite sensitive to the presence of outliers. Through the use
of the CF-tree, Birch is both dynamic and scalable. However, it detects only spherical
type clusters. DBSCAN is a density-based approach. The time complexity of DBSCAN
can be improved to O(n lgn) with appropriate spatial indices. We have not included
the genetic algorithms in this table because their performance totally depends on the
technique chosen to represent individuals, how crossover is done, and the termination
condition used.
5.9 EXERCISES
1. A major problem with the single link algorithm is that clusters consisting of long
chains may be created. Describe and illustrate this concept.
2. Show the dendrogram created by the single, complete, and average link clustering
algorithms using the following adjacency matrix:
Item A B
A 0 1
B 1 0
c 4 2
D 5 6
C D
4 5
2 6
0 3
3 0
3. Construct a graph showing all edges for the data in Exercise 2. Find an MST for
this graph. Is the MST single link hierarchical clustering the same as that found
using the traditional single link algorithm?
4. Convert Algorithm 5.1 to a generic divisive algorithm. What technique would be
used to split clusters in the single link and complete link versions?
5. Trace the results of applying the squared error Algorithm 5.5 to the data from
Example 5.4 into two clusters. Indicate the convergence threshold you have used.
6. Use the K-means algorithm to cluster the data in Example 5.4 into three clusters.
7. Trace the use of the nearest neighbor algorithm on the data of Exercise 2 assuming
a threshold of 3.
8. Determine the cost Cjih for each of the four cases given for the PAM algorithm.
9. Finish the application of PAM in Example 5.6.
10. (Research) Perform a survey of recently proposed clustering algorithms. Identify
where they fit in the classification tree in Figure 5.2. Describe their approach and
performance.
5.1 0 BIBLIOGRAPHIC NOTES
There are many excellent books examining the concept of clustering. In [JD88], a thor
ough treatment of clustering algorithms, including application domains, and statement
of algorithms, is provided. This work also looks at a different classification of cluster
ing techniques. Other clustering and prediction books include [Har75], [JS71], [SS73],
[TB70], and [WI98].
A survey article of clustering with a complete list of references was published
in 1999 [JMF99]. It covers more clustering techniques than are found in this chapter.
Included are fuzzy clustering, evolutionary techniques, and a comparison of the two.
An excellent discussion of applications is also included. Fuzzy clustering associates a
membership function with every item and every cluster. Imagine that each cluster is

represented as a vector of membership values, one for each element. Other clustering
surveys have been published in [NH94] and [HKT01].
Clustering tutorials have been presented at SIGMOD 1999 [HK99] and
PAKDD-02.1
The agglomerative clustering methods are among the oldest techniques. Proposals
include SLINK [Sib73] for single linkage and CLINK [Def77] for complete linkage. An
excellent study of these algorithms can be found in [KR90]. The AGNES and DIANA
techniques are some of the earliest methods. AGNES (AGglomerative NESting) is agglom
erative, while DIANA (Dlvisia ANAlysis) is divisive. Both are known not to scale well.
Articles on single link clustering date back to 195 1 [FLP+51]. The EM algorithm has
frequently been used to perform interative clustering [DLR77]. There have been many
variations of the K-means clustering algorithm. The earliest reference is to a version
by Forgy in 1965 [For65], [McQ67]. Another approach for partitiomil clustering is to
allow splitting and merging of clusters. Here merging is performed based on the dis
tance between the centroids of two clusters. A cluster is split if its variance is above a
certain threshold! One proposed algorithm performing this is called ISODATA [BH65].
CURE, predominantly a hierarchical technique, was first proposed in [GRS98]. Finding
connected components in a graph is a well-known graph technique that is described in
any graph theory or data structures text such as [Har72] .
The MST partitional algorithm was originally proposed by Zahn [Zha71].
A recent work has looked at extending the definition of clustering to be more
applicable to categorical data. CAtegorical ClusT
ering Using Summaries (CACTUS) gen
eralizes the traditional definition of clustering and distance. To perform the clustering, it
uses summary information obtained from the actual data. The summary information fits
into main memory and can be easily constructed. The definition of similarity between
tuples is given by looking at the support of two attribute values within the database D.
Given two categorical attribute Ai , Aj with domains D; , Dj, respectively, the support
of the attribute pair (a; , aj) is defined as:
(5. 1 8)
The similarity of attributes used for clustering is based on the support [GGR99b].
Several techniques have been proposed to scale up clustering algorithms to work
effectively on large databases. Sampling and data compressions are perhaps the most
common techniques. With sampling, the algorithm is applied to a large enough sample to
ideally produce a good clustering result. Both BIRCH and CACTUS employ compression
techniques. A more recent data compression approach, data bubbles, has been applied to
hierarchical clustering [BKKS01]. A data bubble is a compressed data item that represents
a larger set of items in the database. Given a set of items, a data bubble consists of (a
representative item, cardinality of set, radius of set, estimated average k nearest neighbor
distances in the set). Another compression technique for hierarchical clustering, which
actually compresses the dendrogram, has recently been proposed [XD01b].
A recent article has proposed outlier detection algorithms that scale well to large
datasets and can be used to model the previous statistical tests [KN98].
1 0smar Zaiane and Andrew Foss, "Data Clustering Analysis, from Simple Groupings to Scalable Clustering
with Constraints," Tutorial at Sixth Pacific-Asia Conference on Knowledge Discovery and Data Mining, May 6,
2002.
Section 5. 1 0 Bibl iographic Notes 163
Discussion of the bond energy algorithm and its use can be found in [WTMSW72],
[OV99], and [TF82]. It can be used to cluster attributes based on use and then perform
logical or physical clustering.
DBSCAN was first proposed in [EKSX96]. Other density-based algorithms include
DENCLUE [HK98] and OPTICS [].
ROCK was proposed by Guha in [GRS99].
There have been many approaches targeted to clustering of large databases; includ
ing BIRCH [ZRL96], CLARANS [NH94], CURE [GRS98], DBSCAN [EKSX96], and
ROCK [GRS99]. Specifics concerning CF tree maintenance can be found in the lit
erature [ZRL96]. A discussion of the use of R*-trees in DBSCAN can be found in
[EKSX96]. It is possible that a border point could belong to two clusters. A recent
algorithm, CHAMELEON, is a hierarchical clustering algmithm that bases merging of
clusters on both closeness and their interconnection [KHK99]. When dealing with large
databases, the requirement to fix the number of clusters dynamically can be a major
problem. Recent research into online (or iterative) clustering algorithms has been per
formed. "Online" implies that the algorithm can be performed dynamically as the data
are generated, and thus it works well for dynamic databases. In addition, some work has
examined how to adapt, thus allowing the user to change the number of clusters dynam
ically. These adaptive algorithms avoid having to completely recluster the database if
the users' needs change. One recent online approach represents the clusters by profiles
(such as cluster mean and size). These profiles are shown to the user, and the user has
the ability to change the parameters (number of clusters) at any time during process
ing. One recent clustering approach is both online and adaptive, OAK (online adaptive
clustering) [XDOl b]. OAK can also handle outliers effectively by adjusting a viewing
parameter, which gives the user a broader view of the clustering, so that he or she can
choose his or her desired clusters.
NNs have been used to solve the clustering problem [JMF99]. Kohonen's self
organizing maps are introduced in [Koh82]. One of the earliest algorithms was the leader
clustering algorithm proposed by Hartigan [Har75]. Its time complexity is only O (kn)
and its space is only O(k) . A common NN applied is a competitive one such as an
SOFM where the learning is nonsupervised [JMF99].
References to clustering algorithms based on genetic algorithms include [JB91]
and [BRE91].

C H A P T E R 6
Association Ru les
6.1 INTRODUCTION
6.2 LARGE ITEMSETS
6.3 BASIC ALGORITHMS
6.4 PARALLEL AN D DISTRIBUTED ALGORITHMS
6.5 COMPARING APPROACHES
6.6 INCREMENTAL RULES
6.7 ADVANCED ASSOCIATION RULE TECHNQIUES
6.8 MEASURING THE QUALITY OF RULES
6.9 EXERCISES ,
6.1 0 BIBLIOGRAPHic NOTES
6.1 INTRODUCTION
The purchasing of one product when another product is purchased represents an asso
ciation rule. Association rules are frequently used by retail stores to assist in marketing,
advertising, floor placement, and inventory control. Although they have direct applicability
to retail businesses, they have been used for other purposes as well, including predicting
faults in telecommunication networks. Association rilles are used to show the relationships
between data items. These uncovered relationships are not iriherent in the data, as with
functional dependencies, and they do not represent any sort of causality or correlation.
Instead, association rules detect common usage of items. Example 6.1 illustrates this.
EXAMPLE 6.1
A grocery store chain keeps a record of weekly transactions where each transaction
represents the items bought during one cash register transaction. The executives of the
chain receive a summarized report of the transactions indicating what types of items have
sold at what quantity. In addition, they periodically request information about what items
are commonly purchased together. They find that 100% of the time that PeanutButter
is purchased, so is Bread. In addition, 33.3% of the time PeimutButter is purchased,
Jelly is also purchased. However, PeanutButter exists in only about 50% of the overall
transactions.
A database in which an association rule is to be found is viewed as a set
of tuples, where each tuple contains a set of items. For example, a tuple could be
{PeanutButter, Bread, Jelly}, which consists of the three items: peanut butter, bread, and
164
.....
Section 6.1 Introduction 165
jelly. Keeping grocery story cash register transactions in mind, each item represents
an item purchased, while each tuple is the list of items purchased at one time.
In the simplest cases, we are not interested in quantity or cost, so these may be
removed from the records before processing. Table 6.1 is used throughout this chapter
to illustrate different algorithms. Here there are five transactions and five items:
{Beer, Bread, Jelly, Milk, PeanutButter}. Throughout this chapter we list items in alpha
betical order within a transaction. Although this is not required, algorithms often assume
that this sorting is done in a preprocessing step.
The support of an item (or set of items) is the percentage of transactions in which
that item (or items) occurs. Table 6.2 shows the support for all subsets of items from our
total set. As seen, there is an exponential growth in the sets of items. In this case we
could have 31 sets of items from the original set of five items (ignoring the empty set).
This explosive growth in potential sets of items is an issue that most association
TABLE 6. 1 : Sample Data to Illustrate Association Rules
Transaction Items
Bread, Jelly, PeanutButter
Bread, PeanutButter
Bread, Milk, PeanutButter
Beer, Bread
Beer, Milk
TABLE 6.2: Support of All Sets of Items Found in Table 6. 1
Set Support Set
Beer 40 Beer, Bread, Milk
Bread 80 Beer, Bread, PeanutButter
Jelly 20 Beer, Jelly, Milk
Milk 40 Beer, Jelly, PeanutButter
PeanutButter 60 Beer, Milk, PeanutButter
Beer, Bread 20 Bread, Jelly, Milk
Beer, Jelly 0 Bread, Jelly, PeanutButter
Beer, Milk 20 Bread, Milk, PeanutButter
Beer, PeanutButter 0 Jelly, Milk, PeanutButter
Bread, Jelly 20 Beer, Bread, Jelly, Milk
Bread, Milk 20 Beer, Bread, Jelly, PeanutButter
Bread, PeanutButter 60 Beer, Bread, Milk, PeanutButter
Jelly, Milk 0 Beer, Jelly, Milk, PeanutButter
Jelly, PeanutButter 20 Bread, Jelly, Milk, PeanutButter
Milk, PeanutButter 20 Beer, Bread, Jelly, Milk, PeanutButter
Beer, Bread, Jelly 0
Support
0
0
0
0
0
0
20
20
0
0
0
0
0
0
0

1 66 Chapter 6 Association Rules
rule algorithms must contend with, as the conventional approach to generating association
rules is in actuality counting the occurrence of sets of items in the transaction database.
Note that we are dealing with categorical data. Given a target domain, the under
lying set of items usually is known, so that an encoding of the transactions could be
performed before processing. As we will see, however, association rules can be applied
to data domains other than categorical.
DEFINITION 6. 1. Given a set of items I = U1 , h . . . , Im} and a database of
transactions D = {tJ , tz, . . . , tn} where ti = Ui ! , liz• . . . , Iik} and liJ E /, an
association rule is an implication of the form X =} Y where X, Y c I are sets of
items called itemsets and X n Y = 0.
DEFINITION 6.2. The support (s) for an association rule X ::::} Y is the percentage
of transactions in the database that contain X U Y.
DEFINITION 6.3. The confidence or strength (a) for an association rule X ::::} Y
is the ratio of the number of transactions that contain X U Y to the number of
transactions that contain X.
A formal definition, from [AIS93], is found in Definition 6.1. We generally are not
interested in all implications but only those that are important. Here importance usually
is measured by two features called support and con
fidence as defined in Definitions
6.2 and 6.3. Table 6.3 shows the support and confidence for several association rules,
including those from Example 6.1.
The selection of association rules is based on these two values as described in
the definition of the association rule problem in Definition 6.4. Confidence measures the
strength of the rule, whereas support measures how often it should occur in the database.
TYpically, large confidence values and a smaller support are used. For example, look at
Bread ::::} PeanutButter in Table 6.3. With a = 75%, this indicates that this rule holds
75% of the time that it could. That is, 3/4 times that Bread occurs, so does PeanutButter.
This is a stronger rule than Jelly ::::} Milk because there are no times Milk is purchased
when Jelly is bought. An advertising executive probably would not want to base an
advertising campaign on the fact that when a person buys Jelly he also buys Milk.
Lower values for support may be allowed as support indicates the percentage of time
the rule occurs throughout the database. For example, with Jelly ::::} PeanutButter, the
confidence is 100% but the support is only 20%. It may be the case that this association
TABLE 6.3: Support and Confidence for Some Association Rules
X =} Y s a
Bread ==? PeanutButter 60% 75%
PeanutButter ==? Bread 60% 100%
Beer ::::} Bread 20% 50%
PeanutButter ::::} Jelly 20% 33.3%
Jelly ::::} PeanutButter 20% 100%
Jelly ::::} Milk 0% 0%
6.2
Section 6.2 Large ltemsets 167
rule exists only in 20% of the transactions, but when the antecedent Jelly occurs, the
consequent always occurs. Here an advertising strategy targeted to people who purchase
Jelly would be appropriate.
The discussion so far has centered around the use of association rules in the market
basket area. Example 6.2 illustrates a use for association rules in another domain: telecom
munications. This example, although quite simplified from the similar real-world problem,
illustrates the importance of association rules in other domains and the fact that support
need not always be high.
EXAMPLE 6.2
A telephone company must ensure that a high percentage of all phone calls are made
within a certain period of time. Since each phone call must be routed through many
switches, it is imperative that each switch work correctly. The failure of any switch
could result in a call not being completed or being completed in an unacceptably long
period of time. In this environment, a potential data mining problem would be to predict
a failure of a node. Then, when the node is predicted to fail, measures can be taken by
the phone company to route all calls around the node and replace the switch. To this
end, the company keeps a history of calls through a switch. Each call history indicates
the success or failure of the switch, associated timing, and error indication. The history
contains results of the last and prior traffic through the switch. A transaction of the type
(success, failure) indicates that the most recent call could not be handled successfully,
while the call before that was handled fine. Another transaction (ERRl , failure) indicates
that the previous call was handled but an error occurred, ERRl . This error could be
something like excessive time. The data mining problem can then be stated as finding
association rules of the type X ::::} Failure. If these types of rules occur with a high
confidence, we could predict failure and immediately take the node off-line. Even though
the support might be low because the X condition does not frequently occur, most often
when it occurs, the node fails with the next traffic.
DEFINITION 6.4. Given a set of items I = U1 , /z, . . . , Im } and a database of
transactions D = {tJ , tz, . . . , tn} where ti = {Ii i , liz, . . . , Iik} and Iij E /, the
association rule problem is to identify all association rules X =} Y with a mini
mum support and confidence. These values (s, a) are given as input to the problem.
The efficiency of association rule algorithms usually is discussed with respect to the
number of scans of the database that are required and the maximum number of itemsets
that must be counted.
LARGE ITEMSETS
The most common approach to finding association rules is to break up the problem into
two parts:
1. Find large itemsets as defined in Definition 6.5.
2. Generate rules from frequent itemsets.
An itemset is any subset of the set of all items, I.

DEFINITION 6.5. A large (frequent) itemset is an itemset whose number of
occurrences is above a threshold, s. We use the notation L to indicate the complete
set of large itemsets and l to indicate a specific large itemset.
Once the large itemsets have been found, we know that any interesting association
rule, X ::::} Y, must have X U Y in this set of frequent itemsets. Note that the subset of any
large itemset is also large. Because of the large number of notations used in association
rule algorithms, we summarize them in Table 6.4. When a specific term has a subscript,
this indicates the size of the set being considered. For example, lk is a large itemset of
size k. Some algorithms divide the set of transactions into partitions. In this case, we use
p to indicate the number of partitions and a superscript to indicate which partition. For
example, Di is the i1h partition of D.
Finding large itemsets generally is quite easy but very costly. The naive approach
would be to count all itemsets that appear in any transaction. Given a set of items of size
m, there are 2m subsets. Since we are not interested in the empty set, the potential number
of large itemseti is then 2m - 1 . Because of the explosive growth of this number, the
challenge of solving the association rule problem is often viewed as how to efficiently
determine all large itemsets. (When m = 5, there are potentially 31 itemsets. When
m = 30, this becomes 1073741 823.) Most association rule algorithms are based on
smart ways to reduce the number of itemsets to be counted. These potentially large
itemsets are called candidates, and the set of all counted (potentially large) itemsets is
the candidate itemset (c). One performance measure used for association rule algorithms
is the size of C. Another problem to be solved by association rule algorithms is what
data structure is to be used during the counting process. As we will see, several have
been proposed. A trie or hash tree are common.
When all large itemsets are found, generating the association rules is straightfor
ward. Algorithm 6. 1, which is modified from [AS94], outlines this technique. In this
algorithm we use a function support, which returns the support for the input itemset.
TABLE 6.4: Association Rule Notation
Term Description
D Database of transactions
ti Transaction in D
s Support
a Confidence
X, Y Itemsets
X ::::} Y Association rule
L Set of large itemsets
l Large itemset in L
c Set of candidate itemsets
p Number of partitions
ALGORITHM 6.1
Input :
D
I
L
s
a
Output :
//Database of transactions
I / I tems
/ /Large itemsets
/ /Support
//Conf idence
Section 6.3
R //As sociat ion Rules satisfying s and a
ARGen algori thm:
R = 0 ;
for each 1 E L do
for each x c 1 such that x i= 0 do
1
.
f
support(1) ::>
h
support(x) - a t en
R = R U {x ==> (1 - x)} ;
Basic Algorithms 169
To illustrate this algorithm, again refer to the data in Table 6.1 with associated
supports shown in Table 6.2. Suppose that the input support and confidence are s = 30%
and a = 50%, respectively. Using this value of s, we obtain the following set of large
itemsets:
L = {{Beer}, {Bread}, {Milk}, {PeanutButter}{Bread, PeanutButter}} .
We now look a t what association rules are generated from the last large itemset. Here l =
{Bread, PeanutButter}. There are two nonempty subsets of l: {Bread} and {PeanutButter}.
With the first one we see:
support({Bread, PeanutButter})
=
60
= 0.75
support({Bread}) 80
This means that the confidence of the association rule Bread ::::} PeanutButter is 75%,
just as is seen in Table 6.3. Since this is above a, it is a valid association rule and is
added to R. Likewise with the second large itemset
support({Bread, PeanutButter}) = 60 =
1
support({PeanutButter}) 60
This means that the confidence of the association rule PeanutButter ::::} Bread is 100%,
and this is a valid association rule.
All of the algorithms discussed in subsequent sections look primarily at ways to
efficiently discover large itemsets.
6.3 BASIC ALGORITHMS
6.3.1 Apriori Algorithm
The Apriori algorithm is the most well known association rule algorithm and is used
in most commercial products. It uses the following property, which we call the large
itemset property:
Any subset of a large itemset must be large.

The large itemsets are also said to be downward closed because if an itemset satisfies
the minimum support requirements, so do all of its subsets. Looking at the contrapos
itive of this, if we know that an itemset is small, we need not generate any super
sets of it as candidates because they also must be small. We use the lattice shown in
Figure 6.1(a) to illustrate the concept of this important property. In this case there are
four items {A, B, C, D}. The lines in the lattice represent the subset relationship, so the
large itemset property says that any set in a path above an itemset must be large if the
original itemset is large. In Figure 6.1 (b) the nonempty subsets of ACD1 are seen as
{AC, AD, CD, A, C, D}. If ACD is large, so is each of these subsets. If any one of these
subsets is small, then so is ACD.
the basic idea of the Apriori algorithm is to generate candidate itemsets of a
particular size and then scan the database to count these to see if they are large. During
scan i, candidates of size i, C; are counted. Only those candidates that are large are used
to generate candidates for the next pass. That is L; are used to generate C;+l · An itemset
is considered as a candidate only if all its subsets also are large. To generate candidates
of size i + 1, joins are made of large itemsets found in the previous pass. Table 6.5 shows
the process using the data found in Table 6.1 with s = 30% and a = 50%. There are no
candidates of size three because there is only one large itemset of size two.
An algorithm called Apriori-Gen is used to generate the candidate itemsets for each
pass after the first. All singleton itemsets are used as candidates in the first pass. Here
the set of large itemsets of the previous pass, L;- I , is joined with itself to determine
the candidates. Individual itemsets must have all but one item in common in order to be
combined. Example 6.3 further illustrates the concept. After the first scan, every large
itemset is combined with every other large itemset.
A B c D
AB AC AD BC BD CD
ABC ABD ACD BCD
ABCD
(a) Lattice of itemsets for (A, B, C, D)
A B
AB AC AD BC
c
ABC ABD ACD
ABCD
(b) Subsets ofACD
FIGURE 6. 1 : Downward closure.
D
BD CD
BCD
IFollowing the usual convention with association rule discussions, we simply list the items in the set rather
than using the traditional set notation. So here we use ACD to mean {A, C, D).
Pass
2
Transaction
!1
!2
t3
!4
ts
!6
!7
ts
tg
!JO
Section 63 Basic Algorithms 171
TABLE 6.5: Using Apriori with Transactions in Table 6.1
Candidates
{Beer}, {Bread}, {Jelly},
{Milk}, {PeanutButter}
{Beer, Bread}, {Beer, Milk},
Large Itemsets
{Beer}, {Bread},
{Milk}, {PeanutButter}
{Bread, PeanutButter}
{Beer, PeanutButter}, {Bread, Milk},
{Bread, PeanutButter}, {Milk, PeanutButter}
TABLE 6.6: Sample Clothing Transactions
Items Transaction Items
Blouse tu TShirt
Shoes, Skirt, TShirt !J2 Blouse, Jeans, Shoes, Skirt, TShirt
Jeans, TShirt !13 Jeans, Shoes, Shorts, TShirt
Jeans, Shoes, TShirt !]4 Shoes, Skirt, TShirt
Jeans, Shorts tls Jeans, TShirt
Shoes, TShirt !J6 Skirt, TShirt
Jeans, Skirt !J7 Blouse, Jeans, Skirt
Jeans, Shoes, Shorts, TShirt fJS Jeans, Shoes, Shorts, TShirt
Jeans !J9 Jeans
Jeans, Shoes, TShirt t2o Jeans, Shoes, Shorts, TShirt
EXAMPLE 6.3
A woman's clothing store has 10 cash register transactions during one day, as shown in
Table 6.6. When Apriori is applied to the data, during scan one, we have six candidate
itemsets, as seen in Table 6.7. Of these, 5 candidates are large. When Apriori-Gen is
applied to these 5 candidates, we combine every one with all the other 5. Thus, we
get a total of 4 + 3 + 2 + 1 = 10 candidates during scan two. Of these, 7 candidates
are large. When we apply Apriori-Gen at this level, we join any set with another set
that has one item in common. Thus, {Jeans, Shoes} is joined with {Jeans, Shorts} but
not with {Shorts, TShirt}. {Jeans, Shoes} will be joined with any other iteinset con
taining either Jeans or Shoes. When it is joined, the new item is added to it. There
are four large itemsets after scan four. When we go to join these we must match on
two of the three attributes. For example {Jeans, Shoes, Shorts} After scan four, there
is only one large itemset. So we obtain no new itemsets of size five to count in the
next pass. joins with {Jeans, Shoes, TShirt} to yield new candidate {Jeans, Shoes, Shorts,
TShirt}.
The Apriori-Gen algorithm is shown in Algorithm 6.2. Apriori-Gen is guaranteed
to generate a superset of the large itemsets of size i, C; :::> L;, when input L;_ 1 , A

172 Chapter 6 Association Rules
TABLE 6.7: Apriori-Gen Example
Scan
2
3
4
5
Candidates
{Blouse}, {Jeans}, {Shoes},
{Shorts}, {Skirt}, {TShirt}
{Jeans, Shoes}, {Jeans, Shorts}, {Jeans, Skirt},
{Jeans, TShirt}, {Shoes, Shorts}, {Shoes, Skirt},
{Shoes, TShirt}, {Shorts, Skirt}, {Shorts, TShirt},
{Skirt, TShirt}
{Jeans, Shoes, Shorts}, {Jeans, Shoes, TShirt},
{Jeans, Shorts, TShirt}, {Jeans, Skirt, TShirt},
{Shoes, Shorts, TShirt}, {Shoes, Skirt, TShirt},
{Shorts, Skirt, TShirt}
{Jeans, Shoes; Shorts, TShirt}
0
1
Large ltemsets
{Jeans}, {Shoes}, {Shorts}
{Skirt}, {Tshirt}
{Jeans, Shoes}, {Jeans, Shorts},
{Jeans, TShirt}, {Shoes, Shorts},
{Shoes, TShirt}, {Shorts, TShirt},
{Skirt, TShirt}
{Jearts, Shoes, Shorts},
{Jeans, Shoes, TShirt},
{Jeahs, Shorts, TShirt},
{Shoes, Shorts, TShirt}
{Jeans, Shoes, Shorts, TShirt}
0
pruning step, not shown, could be added at the end of this algorithm to prune away any
candidates that have subsets of size i - 1 that are not large.
ALGORITHM 6.2
Input :
Li_1 / /Large itemsets of s i z e i - 1
Output :
Ci / /Candidates of s i ze i
Apriori -gen algorithm:
ci = 0 ;
for each I E Li-1 do
for each J -1 I E Li-1 do
if i - 2 of the element s in I and J are equal then
Ck = Ck U {I U J} ;
Given the large itemset property and Apriori-Gen, the Apriori algorithm itself (see
Algorithm 6.3) is rather straightforward. In this algorithm we use Ci to be the count for
item /i E /.
ALGORITHM 6.3
Input :
I
D
s
Output :
L
/ / Items ets
/ /Database of transactions
/ /Support
/ /Large itemsets
Apriori algorithm:
k = 0 ; I Ik is used as the s can number .
L = 0 ;
Section 6.3 Basic Algorithms 1 73
C1 = I; / / Initial candidates are set to be the items .
repeat
k = k + l ;
Lk = 0 ;
for each Ii E Ck do
ci = 0 ; II Initial counts for each items et are o .
for each tj E D do
for each Ii E Cx do
if Ii E tj then
Ci = Ci + 1 ;
for each Ii E Ck do
if ci 2: (sx I D I) do
Lk = Lk U Ii ;
L = L U Lk ;
Ck+1 = Aprior i - Gen(Lk)
until Ck+1 = 0;
The Apriori algorithm assumes that the database is memory-resident. The maximum
number of database scans is one more than the cardinality of the largest large itemset.
This potentially large number of database scans is a weakness of the Apriori approach.
6.3.2 Sampling Algorithm
To facilitate efficient counting of itemsets with large databases, sampling ofthe database
_ may be used. The original sampling algorithm reduces the number of database scans
to one in the best case and two in the worst case. The database sample is drawn such
that it can be memory-resident. Then any algorithm, such as Apriori, is used to find the
large itemsets for the sample. These are viewed as potentially large (PL) itemsets and
used as candidates to be counted using the entire database. Additional candidates are
determined by applying the negative border function, BD- , against the large itemsets
from the sample. The entire set of candidates is then C = BD- (PL) U PL. The negative
border function is a generalization of the Apriori-Gen algorithm. It is defined as the
minimal set of itemsets that are not in PL, but whose subsets are all in PL. Example 6.4
illustrates the idea.
EXAMPLE 6.4
Suppose the set of items is {A, B, C, D}. The set of large itemsets found to exist in a
sample of the database is PL = {A, C, D, CD}. The first scan of the entire database,
then, generates the set of candidates as follows: C = BD -(PL) U PL = {B, AC , AD} U
{A, C, D, CD}. Here we add AC because both A and C are in PL. Likewise we add AD .
We could not have added ACD because neither AC nor AD is in PL. When looking at
the lattice (Figure 6.2), we add only sets where all subsets are already in PL. Note that
we add B because all its subsets are vacuously in PL.
Algorithm 6.4 shows the sampling algorithm. Here the Apriori algorithm is shown
to find the large itemsets in the sample, but any large itemset algorithm could be used.
Any algorithm to obtain a sample of the database could be used as well. The set of large

<P <P
� �
A B C D A B C D
� �
AB AC AD BC BD CD AB AC AD BC BD CD
� �
ABC ABD ACD BCD ABC ABD ACD BCD
� �
ABCD ABCD
(a) PL (b) PL U BD-(PL)
FIGURE 6.2: Negative border.
t
itemsets is used as a set of candidates during a scan of the entire database. If an itemset
is large in the sample, it is viewed to be potentially large in the entire database. Thus,
the set of large itemsets from the sample is called PL. In an attempt to obtain all the
large itemsets during the first scan, however, PL is expanded by its negative border. So
the total set of candidates is viewed to be C = PLU BD-(PL).
During the first scan of the database, all candidates in C are counted. If all candi
dates that are large are in PL [none in BD-(PL)], then all large itemsets are found. If,
however, some large itemsets are in the negative border, a second scan is needed. Think
of Bv- (PL) as a buffer area on the border of large itemsets. The set of all itemsets is
divided into four areas: those that are known to be large, those that are known to be
small, those that are on the negative border of those known to be large, and the others.
The negative border is a buffer zone between those known to be large and the others. It
represents the smallest possible set of itemsets that could potentially be large. Because
of the large itemset property, we know that if there are no large itemsets in this area,
then there can be none in the rest of the set.
During the second scan, additional candidates are generated and counted. This is
done to ensure that all large itemsets are found. Here ML, the missing large itemsets,
are those in L that are not in PL. Since there are some large itemsets in ML, there may
be some in the rest of the set of itemsets. To find all the remaining large itemsets in the
second scan, the sampling algorithm repeatedly applies the negative border function until
the set of possible candidates does not grow further. While this creates a potentially large
set of candidates (with many not large), it does guarantee that only one more database
scan is required.
ALGORITHM 6.4
Input :
I
D
s
Output :
L
//Itemsets
//Database of transactions
//Support
//Large itemsets
Sampl ing algorithm :
Ds = Sample drawn from D;
PL = Apriori(I, Ds, smalls) ;
C = PL U BD- (PL) ;
L = 0 ;
for each Ii E C do
Section 6.3 Basic Algorithms 175
Ci = 0 ; II Initial counts for each itemset are 0 ;
for each tj E D do II First scan count .
for each Ii E C do
if Ii E tj I then
Ci = Ci + 1 ;
for each Ii E C do
if ci ;::: (sx I D I) do
L = L U ii ;
//Missing large itemsets .
ML = {x I x E BD- (PL) 1 x E L} ;
if ML =f. 0 1 then
II Set candidates to be the large itemsets .
C = L;
repeat
C= C U BD- (C) ; II Expand candidate sets
using negative border.
unt i l no new itemsets are added to C;
for eacll Ii E C do
ci = 0 ; II Initial counts for each itemset are 0 .
for each tj E D do II Second scan count .
for each Ii E C do
if Ii E tj 1 then
Ci = Ci + 1 ;
i f Ci ;::: (sx I D I) do
L = L U ii ;
The algorithm shows that the application of the Apriori algorithm to the sample is
performed using a support called smalls. Here smalls can be any support values less than
s. The idea is that by reducing the support when finding large itemsets in the sample, more
of the true large itemsets from the complete database will be discovered. We illustrate
the use of the sampling algorithm on the grocery store data in Example 6.5.
EXAMPLE 6.5
We use the sampling algorithm to find all large itemsets in the grocery data where
s = 20%. Suppose that the sample database is determined to be the first two transactions:
Ds = {t1 = {Bread, Jelly, PeanutButter}, tz = {Bread, PeanutButter}}
If we reduce s to be smalls = 10%, then for an itemset to be large in the sample it must
occur in at least 0. 1 x 2 transactions. So it must occur in one of the two transactions.
When we apply Apriori to Ds we get:
PL = {{Bread}, {Jelly}, {PeanutButter}, {Bread, Jelly}, {Bread, PeanutButter},
{Jelly, PeanutButter}, {Bread, Jelly, PeanutButter}}

When we apply the negative border, we get
BD-(PL) = {{Beer}, {Milk}}
We thus use the following set of candidates to count during the first database scan:
PL = {{Bread}, {Jelly}, {PeanutButter}, {Bread, Jelly}, {Bread, PeanutButter},
{Jelly, PeanutButter} , {Bread, Jelly, PeanutButter}, {Beer} , {Milk}}
Remember that during this scan we use s = 20% and apply it against all five transactions
in the entire database. For an itemset to be large, then, we must have an itemset in 20% x 5
or at least one transaction. We then find that both {Beer} and {Milk} are large. Thus,
ML = { {Beer}, {Milk}}. Following the algorithm, we first set C = L, which in this case
is also PL. Applying the negative border, we get
C = BD-(C) = {{Beer, Bread}, {Beer, Jelly}, {Beer, Milk} , {Beer, PeanutButter} ,
{Bre�d, Milk}, {Jelly, Milk}, {Milk, PeanutButter}, {Jelly, Milk}}
Since this has uncovered new itemsets, we again apply it and this time find all itemsets
of size three. A last application then finds all itemsets of size four, and we scan the
database using all remaining itemsets not already known to be large.
Example 6.5 illustrates a potential problem with the use of the sampling algorithm;
that is, a very large set of candidates may be used during the second scan. This is required
to ensure that all large itemsets are found during the second scan. However, the set of
candidates generated by successive applications of the negative border function will not
always generate the entire set of itemsets. This happened in Example 6.5 because we
found that all itemsets in PL were large and all itemsets in BD- (PL) also were large.
If instead of using a support of 20%, we had used one of 40%, the results would be
different, as shown in Example 6.6.
EXAMPLE 6.6
Suppose that smalls and Ds are as were used in Example 6.5. We thus find that PL and
Bv- (PL) are the same. During the first scan of the entire database, we identify large
itemsets only if they have a support of at least 40%. Looking at Table 6.2, we see that
from the initial scan we obtain the following large itemsets:
L = {{Bread}, {PeanutButter}, {Bread, PeanutButter} , {Beer}, {Milk}}
Here ML = {{Beer} , {Milk}}, so we need a second scan. When we first apply the negative
border we get
C = BD-(C) = { {Beer, Bread} , {Beer, Milk}, {Beer, PeanutButter} ,
{Bread, Milk}, {Milk, PeanutButter}}
Section 6.3 Basic Algorithms 177
Note that this is smaller than what we found with Example 6.5. Si
.
nce
.
Jelly is �ssing,
we will not generate the entire set of itemsets during repeated apphcatwn of BD . The
second application yields
{{Beer, Bread, Milk}, {Beer, Bread, PeanutButter},
{Beer, Milk, PeanutButter}, {Bread, Milk, PeanutButter}}
The final application obtains
C = BD- (C) = {{Beer, Bread, Milk, PeanutButter}}
6.3.3 Partitioning
Various approaches to generating large itemsets have �ee� �
i'opo
.
sed based
.
o
�
a
partitioning of the set of transactions. In this case, D 1s dtvtded
.
mto p P
:U
tttlons
DL , D2, . . . , DP. Partitioning may improve the performan,�e of findmg large 1temsets
in several ways:
• By taking advantage of the large itemset property, we know that a large it
�
m
set must be large in at least one of the partitions. This idea can help to destgn
algorithms more efficiently than those based on looking at the entire database.
• Partitioning algorithms may be able to adapt better to limited rna��
me
r_n
ory. Each
partition can be created such that it fits into main memory. �
� .
addttwn, 1t would be
expected that the number of itemsets to be counted per partttlon would be smaller
than those needed for the entire database.
• By using partitioning, parallel and/or distributed algorithm
�
can be easily created,
where each partition could be handled by a separate machine.
• Incremental generation of association rules may be easier to perform by �
eating
the current state of the database as one partition and treating the new entnes as a
second partition.
The basic partition algorithm reduces the number of databas
�
scans
.
to
.
two and
divides the database into partitions such that each can pe placed 1
�
to mat
�
memory.
When it scans the database, it brings that partition of the database mto mam me�
ory
and counts the items in that partition alone. During the first database scan, the algonthr_n
finds all large itemsets in each partition. Although ·any algorithm could be used for
.
thi
.
s
purpose, the original proposal assumes that some leve�-
�
ise �pproa�
h, such as Apnon,
is used. Here Li represents the large itemsets from partition D . Dunng the se
�
ond scan,
only those itemsets that are large in at least one p�ition are used as
.
candidates and
counted to determirte if they are large across the entrre database. Algonthm 6.5 shows
this basic partition algorithm. ·

nt
It Bread, Jelly, PeanutButter L1 = {{Bread]. {Jelly), {PeanutButterj, {Bread,Jelly]. {Bread, PeanutButter].
12 Bread, PeanutBptter {Jelly, PeanutButterj, {Bread,Jelly, PeanutButteriJ
13 Bread, Milk, PeanutButter
D2
/4 Beer, Bread
L2 = {{Beer), {Bread]. {Milk]. {PeanutButterj, {Beer, Bread), {B
[Bread, Milk), {Bread, PeanutButterj, {Milk, PeanutBu
eer,Milkj.
tterj,
Is Beer, Milk
{Bread, Milk,PeanutButterjj
FIG URE 6.3: Partitioning example.
ALGORITHM 6.5
Input :
I / / I temsets
D = {D1
, v2, . . . , .oP} I /Database of t ransactions divided
s / /Support
Output :
L //Large itemsets
Partition algorithm:
C= 0 ; 1
into partitions
for i = 1 to p do I/Find l arge i t emsets in each partition .
L
i
= Apriori(I, D
i
, s) ;
C = C U L
i
;
L = 0 ;
for each Ii E C do
ci = 0 ; / / Initial counts for each itemset are 0 .
for each t;;j E D do II Count candidates during second scan .
for each Ii E C do
i f Ii E tj , then
ci = ci + 1 ;
for each Ii E C do
if ci 2: (sx I D I) do
L = L U ii i
Figure 6.3 illustrates the use of the partition algorithm using the market basket data.
Here the database is partitioned into two parts, the first containing two transactions and
the second with three transactions. Using a support of 10%, the resulting large itemsets
L 1 and L2 are shown. If the items are uniformly distributed across the partitions, then a
large fraction of the itemsets will be large. However, if the data are not uniform, there
may be a large percentage of false candidates.
6.4 PARALLEL AND DISTRIBUTED ALGORITHMS
Most parallelor distributed association rule algorithms strive to parallelize either the data,
known as dataparallelism, or the candidates, referred to !iS taskparallelism. With task par
allelism, thecandidates are partitioned and counted sep!ifately at each processor. Obviously,
the partition algorithm would be easy to parallelize using the task parallelism approach.
Other dimensions in differentiating different parallel association rule algorithms are the
load-balancing approach used and the architecture. The data parallelism algorithms have
reduced communication cost over the task, because only the initial candidates (the set of
items) and the local counts at each iteration must be distributed. With task parallelism,
not only the candidates but also the local set of transactions must be broadcast to all other
sites. However, the data parallelism algorithms require that memory at each processor be
Section 6.4 Parallel and D istributed Algorithms 1 79
large enough to store all candidates at each scan (otherwise the performance will degrade
considerably because 1/0 is required for both the database and the candidate set). The task
parallelism approaches can avoid this because only the subset of the candidates that are
assigned to a processor during each scan must fit into memory. Since not all partitions of
the candidates must be the same size, the task parallel algorithms can adapt to the amount
of memory at each'site. The only restriction is that the total size of all candidates be small
enough to fit into the total size of memory in all processors combined. Note that there are
some variations of the•basic algorithms discussed in this section that address these memory
issues. Performance studies have shown that the data parallelism tasks scale linearly with
the number of processors and !lie database size. Because of the reduced memory require
ments, however, the tast p<jfallelism may work where data parallelism may not work.
6.4.1 Data Parallelism
One data parallelism algorithm is the count distribution algorithm (CDA). The dat
_
abase
is divided into p partitions, one for each processor. Each processor counts the candidates
for its data and then broadcasts its counts to all other processors. Each processor then
determines the global counts. These then are used to determine the large itemsets and to
generate the candidates for the next scan. The algprithm is shown in Algorithm 6.6.
ALGORITHM 6.6
Input :
I / / I temsets
pl , p2 , . . . , PP ; / / Processors
D = D1 , D2 , . . . , .oP;
s I/Support
//Database divided into partitions
Output :
L · //Large itemsets
Count dis tributio� algorithm:
1
perform in parallel at each processor P
k = 0 ; I Ik i s used as the s can
L = 0 ;
.
//Count in paral lel .
number .
c1 = I ; // Ini t ial candidates are s e t to be the items .
repeat
k = k + l ;
Lk = 0 ;
for each Ii E Ck do
ci = 0 ; I / Init ial counts for each i temset are 0 .
for each tj E D1 do
for each Ii E CK do
i:fi Ii E t� then
ci = ci + 1 ;
broadcast c� to all other proce ssors ;
� .
for each Ii E Ck do I/Determ�ne global counts .
·
' p 1
Ci = Ll=l ci i
for each Ii E Ck dp
if ci 2: (sx I D1 U v2 U · · · U nP I) do
Lk = Lk U Ii ;
L = L U Lk i
Ck+l = Apriori -Gen(Lk)
until Ck+l = 0 ;

pl p2 p3
nt: Dz: D3:
ft, tz t3, t4 ts
Counts: Counts: Counts:
Beet O Beer l Beer l
Bread 2 Bread 2 Bread 0
Jelly 1 Jelly O Jelly 0
Milk O Milk l Milk l
PeanutButter 2
Broadcast Local Counts
FIGURE 6.4: Data parallelism using CDA (modified from [DXGHOO]).
Figure 6.4, which is modified from [DXGHOO], illustrates the approach used by the
CDA algorithm using the grocery store data. Here there are three processors. The first
two transactions are counted at P1 , the next two at P2, and the last one at P3. When the
local counts are obtained, they are then broadcast to the other sites so that global counts
can be generated.
6.4.2 Task Parallelism
The data distribution algorithm (DDA) demonstrates task parallelism. Here the candi
dates as well as the database are partitioned among the processors. Each processor in
parallel cmmts the candidates given to it using its local database partition. Following our
convention, we use c£ to indicate the candidates of size k examined at processor P1.
Also, Li are the local large k-itemsets at processor l. Then each processor broadcasts
its database prutition to all other processors. Each processor then uses this to obtain a
global count for its data and broadcasts this count to all other processors. Each processor
then can determine globally large itemsets and generate the next candidates. These candi
dates then are divided among the processors for the next scan. Algorithm 6.7 shows this
approach. Here we show that the candidates are actually sent to each processor. How
ever, some prearranged technique could be used locally by each processor to determine
its own candidates. This algorithm suffers from high message traffic whose impact can
be reduced by overlapping communication and processing.
ALGORITHM 6.7
Input :
I //Itemsets
p
l
, p
2
, . . . , PP; //Processors
D = D1 , D2 , . . . , IJP ;
s //Support
Output :
L //Large itemsets
Data dis tribution algorithm :
//Database divided into partitions
Section 6.5 Com paring Approaches 181
for each 1 ::; 1 ::; p do //Distribute size 1 candidates
to each processor.
determine ci and distribute to P1 ;
perform in parallel at each processor P1 //Count in parallel .
k = 0 ; IIk is used as the scan number .
L = 0;
repeat
k = k+ 1 ;
Lfc = 0 ;
for each Ii E cfc do
ci = 0 ; I/Initial counts for each itemset are 0 .
for each tj E D1 do
for each Ii E C� do
if Ii E tj I then
ci = ci + 1 ;
broadcast D1 t o all other
for every other processor
for each tj E rf'l do
for each Ii E ci do
i f Ii E tj I then
ci = ci + 1 ;
//Determine local counts .
processors ;
m =f: 1 do
//Determine
global counts .
if ci ::: (sx I D1 U D2 U . . · U vP I) do
Lfc = Lfc U Ii i
broadcast Lfc to all other processors ;
Lk = Lk U L� U . . · U L�; //Global large
Ck+l = Apriori-gen(Lk)
c�+l c ck+l ;
until cfc+l = 0 ;
k-itemsets .
//Determine next set of
local candidates .
Figure 6.5, which is modified from [DXGHOO], illustrates the approach used by
the DDA algorithm using the grocery store data. Here there are three processors. P 1 is
counting Beer and Bread, P2 is counting Jelly and Milk, and P3 is counting PeanutButter.
The first two transactions initially are counted at P1 , the next two at P2, and the last
one at P3• When the local counts are obtained, the database paititions are then broadcast
to the other sites so that each site can obtain a global count.
6.5 COMPARING APPROACHES
Although we have covered only the major association rule algorithms that have been
proposed, there have been many such algorithms (see Bibliography). Algorithms can be
classified along the following dimensions [DXGHOO]:
• Target: The algorithms we have examined generate all rules that satisfy a given
support and confidence level. Alternatives to these types of algorithms are those
that generate some subset of the algorithms based on the constraints given.

182 Chapter 6
pl
Counts:
Beer O
Bread 2
Association Rules
p2
Counts:
Jelly 0
Milk 1
Broadcast Database Partition
p3
Counts:
PeanutButter 0
FIGURE 6.5: Task parallelism using DDA (modified from [DXGHOO]).
• Type: Algorithms may generate regular association rules or more advanced asso
ciation rules sJch as those introduced in section 6.7 and Chapters 8 and 9.
• Data type: We have examined rules generated for data in categorical databases.
Rules may also be derived for other types of data such as plain text. This concept
is further investigated in Section 6.7 and in Chapter 7 when we look at Web usage
mining.
• Data source: Our investigation has been limited to the use of association rules
for market basket data. This assumes that data are present in a transaction. The
absence of data may also be important.
• Technique: The most common strategy to generate association rules is that of
finding large itemsets. Other techniques may also be used.
• Itemset strategy: Itemsets may be counted in different ways. The most naive
approach is to generate all itemsets and count them. As this is usually too space
intensive, the bottom-up approach used by Apriori, which takes advantage of the
large itemset property, is the most common approach. A top-down technique could
also be used.
• Transaction strategy: To count the itemsets, the transactions in the database must
be scanned. All transactions could be counted, only a sample may be counted, or
the transactions could be divided into partitions.
• Itemset data structure: The most common data structure used to store the can
didate itemsets and their counts is a hash tree. Hash trees provide an effective
technique to store, access, and count itemsets. They are efficient to search, insert,
and delete itemsets. A hash tree is a multiway search tree where the branch to
be taken at each level in the tree is determined by applying a hash function as
opposed to comparing key values to branching points in the node. A leaf node in
the hash tree contains the candidates that hash to it, stored in sorted order. Each
internal node actually contains a hash table with links to children nodes. Figure 6.6
shows one possible hash tree for candidate itemsets of size 3, which were shown
Section 6.5 Comparing Approaches 183
1, 4 2, 5 3, 6
3,4, 6
3, 5, 6
2, 4, 6 2, 5, 6
FIGURE 6.6: Hash tree for C3 shown in Table 6.7.
in Table 6.7. For simplicity we have replaced each item with its numeric value in
order: Blouse is 1, Jeans is 2, and so on. Here items 1, 4 hash to the first entry;
2, 5 hash to the second entry; and 3, 6 hash to the third entry.
• Transaction data structure: Transactions may be viewed as in a flat file or as a
TID list, which can be viewed as an inverted file. TI1e items usually are encoded
(as seen in the hash tree example), and the use of bit maps has also been proposed.
• Optimization: These techniques look at how to improve on the performance of
an algorithm given data distribution (skewness) or amount of main memory.
• Architecture: Sequential, parallel, and distributed algorithms have been proposed.
• Parallelism strategy: Both data parallelism and task parallelism have been used.
Table 6.8 (derived from [DXGHOO]) provides a high-level comparison of the asso
ciation rule algorithms we have covered in this chapter. When m is the number of items,
the maximum number of scans is m + 1 for the level-wise algorithms. This applies to
the parallel algorithms because they are based on Apriori. Both sampling algorithms and
TABLE 6.8: Comparison of Association Rule Algorithms (modified from [DXGHOO])
Partitioning Scans Data Structure Parallelism
Apriori m + 1 hash tree none
Sampling 2 not specified none
Partitioning 2 hash table none
CDA m + l hash tree data
DDA m + 1 hash tree task

partitioning algorithms require at most two complete scans of the transaction database.
However, remember that the sampling algorithm must access the database to read the
sample into memory and then many scans of it into memory may be required. Similarly,
for the partitioning algorithm, each partition must be read into memory and may be
scanned there multiple times.
6.6 INCREMENTAL RULES
All algorithms discussed so far assume a static database. However, in reality we cannot
assume this. With these prior algorithms, generating association rules for a new database
state requires a complete rerun of the algorithm. Several approaches have been proposed
to address the issue of how to maintain the association rules as the underlying database
changes. Most of the proposed approaches have addressed the issue of how to modify the
association rules as inserts are performed on the database. These incremental updating
approaches concentrate on determining the large itemsets for D U db where D is a
database state and db are updates to it and where the large itemsets for D, L are known.
1
One incremental approach, f
ast update (FUP), is based on the Apriori algorithm.
Each iteration, k, scans both db and D with candidates generated from the prior iter
ation, k - 1, based on the large itemsets at that scan. In addition, we use as part of
the candidate set for scan k to be Lk found in D. The difference is that the num
ber of candidates examined at each iteration is reduced through pruning of the candi
dates. Although other pruning techniques are used, primary pruning is based on the fact
that we already know L from D. Remember that according to the large itemset prop
erty, an itemset must be large in at least one of these partitions of the new database.
For each scan k of db, Lk plus the counts for each itemset in Lk are used as input.
When the count for each item in Lk is found in db, we automatically know whether it
will be large in the entire database without scanning D. We need not even count any
items in Lk during the scan of db if they have a subset that is not large in the entire
database.
6.7 ADVANCED ASSOCIATION RULE TECHNIQUES
In this section we investigate several techniques that have been proposed to generate
association rules that are more complex than the basic rules.
6.7.1 Generalized Association Rules
Using a concept hierarchy that shows the set relationship between different items, gen
eralized association rules allow rules at different levels. Example 6.7 illustrates the
use of these generalized rules using the concept hierarchy in Figure 6.7. Association
rules could be generated for any and all levels in the hierarchy. A generalized associ
ation rule, X =? Y, is defined like a regular association rule with the restriction that
no item in Y may be above any item in X. When generating generalized association
rules, all possible rules are generated using one or more given hierarchies. Several
algorithms have been proposed to generate generalized rules. The simplest would be
to expand each transaction by adding (for each item in it) all items above it in any
hierarchy.
6.7.2
6.7.3
Section 6.7 Advanced Association Rule Techniques 185
Food
�
Meat Dairy
��
Vegetables Grain
A
Fruit
Bread Rice Yogurt Milk Cheese
� ��
Wheat White Rye Whole 2% Skim
FIGURE 6.7: Concept hierarchy.
EXAMPLE 6.7
Figure 6.7 shows a partial concept hierarchy for food. This hierarchy shows that Wheat
Bread is a type of Bread, which is a type of grain. An association rule of the
.
form
Bread =? PeanutButter has a lower support and threshold than one of the form Gram =?
PeanutButter. There obviously are more transactions containing any type of grain than
transactions containing Bread. Likewise, Wheat Bread =
.>- Peanutbutter has a lower
threshold and support than Bread =? PeanutButter.
Multiple-Level Association Rules
A variation of generalized rules are multiple-level association rules. With multiple-level
rules, itemsets may occur from any level in the hierarchy: Using a variation of the Apriori
algorithm, the concept hierarchy is traversed in a top-down manner and large itemsets
are generated. When large itemsets are found at level i, large itemsets are generat�d for
level i + 1. Large k-itemsets at one level in the concept hierarchy are used as candtdates
to generate large k-itemsets for children at the next level.
Modification to the basic association rule ideas may be changed. We expect that
there is more support for itemsets occurring at higher levels in the concept hierarch!.
Thus, the minimum support required for association rules may vary based on level m
the hierarchy. We would expect that the frequency of itemsets at higher levels i
.
s
.
much
greater than the frequency of itemsets at lower levels. Thus, for the reduced nurumum
support concept, the following rules apply:
• The minimum support for all nodes in the hierarchy at the same level is iqentical.
• If o:; is the minimum support for level i in the hierarchy and a; -1 is the minimum
support for level i - 1, then a; - 1 > ct; .
Quantitative Association Rules
The association rule algorithms discussed so far assume that the data are categorical. A
quantitative association rule is one that involves categorical and quantitative data. An

example of a quantitative rule is:
A customer buys wine for between $30 and $50 a bottle => she also buys caviar
This differs from a traditional association rule such as:
A customer buys wine => she also buys caviar.
The cost quantity has been divided into an interval (much as was done when we looked
at handling numeric data in clustering and classification). In these cases, the items are
not simple literals. For example, instead of having the items {Bread, Jelly}, we might
have the items {(Bread:[O . . . 1]), (Bread:(l . . . 2]) , (Bread:(2 . . . oo)) , (Jelly:[O . . . 1 .5]),
(Jelly:( l .5 . . . 3]), (Jelly:(3 . . . oo))}.
The basic approach to finding quantitative association rules is found in Algo
rithm 6.8. Here we show the Jpriori algorithm being used to generate the large itemsets,
but any such algorithm could pe used.
1
. -
ALGORITHM 6.8
Input :
I //Itemsets
p
l
,
p
2
, . . . , PP ; //Processors
D = D
1
, D
2
, .
. . , IJP;
s //Support
Output :
L //Large itemsets
/ /Database divided into part itions
Quanti tative as sociation rule algorithm :
for each Ij E I do I/Partition items .
if Ij i s to be partit ioned, then
determine number of partitions ;
map attribute value s into new partitions creating new items ;
replace Ij in I wi th the new items Ij1 , . . . , Ijm ;
Apriori ( I , D , s ) ;
Because we have divided what was one item into several items, the minimum
support and confidence used for quantitative rules may need to be lowered. The min
imum support pro�lem obviously is worse with a large number of intervals. Thus,
an alternative solution would be to combine adjacent intervals when calculating sup
port. Similarly, when there are a small number of intervals, the confidence thresh
old may need to be lowered. For example, look at X => Y. Suppose there are only
two intervals for X . Then the count for those transactions containing X will be quite
high when compared to those containing Y (if this is a more typical item with many
intervals).
6.7.4 Using Multiple Minimum Supports
When looking at large databases with many types of data, using one minimum support
value can be a problem. Different items behave differently. It certainly is easier to obtain
a given support tllreshold with an attribute that has only two values than it is with an
Section 6.7 Advanced Association Rule Techn iques 187
attribute that has hundreds of values. It might be more meaningful to find a rule of
the form
SkimMilk ==> WheatBread
with a support of 3% than it is to find
Milk ==> Bread
with a support of 6%. Thus, having only one support value for all association rules may
not work well. Some useful rules could be missed. This is particularly of interest when
looking at generalized association rules, but it may arise in other situations as well. Think
of generating association rules from a non-market basket database. As was seen with
quantitative rules, we may partition attribute values into ranges. Partitions that have a
small number of values obviously will produce lower supports than those with a large
number of values. If a larger support is used, we might miss out on generating meaningful
association rules.
This problem is called the rare item problem. If the minimum support is too high,
then rules involving items that rarely occur will not be generated. If it is set too iow,
then too many rules may be generated, many of which (particularly for the frequently
occurring items) are not important. Different approaches have been proposed to handle
this. One approach is to partition the data based on support and generate association
rules for each partition separately. Alternatively, we could group rare items together and
generate association rules for these groupings. A more recent approach to handling this
problem is to combine clustering and association rules. First we cluster the qata together
based on some clustering criteria, and then we generate rules for each cluster separately.
This is a generalization of the partitioning of the data solution.
One approach, M/Sapriori, allows a different support threshold to be indicated for
each item. Here MIS stands for minimum item support. The minimum support for a rule
is the minimum of all the minimum supports for each item in the rule. An interest
ing problem occurs when multiple minimum supports are used. The minimum support
requirement for an itemset may be met even though it is not met for some of its subsets.
This seems to violate the large itemset property. Example 6.8, which is adapted from
[LHM99], illustrates this. A variation of the downward closure property, called the sorted
downward closure property, is satisfied and used for the MISapriori algorithm. First the
items are sorted in ascending MIS value. Then the candidate generation at scan 2 looks
only at adding to a large item any item following it (larger than or equal to MIS value)
in the sorted order.
EXAMPLE 6.8
Suppose we have three items, {A, B, C}, with nummum supports MIS(A) = 20%,
MIS(B) = 3%, and MIS(C) = 4%. Because the support for A is so large, it may be
small, while both AB and AC may be large because the required support for AB
min(M/S (A), MIS (B)) = 3% and AC = min(MIS (A), MIS (C)) = 4%.
6.7.5 Correlation Rules
A correlation rule is defined as a set of itemsets that are correlated. The motiva
tion for developing these correlation rules is that negative correlations may be useful.

6.8
Example 6.9, which is modified from [BMS77], illustrates this concept. In this example,
even though the probability of purchasing two items together seems high, it is much
higher if each item is purchased without the other item. Correlation satisfies upward
closure in the itemset lattice. Thus, if a set is correlated, so is every superset of it.
EXAMPLE 6.9
Suppose there are two items, {A, B} where A ::::} B has a support of 15% and a confidence
of 60%. Because these values are high, a typical association rule algorithm probably
would deduce this to be a valuable rule. However, if the probability to purchase item
B is 70%, then we see that the probability of purchasing B has actually gone down,
presumably because A was purchased. Thus, there appears to be a negative correlation
between buying A and buying B. The correlation can be expresed as
P (A, B)
correlation(A ===> B) =
P(A) P (B)
(6.1)
1
which in this case is: 0.2°5�50.7 = 0.857. Because this correlation value is lower than 1, it
indicates a negative correlation between A and B.
MEASURING THE QUALITY O F RULES
Support and confidence are the normal methods used to measure the quality of an asso
ciation rule:
s(A ===> B) = P (A , B) (6.2)
and
a(A ===> B) = P (B I A) (6.3)
However, there are someproblems associated with these metrics. For example, confidence
totally ignores P(B). A rule may have a high support and confidence but may be an
obvious rule. For example, if someone purchases potato chips, there may be a high
likelihood that he or she would also buy a cola. This rule is not really of interest because
it is not surprising. Various concepts such as surprise and interest have been used to
evaluate the quality or usefulness of rules. We briefly examine some of these in this
section.
With correlation rules, we saw that correlation may be used to measure the rela
tionship between items in a rule. This may also be expressed as the lift or interest
P(A, B)
interest(A ===> B) =
P(A) P (B)
(6.4)
This measure takes into account both P(A) and P (B). A problem with this measure is
that it is symmetric. Thus, there is no difference between the value for interest(A ::::} B)
and the value for interest(B ::::} A).
As with lift, conviction takes into account both P (A) and P(B). From logic we
know that implication A � B = -.(A 1 -. B). A measure of the independence of
the negation of implication, then, is Pf1�;(�k). To take into account the negation, the
Section 6.8 Measuring the Qua lity of Rules 189
conviction measure inverts this ratio. The formula for conviction is [BMS77]
. . P(A) P(-•B)
convtctwn(A ===> B) =
P(A,
-.
B
) (6.5)
Conviction has a value of 1 if A and B are not related. Rules that always hold have a
value of oo.
The usefulness of discovered association rules may be tied to the amount of surprise
associated with the rules or how they deviate from previously known rules. Here surprise
is a measure of the changes of correlations between items over time. For example, if you
are aware that beer and pretzels are often purchased together, it would be a surprise if
this relationship actually lowered significantly. Thus, this rule beer ::::} pretzel would be
of interest even if the confidence decreased.
Another technique to measure the significance of rules by using the chi squared
test for independence has been proposed. This significance test was proposed for use
with correlation rules. Unlike the support or confidence measurement, the chi squared
significance test takes into account both the presence and the absence of items in sets.
Here it is used to measure how much an itemset (potential correlation rule) count differs
from the expected. The chi squared test is well understood because it has been used in the
statistics community for quite a while. Unlike support and confidence, where arbitrary
values must be chosen to determine which rules are of interest, the chi squared values are
well understood with existing tables that show the critical values to be used to determine
relationships between items.
The chi squared statistic can be calculated in the following manner. Suppose the
set of items is I = {!1 , h . . . , Im}. Because we are interested in both the occurrence
and the nonoccurrence of an item, a transaction t1 can be viewed as
(6.6)
Given any possible itemset X, it also is viewed as a subset of the Cartesian product. The
chi squared statistic is then calculated for X as
2
_ "" (O(X) - E[X])2
X - � E[X]
XE/
(6.7)
Here O (X) is the count of the number of transactions that contain the items in X. For
one item Ii , the expected value is E[Ii] = 0 Ui), the count of the number of transactions
that contain Ii . E[Ji] = n - O(li). The expected value E[X] is calculated assuming
independence and is thus defined as
m E[I
; ]
E[X] = n X
n
-
n
-
i=l
Here n is the number of transactions.
(6.8)
Table 6.9, which is called a contingency table, shows the distribution of the data
in Example 6.9 assuming that the sample has 100 items in it. From this we find

6.9
TABLE 6.9: Contingency Table for Example 6.9
A
A
Total
B
15
55
70
10
20
30
Total
25
75
100
E[AB] = 17.5, E[AB] = 7.5, E[AB] = 52.5, and E[AB] = 22.5. Using these val
ues, we calculate x2 for this example as
'"'
(O(X) - E[X])2
� . E[X]
Xe/ .;
(20 - 22.5)2
= 1.587
+ 22.5
(15 - 17.5)2 (10 - 7.5)2 (55 - 52.5)2
17.5 + 7.5 + 52.5
(6.9)
If all values were independent, then the chi squared statistic should be 0. A chi squared
table (found in most statistics books) can be examined to interpret this value. Examining
a table of critical values for the chi squared statistic, we see that a chi squared value
less than 3.84 indicates that we should not reject the independent assumption. This is
done with 95% confidence. Thus, even though there appears to be a negative correlation
between A and B, it is not statistically significant.
EXERCISES
1. Trace the results of using the Apriori algorithm on the grocery store example with
s = 20% and a = 40%. Be sure to show the candidate and large itemsets for each
database scan. Also indicate the association rules that will be generated.
2. Prove that all potentially large itemsets are found by the repeated application of
B v- as is used in the sampling algorithm.
3. Trace the results of using the sampling algorithm on the clothing store exam
ple with s = 20% and a = 40%. Be sure to show the use of the negative
border function as well as the candidates and large itemsets for each database
scan.
4. Trace the results of using the partition algorithm on the grocery store example with
s = 20% and a = 40%. For the grocery store example, use two partitions of size
2 and 3, respectively. You need not show all the steps involved in finding the large
itemsets for each partition. Simply show the resulting large itemsets found for each
partition.
5. Trace the results of using the count distribution algorithm on the clothing data with
s = 20%. Assume that there are three processors with partitions created from the
beginning of the database of size 7, 7, and 6, respectively.
6. Trace the results of using the data distribution algorithm on the clothing data
with s = 20%. Assume that there are three processors with partitions created
Section 6. 1 0 Bibliographic Notes 191
from the beginning of the database of size 7, 7, and 6, respectively. Assume that
candidates are distributed at each scan by dividing the total set into subsets of equal
size.
7. Calculate the lift and conviction for the rules shown in Table 6.3. Compare these
to the shown support and confidence.
8. Perform a survey of recent research examining techniques to generate rules
incrementally.
The development of association rules can be traced to one paper in 1993 [AIS93].
Agrawal proposed the AIS algorithm before Apriori [AIS93]. However, this algorithm
and another, SETM [HS95], do not take advantage of the large itemset property and thus
generate too many candidate sets. The a priori algorithm is still the major technique used
by commercial products to detect large itemsets. It was proposed by 1994 in [AS94].
Another algorithm proposed about the same time, OCD, uses sampling [MTV94]. It
produces fewer candidates than AIS.
There have been many proposed algorithms that improve on Apriori. Apriori-TID
does not use the database to count support [AS94]. Instead, it uses a special encoding
fo the candidates from the previous pass. Apriori has better performance in early passes
of the database while Apriori-T/D has better performance in later. A combination of the
two, Apriori-Hybrid, has been proposed [Sri96]. The dynamic itemset counting (DIC)
algorithm divides the database into intervals (like the partitions in the partition algorithm)
[BMUT77]. The scan of first interval counts the 1-itemsets. Then candidates of size 1 are
generated. The scan of the second interval, then, counts the 1-itemsets as well as those
2-itemsets. In this manner itemsets are counted earlier. However, more memory space
may be required. The partition algorithm was first studied in 1995 by Savasere [SON95],
while the sampling algorithm is attributed to Toivonen in 1996 [Toi96]. The problem of
uneven distribution of data in the partition algorithm was addressed in [LD98], where
a set of algorithms were proposed that better prune away false candidates before the
second scan.
The CDA and DDA algorithms were both proposed in [AS96]. Other data parallel
algorithms include PDM [PCY95], DMA [CHN+96], and CCPD [ZOPL96]. Additional
task parallel algorithms include IDD [HKK97], HPA [SK96], and PAR [ZPOL97]. A
hybrid approach, hybrid distribution (HD), which combines the advantages of each tech
nique has a speed-up close to the data parallelism approach [HKK97]. For a discussion
of other parallel algorithms, see either [DXGHOO] or [Zak99].
Many additional algorithms have been proposed. CARMA (continuous association
rule mining algorithm) [Hid99] proposes a technique that is dynamic in that it allows
the user to change the support and confidence while the algorithm is running. Some
recent work has examined the use of an AI type search algorithm called OPUS [WebOO].
OPUS prunes out portions of the search tree based on the desired rule characteristics.
However, many scans of the database are required and, thus it assumes that the database
is memory-resident.
An approach to determining if an item should be partitioned when generating
quantitative rules has been proposed [SA96a]. Variations of quantitative rules include
profile association rules where the left side ofthe rule represents some profile information

about a customer while the right side of the rule contains the purchase information
[ASY98]. Another variation on quantitative rules is a ratio rule [KLKF98]. These rules
indicate the ratio between the quantitative values of individual items. When fuzzy regions
are used instead ofdiscrete partitions, we obtainfUzzy association rules [KFW98]. Recent
research has examined association rules for multimedia data [ZHLH98].
After the initial work in [CHNW96], much additional work has examined associ
ation rules in an incremental environment. Several improvements on the original FUP
have been proposed [CNT96] [CLK97]. Another technique aims at reducing the number
of additional scans of the original database [TBAR97].
Generalized association rules were studied in [SA95]. Multiple-level association
rules were proposed in [HF95]. Quantitative association rules were studied in [Sri96].
Algorithm 6.8 does not show how to determine whether an item should be partitioned.
One technique proposed to do this is a metric called partial completeness [SA96a]. The
rare item problem was investigated [Man98]. The MISapriori approach was subsequently
proposed in [LHM99]. Correlation rules were first examined in [BMS77]. Many of the
additional measures for rules were investigated in [BMS77].
A survey of aslociation rules has recently appeared [DXGHOO]. A survey ofparallel
and distribution association rule algorithms has also been published [Zak99]. One recent
textbook is devoted to the study of association rules and sequential patterns [AdaOO]. A
recent tutorial [HLPOl ] has examined association rules and sequential patterns.
P A R T T H R E E
ADVANCE D TOPICS

C H A P T E R 7
Web M i n i ng
7.1 INTRODUCTION
7.2 WEB CONTENT MINING
7.3 WEB STRUCTURE MINING
7.4 WEB USAGE MINING
7.5 EXERCISES
7.1 INTRODUCTION
Determining the size of the World Wide Web is extremely difficult. It 1999 it was
estimated to contain over 350 million pages with growth at the rate of about 1 million
pages a day [CvdBD99]. Google recently announced that it indexes 3 billion Web docu
ments [Goo01]. The Web can be viewed. as the the largest database available and presents
a challenging task for effective design and access, Here we use the term database quite
loosely because, of course, there is no real structure or schema to the Web. Thus, data
mining applied to the Web has the potential to be quite beneficial. W
eb mining is min
ing of data related to the World Wide Web. This may be the data actually present in
Web pages or data related to Web activity. Web data can be classified into the following
classes [SCDTOO]:
• Content of actual Web pages.
• Intrapage structure includes the HTML or XML code for the page.
• Interpage structure is the actual linkage structure between Web pages.
• Usage data that describe how Web pages are accessed by visitors.
• User profiles include demographic and registration information obtained about
users. This could also include information found in cookies.
Web mining tasks can be divided into several classes. Figure 7.1 shows one taxo
nomy of Web mining activities [Za199]. Web content mining examines the content of
Web pages as well as results of Web searching. The content includes text as well as
graphics data. Web content mining is further divided into Web page content mining and
search results mining. The first is traditional searching of Web pages via content, while
195

1 96 Chapter 7 Web Mining
FIGURE 7. 1 : Web mining taxonomy (modified from [Za199]).
the second is a further search of pages found from a previous search. Thus, some mining
activities have beep built on top of traditional search engines, using their result as the
data to be mined. With Web structure mining, information is obtained from the actual
organization of pages on the Web.
·
Content mining is similar to the work performed by
basic IR techniques, but it usually goes farther than simply employing keyword searching.
For example, clustering may be applied to Web pages to identify similar pages. The
intrapage structure includes links within the page as well as the code (HTML, XML) for
the page. Web usage mining looks at logs of Web access. General access pattern tracking
is a type of usage mining that looks at a history of Web pages visited. This usage may be
general or may be targeted to specific usage or users. Besides identifying what the traffic
patterns look like, usage mining also involves the mining of these sequential patterns.
For example, patterns can be clustered based on their similarity. This in turn can be used
to cluster users into groups based on similar access behavior.
There are many applications for Web mining. One application is targeted advertis
ing. T
argeting is any technique that is used to direct business marketing or advertising
to the most beneficial subset of the total population. The objective is to maximize the
results of the advertising; that is, send it to all (and only) the set of potential customers
who will buy. In this manner, the cost of sending an advertisement to someone who will
not purchase that product can be avoided. Targeting attempts to send advertisements to
people who have not been to a Web site to entice them to visit it. Thus, a targeted ad
is found on a different Web site. All of the data mining techniques we have seen so far
could be used to target advertising to a subset of the audience. In this manner, advertising
costs can be reduced while either not impacting results or improving results. On the Web,
targeting can be used to display advertising at Web sites visited by persons that fit into
a business' target demographic area. By examining the Web log data to see what source
sites access a Web site, information about the visitors can be obtained. This in turn can
be used to sell advertising space to those companies that would benefit the most.
Although the different Web mining activities may be described separately, they
are intrinsically related. A Webmaster usually wants to create the best set of pages
to accomplish the desired objective for the pages (advertising, marketing, information
dissemination, etc.). The effectiveness of a set of Web pages depends not only on the
content and organization of individual Web pages, but also on the structure of the pages
and their ease of use. Although there are many issues that impact the effectiveness of
Section 7.2 Web Content Mining 1 97
Web sites (user interface, effective use of graphics, response time, etc.), in this chapter
we cover only techniques that involve data mining.
7.2 WEB CONTENT MINING
Web content mining can be thought of as extending the work performed by basic search
engines. There are many different techniques that can be used to search the Internet.
Only a few of these techniques are discussed here. Most search engines are keyword
based. Web content mining goes beyond this basic IR technology. It can improve on
traditional search engines through such techniques as concept hierarchies and synonyms,
user profiles, and analyzing the links between pages. Traditional search engines must
have crawlers to search the Web and gather information, indexing techniques to store the
information, and query processing support to provide fast and accurate information to
users. Data mining techniques can be used to help search engines provide the efficiency,
effectiveness, and scalability needed.
One taxonomy of Web mining divided Web content mining into agent-based and
database approaches [CMS97]. Agent-based approaches have software systems (agents)
that perform the content mining. In the simplest case, search engines belong to this class,
as do intelligent search agents, information filtering, and personalized Web agents. Intelli
gent search agents go beyond the simple search engines and use other techniques besides
keyword searching to accomplish a search. For example, they may use user profiles or
knowledge concerning specified domains. Information filtering utilizes IR techniques,
knowledge of the link structures, and other approaches to retrieve and categorize doc
uments. Personalized Web agents use information about user preferences to direct their
search. The database approaches view the Web data as belonging to a database. There
have been approaches that view the Web as a multilevel database, and there have been
many query languages that target the Web.
Basic content mining is a type of text mining. As seen in Figure 7.2, a modified
version of [Za199, Figure 2.1], text mining functions can be viewed in a hierarchy with the
simplest functions at the top and the more complex functions at the bottom. Much research
is currently under way that investigates the use of natural language processing techniques
in text mining to uncover hidden semantics, such as question and answer systems. More
Keyword
Term association
Similarity search �
Classification Clustering
Natural language processing
FIG U RE 7.2: Text mining hierarchy (modified version of [Za199, Figure 2. 1]).

1 98 Chapter 7 Web Mining
traditional mining operations involve keyword searching, similarity measures, clustering,
and classification.
Many Web content mining activities have centered around techniques to summa
rize the information found. In the simplest case, inverted file indices are created on
keywords. Simple search engines retrieve relevant documents usually using a keyword
based retrieval technique similar to those found in traditional IR systems. While these
do not perform data mining activities, their functionality could be extended to include
more mining-type activities.
One problem associated with retrieval of data from Web documents is that they are
not structured as in traditional databases. There is no schema or division into attributes.
Traditionally, Web pages are defined using hypertext markup language (HT
ML). Web
pages created using HTML are only semistructured, thus making querying more difficult
than with well-formed databases containing schemas and attributes with defined domains.
HTML ultimately will be replaced by extensible markup language (XML), which will
provide structured documents and facilitate easier mining.
7.2.1 Crawlers
A robot (or spider or crawler) is a program that traverses the hypertext structure in
the Web. The page (or set of pages) that the crawler starts with are referred to as the
seed URLs. By starting at one page, all links from it are recorded and saved in a queue.
These new pages are in turn searched and their links are saved. As these robots search
the Web, they may collect information about each page, such as extract keywords and
store in indices for users of the associated search engine. A crawler may visit a certain
number of pages and then stop, build an index, and replace the existing index. This
type of crawler is referred to as a periodic crawler because it is activated periodically.
Crawlers are used to facilitate the creation of indices used by search engines. They allow
the indices to be kept relatively up-to-date with little human intervention. Recent research
has examined how to use an incremental crawler. Traditional crawlers usually replace
the entire index or a section thereof. An incremental crawler selectively searches the Web
and only updates the index incrementally as opposed to replacing it.
Because of the tremendous size of the Web, it has also been proposed that afocused
crawler be used. A focused crawler visits pages related to topics of interest. This concept
is illustrated in Figure 7.3. Figure 7.3(a) illustrates what happens with regular crawling,
while Figure 7.3(b) illustrates focused crawling. The shaded boxes represent pages that
are visited. With focused crawling, if it is determined that a page is not relevant or its
links should not be followed, then the entire set of possible pages underneath it are pruned
and not visited. With thousands of focused crawlers, more of the Web can be covered
than with traditional crawlers. This facilitates better scalability as the Web grows. The
focused crawler architecture consists of three primary components [CvdBD99]:
• A major piece of the architecture is a hypertext classifier that associates a relevance
score for each document with respect to the crawl topic. In addition, the classifier
determines a resource rating that estimates how beneficial it would be for the
crawler to follow the links out of that page.
• A distiller determines which pages contain links to many relevant pages. These are
called hub pages. These are thus highly important pages to be visited. These hub
Section 7.2 Web Content Mining 1 99
(a) Regular crawling (b) Focused crawling
FIGURE 7.3: Focused crawling.
pages may not contain relevant information, but they would be quite important to
facilitate continuing the search.
• The crawler performs the actual crawling on the Web. The pages it visits are
determined via a priority-based structure governed by the priority associated with
pages by the classifier and the distiller.
A performance objective for the focused crawler is a high precision rate or harvest rate.
To use the focused crawler, the user first identifies some sample documents that
are of interest. While the user browses on the Web, he identifies the documents that
are of interest. These are then classified based on a hierarchical classification tree, and
nodes in the tree are marked as good, thus indicating that this node in the tree has
associated with it document(s) that are of interest. These documents are then used as the
seed documents to begin the focused crawling. During the crawling phase, as relevant
documents are found it is determined whether it is worthwhile to follow the links out
of these documents. Each document is classified into a leaf node of the taxonomy tree.
One proposed approach, hardf
ocus, follows links if there is an ancestor of this node that
has been marked as good. Another technique, softf
ocus, identifies the probability that a
page, d, is relevant as
R(d) = L P(c I d) (7. 1)
good(c)
Here c is a node in the tree (thus a page) and good(c) is the indication that it has been
labeled to be of interest. The priority of visiting a page not yet visited is the maximum
of the relevance of pages that have been visited and point to it.
The hierarchical classification approach uses a hierarchical taxonomy and a naive
Bayes classifier. A hierarchical classifier allows the classification to include information
contained in the document as well as other documents near it (in the linkage struc
ture). The objective is to classify a document d to the leaf node c in the hierarchy
with the highest posterior probability P(c I d). Based on statistics of a training set,
each node c in the taxonomy has a probability. The probability that a document can
be generated by the root topic, node q , obviously is 1. Following the argument found
in [CDAR98], suppose q , . . . , Ck = c be the path from the root node to the leaf c. We

200 Chapter 7 Web Mining
thus know
P(c; I d) = P (ci-1 I d) P(c; I c;- J , d)
Using Bayes rule, we have
P(c; I c;- J , d) =
P(c; I c;-J)P(d I c;)
2:: P (d I s)
s is a sibling of c;
(7.2)
(7.3)
P(d I c;) can be found using the Bernoulli model, in which a document is seen as a bag
of words with no order [CvdBD99].
More recent work on focused crawling has proposed the use of context graphs. The
contextfocused crawler (CFC) performs crawling in two steps. In the first phase, context
graphs and classifiers are constructed using a set of seed documents as a training set. In
the second phase, crawling is performed using the classifiers to guide it. In addition, the
context graphs are updated as the crawl takes place. This is a major difference from the
focused crawler, where the classifier is static after the learning phase. The CFC approach
is designed to overcome problems associated with previous crawlers:
• There may be some pages that are not relevant but that have links to relevant
pages. The links out of these documents should be followed.
• Relevant pages may actually have links into an existing relevant page, but no links
into them from relevant pages. However, crawling can really only follow the links
out of a page. It would be nice to identify pages that point to the current page. A
type of backward crawling to determine these pages would be beneficial.
The CFC approach uses a context graph, which is a rooted graph in which the root
represents a seed document and nodes at each level represent pages that have links to
a node at the next higher level. Figure 7.4 contains three levels. The number of levels
in a context graph is dictated by the user. A node in the graph with a path of length n
to the seed document node represents a document that has links indirectly to the seed
document through a path of length n. The number of back links followed is designated
as input to the algorithm. Here n is called the depth of the context graph. The context
graphs created for all seed documents are merged to create a mergedcontext graph. The
Seed document
Level l documents
Level 2 documents
Level 3 documents
FIGURE 7.4: Context graph.
Section 7.2 Web Content Mining 201
context graph is used to gather information about topics that are related to the topic being
explored.
Backward crawling finds pages that are not pointed to by relevant documents but
are themselves relevant. These types of pages may be new and may not yet have been
discovered and linked to from other pages. Although backward links do not really exist
in the Web, a backward crawl can be performed relatively easily because most search
engines already maintain information about the back links. This type of information is
similar to that often used by commercial citation servers, which find documents that
cite a given document. The value of the Science Citation Index in performing tradi
tional literature searches is well known. The use of backlinks on the Web can provide
similar benefits.
CFC performs classification using a term frequency-inverse document frequency
(TF-IDF) technique. The vocabulary used is formed from the documents in the seed set
and is shown in the merged context graph. Each document is represented by a TF-IDF
vector representation and is assigned to a particular level in the merged context graph.
7.2.2 Harvest System
The Harvest system is based on the use of caching, indexing, and crawling. Harvest is
actually a set of tools that facilitate gathering of information from diverse sources. The
Harvest design is centered around the use of gatherers and brokers. A gatherer obtains
information for indexing from an Internet service provider, while a broker provides the
index and query interface. The relationship between brokers and gatherers can vary.
Brokers may interface directly with gatherers or may go through other brokers to get to
the gatherers. Indices in Harvest are topic-specific, as are brokers. This is used to avoid
the scalability problems found without this approach.
Harvest gatherers use the Essence system to assist in collecting data. Although not
designed explicitly for use on the Web, Essence has been shown to be a valid technique for
retrieving Web documents [HS93]. Essence classifies documents by creating a semantic
index. Semantic indexing generates different types of information for different types of
files and then creates indices on this information. This process may first classify files
based on type and then summarize the files typically based on keywords. Essence uses
the file extensions to help classify file types.
7.2.3 Virtual Web View
One proposed approach to handling the large amounts of somewhat unstructured data
on the Web is to create a multiple layered database (MLDB) on top of the data in
the Web (or a portion thereof). This database is massive and distributed. Each layer of
this database is more generalized than the layer beneath it. Unlike the lowest level (the
Web), the upper levels are structured and can be accessed (and mined) by an SQL-like
query language. The MLDB provides an abstracted and condensed view of a portion
of the Web. A view of the MLDB, which is called a V
irtual Web View (VWV), can be
constructed.
The indexing approach used by MLDB does not require the use of spiders. The
technique used is to have the Web servers (masters, administrators) themselves send their
indices (or changes to indices) to the site(s) where indexing is being performed. This
process is triggered when changes to the sites are made. Each layer of the index is smaller

than that beneath it and to which it points. To assist in the creation of the first layer of the
MLDB, both extraction and translation tools are proposed. Translation tools are used to
convert Web documents to XML, while extraction tools extract the desired information
from the Web pages and insert it into the first layer of the MLDB . Web documents that
use XML and follow a standard format would not need any tools to create the layers. It is
proposed that any translation functions be performed directly by the local administrators.
The layer-1 data can be viewed as a massive distributed database.
The higher levels of the database become less distributed and more summarized as
they move up the hierarchy. Generalization tools are proposed, and concept hierarchies
are used to assist in the generalization process for constructing the higher levels of the
MLDB. These hierarchies can be created using the W
ordNet Semantic Network. WordNet
is a database of the English language. Nouns, adjectives, verbs, and adverbs are listed,
divided into groups of synonyms, and linked together using both lexical and semantic
relationships.
A Web data mining query language, WebML is proposed to provide data mining
operations on the ¥LDB. WebML is an extension of DMQL. Documents are accessed
using data mining operations and lists of keywords. A major feature of WebML are four
primitive operations based on the use of concept hierarchies for the keywords [ZaY99] :
1. COVERS: One concept covers another if it is higher (ancestor) in the hierarchy.
This coverage is extended to include synonyms as well.
2. COVERED BY: This is the reverse of COVERS in that it reverses to descendents.
3. LIKE: The concept is a synonym.
4. CLOSE TO: One concept is close to another if it is a sibling in the hierarchy.
Again, this is extended to include synonyms.
The following example illustrates WebML. The query finds all documents at the level of
"www.engr.smu.edu" that have a keyword that covers the keyword cat:
SELECT *
FROM document in ' ' www . engr . srnu . edu ' '
WHERE ONE OF keywords COVERS ' ' cat ' '
WebML allows queries to be stated such that the WHERE clause indicates selection
based on the links found in the page, keywords for the page, and information about the
domain Where the document is found. Because WebML is an extension of DMQL, data
mining functions such as classification, summarization, association rules, clustering, and
prediction are included.
7. 2.4 Personalization
Another example of Web content mining is in the area of personalization. With person
alization, Web access or the contents of a Web page are modified to better fit the desires
of the user. This may involve actually creating Web pages that are unique per user or
using the desires of a user to determine what Web documents to retrieve.
With personalization, advertisements to be sent to a potential customer are chosen
based on specific knowledge concerning that customer. Unlike targeting, personalization
may be performed on the target Web page. The goal here is to entice a current customer
Section 7.2 Web Content Mining 203
to purchase something he or she may not have thought about purchasing. Perhaps
.
t?e
simplest example of personalization is the use of a visitor' s name when he or she VISits
a page. Personalization is almost the opposite of targeting. With targeting, businesses
display advertisements at other sites visited by their users. With personalization, when
a particular person visits a Web site, the advertising can be designed specifically for
that person. MSNBC, for example, allows personalization by asking the user to enter
his or her zip code and favorite stock symbols [msnOO]. Personalization includes such
techniques as use of cookies, use of databases, and more complex data mining and
machine learning strategies [BDH+95]. Example 7.1 illustrates a more complex use
of personalization. Personalization may be perfprmed in many ways-so�
e
_
are not �ata
mining. For example, a Web site may require that a visitor log on and provide mformat10n.
This not only facilitates storage of personalization information (by ID), but also avoids a
common problem of user identification with any type of Web mining. Mining activities
related to personalization require examining Web log data to uncover patterns of access
behavior by use. This may actually fall into the category of Web usage mining.
EXAMPLE 7.1
Wynette Holder often does online shopping through XYZ.com. Every time she visits
their site, she must first log on using an ID. This ID is used to track what she purchases
as well as what pages she visits. Mining of the sales and Web usage data is performed
by XYZ to develop a very detailed user profile for Wynette. This profile in tum is us
�
d
to personalize the advertising they display. For example, Wynette loves chocolate. This
is evidenced by the volume of chocolate she has purchased (and eaten) during the past
year. When Wynette logs in, she goes directly to pages containing the clothes she is
interested in buying. While looking at the pages, XYZ shows a banner ad about some
special sale on Swiss milk chocolate. Wynette cannot resist. She immediately follows
the link to this page and adds the chocolate to her shopping cart. She then returns to the
page with the clothes she wants.
Personalization can be viewed as a type of clustering, classification, or even pre
diction. Through classification, the desires of a user are determined based on tho
_
se for
the class. With clustering, the desires are determined based on those users to which he
or she is determined to be similar. Finally, prediction is used to predict what the user
really wants to see. There are three basic types of Web page: personalization [MCSOO] :
• Manual techniques perform personalization through user registration preferences
or via the use of rules that are used to classify individuals based on profiles or
demographics.
• Collaborativefiltering accomplishes personalization by recommending information
(pages) that have previously been given high ratings from similar users.
• Content-based filtering retrieves pages based on similarity between them and user
profiles.
One of the earliest uses of personalization was with My Yahoo ! [MPROO]. With
My Yahoo! a user himself personalizes what the screen looks like [YahOO]. He can

io4 Chapter 7 Web Mining
provide preferences in such areas as weather, news, stock quotes, movies, and sports.
Once the preferences are set up, each time the user logs in, his page is displayed. The
personalization is accomplished by the user explicitly indicating what he wishes to see.
Some observations about the use of personalization with My Yahoo! are [MPROO]:
• A few users will create very sophisticated pages by utilizing the custorilization
provided.
• Most users do not seem to understand what personalization means and use only
use the default page.
• Any personalization system should be able to support both types of users.
This personalization is not automatic, but more sophisticated approaches to personaliza
tion actually use data mining techniques to determine the user preferences. An automated
personalization technique predicts future needs based on past needs or the needs of simi
lar users.
News Dude uses the interestingness of a document to determine if a user is inter
ested in it [BP99]. fere interestingness is based on the similarity between the document
and that of what the user wishes. Similarity is measured by the co-occurrence of words
in the documents and a user profile created for the user. The target application for News
Dude is news stories. News Dude actually prunes out stories that are too close to stories
the user has already seen. These are determined to be redundant articles. News Dude uses
a two-level scheme to determine interestingness. One level is based on recent articles
the user has read, while the second level is a more long-term profile of general interests.
Thus, a short-term profile is created that summarizes recent articles read, and a long
term profile is created to summarize the general interests. A document is found to be
interesting if it is sufficiently close to either. It was shown that the use of this two-level
approach works better than either profile by itself [MPROO].
Another approach to automatic personalization is that used by Firefly. Firefly is
based on the concept that humans often base decisions on what they hear from others. If
someone likes a TV show, a friend of that person may also like the program. User profiles
are created by users indicating their preferences. Prediction of a user's desires are then
made based on what similar users like. This can be viewed as a type of clustering. This
approach to Web mining is referred to as collaborativefiltering. The initial application of
Firefly has been to predict music that a user would like. Note that there is no examination
of the actual content of Web documents, simply a prediction based on what similar users
like. (One might argue whether this is really a content based Web mining approach.)
Another collaborative approach is called W
eb W
atcher. Web Watcher prioritizes
links found on a page based on a user profile and the results of other users with similar
profiles who have visited this page [JFM97]. A user is required to indicate the intent of
the browsing session. This profile is then matched to the links the user follows.
7.3 WEB STRUCTURE MINING
Web structure mining can be viewed as creating a model of the Web organization or a
portion thereof. This can be used to classify Web pages or to create similarity measures
between documents. We have already seen some structure mining ideas presented in the
content mining section. These approaches used structure to improve on the effectiveness
of search engines and crawlers.
7.3.1
7.3.2
Section 7.3 Web Structure Mining 205
PageRank
The PageRank technique was designed to both increase the effectiveness of search
engines and improve their efficiency [PBMW98]. PageRank is used to measure the
importance of a page and to prioritize pages returned from a traditional search engine
using keyword searching. The effectiveness of this measure has been demonstrated by
the success of Google [GooOO]. (The name Google comes from the word googol, which
is 10100.) The PageRank value for a page is calculated based on the number of pages
that point to it. This is actually a measure based on the number of backlinks to a page. A
backlink is a link pointing to a page rather than pointing out from a page. The measure
is not simply a count of the number of backlinks because a weighting is used to provide
more importance to backlinks coming from important pages. Given a page p, we use
Bp to be the set of pages that point to p, and Fp to be the set of links out of p. The
PageRank of a page p is defined as [PBMW98]
PR(p) = c L
P
�(q)
qEBp q
(7.4)
Here Nq = I Fq 1. The constant c is a value between 0 and 1 and is used for normalization.
A problem, called ranksink, that exists with this PageRank calculation is that when
a cyclic reference occurs (page A points to page B and page B points to page A), the
PR value for these pages increases. This problem is solved by adding an additional term
to the formula:
I ""
PR(q)
PR (p) = c � ----;:;-- + cE(v)
qEBp q
(7.5)
where c is maximized. Here E(v) is a vector that adds an artificial link. This simulates
a random surfer who periodically decides to stop following links and jumps to a new
page. E(v) adds links of small probabilities between every pair of nodes.
The PageRank technique is different from other approaches that look at links. It
does not count all links the same. The values are normalized by the number of links in
the page.
Clever
One recent system developed at IBM, Clever, is aimed at finding both authoritative
pages and hubs [CDK+99]. The authors define an authority as the "best source" for
the requested information [CDK+99]. In addition, a hub is a page that contains links to
authoritative pages. The Clever system identifies authoritative pages and hub pages by
creating weights. A search can be viewed as having a goal of finding the best hubs and
authorities.
Because ofthe distributed and unsupervised development of sites, a user has no way
of knowing whether the information contained within a Web page is accurate. Currently,
there is nothing to prevent someone from producing a page that contains not only errors,
but also blatant lies. In addition, some pages might be of a higher quality than others.
These pages are often referred to as being the most authoritative. Note that this is different
from relevant. A page may be extremely relevant, but if it contains factual errors, users
certainly do not want to retrieve it. The issue of authority usually does not surface in
traditional IR.

Hyperlink-inducedtopic search (HITS) finds hubs and authoritative pages [Kle99a].
The HITS technique contains two components:
• Based on a given set of keywords (found in a query), a set of relevant pages
(perhaps in the �housands) is found.
• Hub and authority measures are associated with these pages. Pages with the highest
values are returned.
The HITS algorithm is outlined in Algorithm 7. 1 . A search engine, SE, is used to find
a small set, root set (R), of pages, P, which satisfy the given query, q. This set is then
expanded into a larger set, base set (B), by adding pages linked either to or from R. This
is used to induce a subgraph of the Web. This graph is the one that is actually examined
to find the hubs and authorities. In the algorithm, we use the notation G(B, L) to indicate
that the graph (subgraph) G is composed of vertices (pages in this case) B and directed
edges or arcs (links) L. The weight used to find authorities, xp. and the weight used
to find hubs, Yp. are then calculated on G. Because pages at the same site often point
to each other, we shopld not really use the structure of the links between these pages
to help find hubs and authorities. The algorithm therefore removes these links from the
graph. Hubs should point to many good authorities, and authorities should be pointed
to by many hubs. This observation is the basis for the weight calculations shown in the
algorithm. An implementation of the weight calculations using an adjacency matrix is
found in the literature [Kle99a]. The approach is basically to iteratively recalculate the
weights until they converge. The weights are normalized so that the sum of the squares of
each is 1 . Normally, the number of hubs and authorities found is each between 5 and 10.
ALGORITHM 7.1
Input :
w
q
s
Output :
A
H
//NWW viewed as a directed graph
//Query
//Support
//Set of authority pages
//Set of hub pages
HITS algorithm
R = SE(W, q)
B = R U {pages l inked to from R) U {pages that l ink to pages in R) ;
G(B, L) = Subgraph of W induced by B;
G(B, L1) = Delete links in G within same site ;
Xp = Lq where (q,p)ELl Yq; II Find authority weights ;
Yp = Lq where (p,q)ELl Xq; II Find hub weights ;
A = {p I p has one of the highest xp) ;
H = {p I p has one of the highest Yp) ;
7.4 WEB USAGE MINING
Web usage mining performs mining on Web usage data, or Web logs. A W
eb log is a
listing of page reference data. Sometimes it is referred to as clickstream data because
each entry corresponds to a mouse click. These logs can be examined from either a client
Section 7.4 Web Usage Mining 207
perspective or a server perspective. When evaluated from a server perspective,
_
mining
uncovers information about the sites where the service resides. It can be used to Improve
the design of the sites. By evaluating a client's sequence of clicks, information about
a user (or group of users) is detected. This could be used to perform prefetching and
caching of pages. Example 7.2 from [XDOla] illustrates Web usage mining.
EXAMPLE 7.2
The webmaster at ABC Corp. learns that a high percentage of users have the following
pattern of reference to pages: (A, B , A, C). This means that a user accesses page A,
then page B, then back to page A, and finally to page C. Based on this observation, he
determines that a link is needed directly to page C from page B. He then adds this link.
Web usage mining can be used for many different purposes. By looking at the
sequence of pages a user accesses, a profile about that user could be developed, thus
aiding in personalization. With site mining, the overall quality and effectiveness of the
pages at the site can be evaluated. One taxonomy of Web usage mining applications has
included [SCDTOO] :
• Personalization for a user can be achieved by keeping track of previously accessed
pages. These pages can be used to identify the typical browsing behavior of a user
and subsequently to predict desired pages.
• By determining frequent access behavior for users, needed links can be identified
to improve the overall performance of future accesses.
• Information concerning frequently accessed pages can be used for caching.
• In addition to modifications to the linkage structure, identifying common access
behaviors can be used to improve the actual design ofWeb pages and to make other
modifications to the site. For example, suppose that visitors to an e-commerce site
can be identified as customers or noncustomers. The behavior of customers can
be compared with that for those who do not purchase anything. This can be used
to identify changes to the overall design. It may be determined that many visitors
never get past a particular page. That target page can be improved in an attempt
to tum these visitors into customers.
• Web usage patterns can be used to gather business intelligence to improve sales
and advertisement.
• Gathering statistics concerning how users actually access Web pages may or may
not be viewed as part of mining.
Web usage mining actually consists of three separate types of activities [SCDTOO]:
• Preprocessing activities center around reformatting the Web log data before pro
cessing.
• Pattern discovery activities form the major portion of the mining activities because
these activities look to find hidden patterns within the log data.

• Pattern analysis is the process of looking at and interpreting the results f th
discovery activities. 0 e
There are many issues associated with using the Web log for mining purposes:
• Identification of the exact user is not possible from the log alone.
• With a Web client cache, the exact sequence of pages a user actually visits ·
�ifficult to uncover from the server site. Pages that are rereferenced may be fou��
m the cache.
• There are many security, privacy, and legal issues yet to be solved. For example, is
the set of pages a person visits actually private information? Should a Web browse
actually divulge information to other companies about the habits of its users? Afte
r
all, this information could be valuable to potential advertisers.
r
7.4. 1 Preprocessing
1
T�e Web usage log probably is not in a format that is usable by mining applications. As
With any data to be used in a mining application, the data may need to be reformatted
and cleansed. There are, in addition, some issues specifically related to the use of Web
logs. Steps that are part of the preprocessing phase include cleansing, user identification
session identification, path completion, and formatting [CMS99].
'
DEFINITION 7.1. Let P be a set of literals, called pages or clicks, and U be a
set of users. A log is a set of triples {(U J , p, , t1) , . . . , (un , Pn , t")} where u; E U
p; E P, and t; is a timestamp.
'
Standard log data consist of the following: source site, destination site, and time
stamp, as shown in Definition 7. 1. The source and destination sites could be listed as
a URL or an IP address. The definition assumes that the source site is identified by
a user ID and the destination site is identified by a page ID. Additional data such as
Web browser information also may be included. Before processing the log, the data
may be changed in several ways. For security or privacy reasons, the page addresses
may be changed into unique (but nonidentifying) page identifications (such as alphabetic
characters). This conversion also will save storage space. In addition, the data may be
cleansed by removing any irrelevant information. As an example, the log entries with
figures (gif, jpg, etc.) can be removed.
Data from the log may be grouped together to provide more information. All pages
visited from one source could be grouped by a server to better understand the patterns
of page r�ferences from each user (source site). Similarly, patterns from groups of sites
may be discovered. References to the same site may be identified and examined to better
understand who visits this page.
A common technique is for a server site to divide the log records into sessions.
As shown in Definition 7.2 from [XDOl a], a session is a set of page references from
one source
.
site
.
during one logical period. Historically, a session would be identified by
a user loggmg mto a c?mputer, performing work, and then logging off. The login and
logoff represent the logical start and end of the session. With Web log data, this is harder
to determine. Several approaches can be used to identify these logical periods:
• Combine all records from the same source site that occur within a time period.
• Add records to a session if they are from the same source site and the time between
two consecutive tiinestamps is less than a certain threshold value.
NCR uses an approach based on the second concept. Any inactive period of 30 min
utes or more ends a session [SPOO]. Empirical results have shown that 25.5 minutes is
appropriate [CP95].
DEFINITION 7.2. Let L be a log. A session S is an ordered list of pages accessed
by a user, i.e., S = ((p J , tJ ) , (p2, t2) , . . . , (pn , tn) ) , where there is a user u; E U
such that {(u; , pJ , t!), (u; , p2, !2) , . . . , (u; , Pn, t")} � L. Here t; .:S t; iff i .:S j.
Since only the ordering of the accesses is our main interest, the access time is often
omitted. Thus, we write a session S as (PI , P2, . . . , Pn).
Associated with each session is a unique identifier, which is called a session ID.
The length of a session S is the number of pages in it, whilch is denoted as len(S). Let
database D be a set of such sessions, and the total length of D be len(D) = LseD len(S).
There are many problems associated with the preprocessing activities, and most
of these problems center around the correct identification of the actual user. User iden
tification is complicated by the use of proxy servers, client side caching, and corporate
firewalls. Tracking who is actually visiting a site (and where they come from) is diffi
cult. Even though a visit to a Web page will include a source URL or IP address that
indicates the source of the request, this may not always be accurate in determining the
source location of the visitor. Users who access the Internet through an Internet service
provider (ISP) will all have the source location of that provider. It is not unique to the
individual. In addition, the same user may use different ISPs. Also, there will be many
users accessing the Web at the same time from one machine. Cookies can be used to
assist in identifying a single user regardless of machine used to access the Web. A cookie
is a file that is used to maintain client-server information between accesses that the client
makes to the server. The cookie file is stored at the client side and sent to the server
with each access.
Identifying the actual sequence of pages accessed by a user is complicated by the
use of client side caching. In this case, actual pages accessed will be missing from the
server side log. Techniques can be used to complete the log by predicting missing pages.
Path completion is an attempt to add page accesses that do not exist in the log but that
actually occurred. Some missing pages can be easily added. For example, if a user visits
page A and then page C, but there is no link from A to C, then at least one page in
this path is missing. Algorithms are used both to infer missing pages and to generate an
approximate timestamp.
7.4.2 Data Structures
Several unique data structures have been proposed to keep track of patterns identified
during the Web usage mining process. A basic data structure that is one possible alter
native is called a trie. A trie is a rooted tree, where each path from the root to a leaf
represents a sequence. Tries are used to store strings for pattern-matching applications.
Each character in the string is stored on the edge to the node. Common prefixes of strings
are shared. A problem in using tries for many long strings is the space required. This is

B
0
u
T G
y
ABOU
AA
T
-
AGORY
(a) uie (b) Suffix tree
FIG URE 7.5: Sample tries.
illustrated in Figure 7.5(a), which shows a standard trie for the three strings {ABOUT,
CAT, CATEGORY}. Note that there are many nodes with a degree of one. This is a
waste of
�
pace that is solved by compressing nodes together when they have degrees
of one. Ftgure 7.5(b) shows a compressed version of this trie. Here a path consisting
of nodes with single children is compressed to one edge. Note in both trees the extra
edge labeled " $." This symbol (or any symbol that is not in the alphabet and is used to
constmct the strings) is added to ensure that a string that is actually a prefix of another
(CAT is a prefix of CATEGORY) terminates in a leaf node.
The compressed trie is called a su
ffix tree. A suffix tree has the following
characteristics:
• Each internal node except the root has at least two children.
• Each edge represents a nonempty subsequence.
• The subsequences represented by sibling edges begin with different symbols.
With the help of a suffix tree, it is efficient not only to find any subsequence in a
sequence, but also to find the common subsequences among multiple sequences. A
suffix tree can also be constructed from a sequence in time and space linear in the
length of the sequence. When given one session of page references, many different
patterns may be found. The exact number of patterns depends on the exact defini
tion of the pattern to be found (discussed in subsection 7.4.3). Example 7.3 illustrates
this idea.
Section 7.4 Web Usage Mining 21 1
ACCfCGTCf
TCGTC
FIGURE 7.6: Sample suffix tree.
EXAMPLE 7.3
Suppose that one session has been identified to be (C, A, C, C, T, C, G, T, C, T). Many
different patterns exist in this session. As a matter of fact, we could identify patterns
starting at the first character, or the second, or any other. The suffix tree created for this
session is shown in Figure 7.6. This tree does ncit contain the special "$" edges.
A slight variation on this suffix tree that is used to build a suffix tree for multiple
sessions is called a generalized suffix tree (GST).
7.4.3 Pattern Discovery
The most common data mining technique used on clickstream data is that of uncovering
traversal patterns. A traversal pattern is a set of pages visited by a user in a session. Other
types of patterns may be uncovered by Web usage mining. For example, association
mles can look at pages accessed together in one session independent of ordering. Similar
traversal patterns may be clustered together to provide a clustering of the users. This is
different from clustering of pages, which tends to identify similar pages, not users.
Several different types of traversal patterns have been examined. These patterns
differ in how the patterns are defined. The differences between the different types of
patterns can be described by the following features:
• Duplicate page references (backward traversals and refreshes/reloads) may or may
not be allowed.
• A pattern may be composed only of contiguous page references, or alternatively
of any pages referenced in the same session.
• The pattern of references may or may not be maximal in the session. A frequent
pattern is maximal if it has no subpattern that is also frequent.
Patterns found using different combinations of these three properties may be used
to discov�r different features and thus may be used for dif
ferent purposes. Knowledge
of contiguous page references frequently made can be useful to predict future references

212 Chapter 7 Web Min ing
TABLE 7. 1 : Comparison of Different Types of Traversal Patterns (from [XDOla])
Ordering Duplicates Consecutive Maximal Support
Association
rules N N N N
freq(X)
# transactions
Episodes y
i
N N N
freg(X)
# time windows
Sequential
patterns y N N y
freq(X)
# customers
Forward
sequences y N y y
freg(X)
# forward sequences
Maximal
frequent
sequences y y y y
freq(X)
# clicks
1
Serial episodes are ordered, parallel episodes are riot, and general episodes are partially ordered.
and thus for prefetching and caching purposes. Knowledge of backward traversals often
followed can be used to improve the design of a set of Web pages by adding new links to
shorten future traversals. The maximal property is used primarily to reduce the number
of meaningful patterns ciscovered. The use of such performance improvements as user
side caching may actually alter the sequences visited by a user and impact any mining
of the Web log data at the server side.
The different types of traversal patterns that have been studied and how they view
these three features are shown in Table 7.1 (from [XDOla]). Example 7.4 illustrates a set
of sessions to be used throughout this section. The sessions are listed in order, and all
timestamps have been removed.
EXAMPLE 7.4
The XYZ Corporation maintains a set of five Web pages: {A, B, C, D, E}. The following
sessions (listed in timestamp order) have been created: D = {S1 = {U1 , (A, B, C)}, s2 =
{U2, (A , C)}, S3 = {U1 , (B, C, E)}, S4 = {U3 , (A , C, D , C, E)}}. Here we have added
to each session the user Ib. Suppose the support threshold is 30%.
Association Rules. Association rules can be used to find what pages are accessed
together. Here we are ·really finding large itemsets. A page is regarded as an item,
and a session is regarded as a transaction with both duplicates and ordering ignored.
The support is defined to be the number of occurrences of the itemset divided by the
number of transactions or sessions. The application of the Apriori algorithm to the data
in Example 7.2 is shown in Example 7.5.
EXAMPLE 7.5
Since there are four transactions and the support is 30%, an itemset must occur in at least
two sessions. During the first scan, we find that L 1 = {{A}, {B}, {C}, {E}}, so C2 =
{{A, B}, {A, C}, {A, E}, {B, C}, {B, E}, {C, E}}. Counting these is scan two, we find
L2 = {{A, C}, {B, C}, {C, E}} and then generate C3 = {{A, B, C}, {A , C, E}, {B, C, E}}.
Counting, we find that none of these are large. The large itemsets are then
L = {{A}, {B}, {C}, {E}, {A, C}, {B, C) , {C, E}}
Sequential Patterns. Although initially proposed for use with market basket data,
sequential patterns have also been applied to Web access logs. A sequential pattern (as
applied to Web usage mining) is defined as an ordered set of pages that satisfies a given
support and is maximal (i.e., it has no subsequence that is also frequent). Support is
defined not as the percentage of sessions with the pattern, but rather the percentage of
customers who have the pattern. Since a user may have many sessions, it is possible
that a sequential pattern could span sessions. It also need not be contiguously accessed
pages. A k-sequence is a sequence of length k (i.e., is it has k pages in it).
Algorithm 7.2 outlines the steps needed to find sequential patterns. After the sort
step to put the data in the correct order, the remaining steps are somewhat similar to those
of the Apriori algorithm. The sort step creates the actual customer sequences, which are
the complete reference sequences from one user (across transactions). During the first
scan it finds all large 1-itemsets. Obviously, a frequent 1-itemset is the same as a frequent
1-sequence. In subsequent scans, candidates are generated from the large itemsets of the
previous scans and then are counted. In counting the candidates, however, the modified
definition of support must be used. In the algorithm we show that AprioriAll is used to
perform this step.
ALGORITHM 7.2
Input :
D = {S1, S2 , . . . , Sk} I/Database of sess ions
s / /Support .
Output : Sequential patterns
Sequential patterns algorithm:
D = sort D on user- ID and time of f irst pa9e reference
in each session ;
find L1 in D;
L = AprioriAl l (D, s, L1 ) ;
f ind maximal reference sequences from L ;
Generating sequential patterns for Example 7.5 is shown in Example 7.6. Here Ci
represents the candidate i-sequences and Li are the large i-sequences.
EXAMPLE 7.6
In this example, user U1 actually has two transactions (sessions). To find his sequential
patterns, we must think of his sequence as the actual concatenation of those pages in

214 Chapter 7 Web Mi ning
S1 and S3. Also, since support is measured not by transactions but by users, a sequence
is large if it is contained in at least one customer' s sequence. After the sort step, we
have that D = (S1 = {U1 , (A, B, C)}, S3 = {U1 , (B, C, E)}, S2 = {U2, (A, C)}, S4 =
{U3, (A, C, D, C, E) }. We find L 1 {{A}, {B}, {C}, {D}, {E}} since each page is referenced
by at least one customer. The following table outlines the steps taken by AprioriAll:
There are variations of this algorithm and several techniques used to improve the
performance. The set of customer sequences is reformatted after L1 is found. Each trans
action is replaced with one that consists only of pages from L 1 . Candidates may be pruned
before counting by removing any candidates that have subsequences that are not large.
Variations on AprioriAll are proposed to avoid generating so many candidates. In effect,
these improvements are used only to avoid generating sequences that are not maximal.
C1 = { (A), (B), (C) , (D), (E) }
L1 = {(A) , (B), (C) , (D), (E) }
C2 = {(A,, B), (A, C), (A, D), (A, E), (B, A), (B, C), (B, D),
(B, E)", (C, A), (C, B) , (C, D), (C, E), (D, A), (D, B),
(D, C), (D, E), (E, A), (E, B), (E, C) , (E, D) }
L2 = {(A, B), (A, C) , (A, D), (A, E), (B, C),
(B, E), (C, B), (C, D), (C, E), (D, C), (D, E)}
C3 = {(A, B, C) , (A, B, D), (A, B, E), (A, C, B), (A, C, D), (A, C, E),
(A , D, B), (A, D, C), (A, D, E), (A, E, B), (A, E, C) , (A, E, D),
(B, C, E), (B, E, C), (C, B, D), (C, B, E), (C, D, B) , (C, D, E),
(C, E, B), (C, E, D), (D, C, B), (D, C, E), (D, E, C) }
L3 = {(A, B, C) , (A, B, E), (A, C, B), (A, C, D), (A, C, E), (A, D, C),
(A , D, E), (B, C, E), (C, B, E), (C, D, E), (D, C, E)}
C4 = {(A , B, C, E), (A, B, E, C) , (A, C, B, D) , (A , C, B, E),
(A , C, D, B), (A, C, D, E), (A, C, E, B), (A, C, E, D),
(A, D, C, E), (A, D, E, C)}
L4 = {(A, B, C, E) , (A, C, B, E), (A, C, D, E), (A , D, C, E)}
Cs = 0
The W
AP-tree (web accesspattern) has been proposed to facilitate efficient count
ing. This tree is used to store the sequences and their counts. Once the tree is built, the
original database of patterns is not needed. Each node in the tree is associated with an
event (a page found at a particular time by a user). The node is labeled with the event and
a count that is associated with the pattern prefix that ends at that event. Only individual
frequent events are added to the tree.
Frequent Episodes. Episodes, which originally were proposed for telecommu
nication alarm analysis, can also be applied to Web logs. All pages (corresponding to
events) are ordered by their access time, and the users usually need not be identified (i.e.,
no sessions). By definition, an episode is a partially ordered set of pages [MTV95]. In
addition, the individual page accesses must occur within a particular time frame. A serial
episode is an episode in which the events are totally ordered. Note that they need not be
contiguous, however. A parallel episode is a set of events where there need not be any
particular ordering. They still do need to satisfy the time constraint, however. Finally, a
general episode is one where the events satisfy some partial order. Note that even though
these seem similar to the idea of sequential patterns and association rules, the added con
straint of a time window does make an episode different from either of these. The original
definition has no concept of user, but, of course, the idea of an episode could be applied
to events by one user or across users. In addition, episodes need not be maximal.
Example 7.7 illustrates the concept of episodes applied to the data in Example 7.4.
Here we keep the original ordering of the events.
EXAMPLE 7.7
The XYZ Corporation maintains a set offive Web pages {A, B, C, D, E}. Assume that the
data in Example 7.2 have the following sequence (independent of user): (A, B, C, A, C,
B, C, A, C, D, C, E). To find episodes, we must know the time window, so the follow
ing sequence shows each event with an integer timestamp: ((A, 1), (B, 2), (C, 2) , (A , 7) ,
(C, 10), (B, 10), (C, 12), (A, 12), (C, 13), (D, 14), (C, 14), (E, 20)) . Suppose that we
wish to find general episodes where the support threshold is 30% and the time win
dow is 3. This means that only events that occur within a time window of 3 are valid.
We assume that the time window is the difference between the last event and the first
event in the episode. To illustrate episodes, Figure 7.7 illustrates the ordering of events
as shown in a DAG (directed acyclic graph) where arcs are used to represent temporal
ordering. The arcs are labeled with the time between successive events. Starting at the
first event and looking at the maximum window size of 3, we see that we have two serial
episodes: AC and AB. B and C occur as parallel episodes. Stmting at the event looking
at time 12, we have the following serial episodes: ACD, ACC, CCC, CCD, AC, CC,
CC, CD. We have two parallel episodes: A and C, and C and D. There also is a general
episode that can be seen as the subgraph from time 12 to time 14. When taking the
frequency into account, an episode must occur a certain number of times in all windows.
Maximal Frequent Forward Sequences. One approach to mining log traversal
patterns is to remove any backward traversals [CPY98]. Each raw session is transformed
into forward reference (i.e., removes the backward traversals and reloads/refreshes), from
which the traversal patterns are then mined using improved level-wise algorithms. For
FIGURE 7.7: DAG for episodes.

example, for the session (A, B, A, C) in Example 7.2, the resulting forward sequences
are (A, B) and (A, C). Looking at Example 7.8 and the sequence (A, B, C, A, C, B,
C, A, C, D, E), we find the following maximal forward references:
(A, B, C), (A, C), (A, C, D), (A, C, E)
As observed by the authors, the "real" access patterns made to get to the really used pages
would not include these backward references. They assume that the backward reference
is included only because of the structure of the pages, not because they really want to do
this. The resulting set of forward references are called maximalf
orward references. They
are called maximal because they are not subsequences of other forward references. The
set of important reference patterns are those that occur with frequency above a desired
threshold. In actuality, we are interested in finding consecutive subsequences within the
forward references. A large reference sequence is a reference sequence (consecutive sub
sequence of a maximal forward reference) that appears more than a minimum number
of times. The minimup1 number threshold is called the support.
Algorithm 7.3
"
outlines the steps needed to mine maximal reference
sequences [CPY98] . After the maximal forward references are found, all subsequences of
these references that occur greater than the required support are identified. Those large
reference sequences that are not subsequences of other large reference sequences are
found in step 3 and become the large reference sequences.
ALGORITHM 7.3
Input :
D = {81 , s2 , . . . , sk) I /Database of sessions
s //Support
Output :
Maximal reference sequences
Maximal frequent forward sequences algorithm :
f ind maximal forward referenc es f rom D;
f ind large ref erence sequences from the maximal ones ;
f ind maximal reference sequences f rom the large ones ;
Maximal Frequent Sequences. The transformation used to remove backward
.
references also loses potentially useful information; for example, from the two forward
sequences (A, B ) and (A, C), we could not tell whether a direct link to page C from
page B is needed, as shown in Example 7.2.
With maximalfrequent sequences (MFS), all four properties in Table 7.1 are re
quired. Since an MFS could potentially start with any page (click) in any session, the
definition of support assumes that the number of clicks is in the denominator. Thus, the
support of a sequence X is defined to be
freq(X) freq
--- = ---
len(D) # clicks
A sequence X is frequent if its support is above a minimum threshold. An MFS must be
maximal. Example 7.8 (from [XDOla]) shows the mining of MFS.
EXAMPLE 7.8
Given D = { (A, B, C, D, E, D, C, F),(A, A, B, C, D, E),(B, G, H, U, V),(G, H, W)}.
The first session has backward traversals, and the second session has a reload/refresh
on page A. Here len(D) = 22. Let the minimum support be Srnin = 0.09. This means
that we are looking at finding sequences that occur at least two times. There are two
maximal frequent sequences: (A, B, C, D, E) and (G, H). Both sequences occur two
times.
Algorithm 7.4 (from [XDOla]) shows the OAT (Online Adaptive Traversal Pat
terns) algorithm designed to find MFS. It utilizes a suffix tree to store patterns. One
suffix tree is created for all sessions. Counts of patterns are maintained in the tree. A
unique feature of OAT is its ability to adapt to the availabe amount of main memory. If
the suffix tree is too big to fit into memory, it is compressed and the algorithm contin
ues. Details concerning the exact techniques used for compression can be found in the
literature [XDOl a].
ALGORITHM 7.4
Input :
81, 82 , . . . , Sn : s e s s ions
Smin : minimum support thre shold
M: main memory s i ze
Output :
Al l maximal f requent s equences (MFS s )
OAT algorithm:
ST = an empty suf f ix t ree ;
/ / f irst scan
for i from 1 to n do
endfor
/ / i f insu f f i c ient main memory with inclus ion o f Si ;
/ / compres s the suf f ix tree us ing f requent sequences ;
if mem(ST U Si) > M, then
ST = OAT_compre s s (ST) ;
endif
//update the suf fix t ree with inclus ion of si
ST = update(ST, Si) ;
if interrupted by the user , then
//do a depth - f i rst traversal of ST and output the MFS s .
MFS_depth_f irs t(ST.root) ;
endif
/ / second sc an
if there are sequences not completely count ed, then
count them in an addit i onal sc an ,
endif
output the MFSs in the suf f ix t ree .

7.4.4 Pattern Analysis
Once patterns have been identified, they must be analyzed to detennine how that infor
mation can be used. Some of the generated patterns may be deleted and determined not
to be of interest.
Recent work has proposed examining Web logs not only to identify frequent types
of traversal patterns, but also to identify patterns that are of interest because of their
uniqueness or statistical properties [WUMOO]. Patterns found need not have contiguous
page references. A Web mining query language, MINT, facilitates the statement of inter
esting properties. The idea of a sequence is expanded to the concept of what the authors
call a g-sequence. A g-sequence is a vector that consists not only of the pages visited
(events) but also of wildcards. For example, the g-sequence b * c stands for a sequence
consisting of b, any number of pages, then c. With the use of wildcards, it is indicated
that the events need not be contiguous. More complicated g-sequences can indicate spe
cific constraints on the number of events that replace the wildcard. With MINT, selection
of patterns that satisfy a g-sequence template are accomplished. The selection constraints
may also include restrictiOJlS on support.
Some of the thrust 6f this work has been in comparing the differences between
traversal patterns of the customers of an e-business site and those that are not cus
tomers [SPFOO]. Visitors to a site have been classified as short-time visitors, active
investigators, and customers [BPW96]. Preprocessing first filters out the visitors who are
short-time. Using concept hierarchies, the contents of the Web pages are then abstracted
to more general concepts. The log is then divided into those for customers and those for
noncustomers. Each log is then examined to find patterns based on any desired require
ments (such as frequency). The patterns found across the two logs are then compared
for similarity. Similarity is detennined using the following rule [SPFOO]:
• Two patterns are comparable if their g-sequences have at least the first n pages the
same. Here n is supplied by the user.
In addition, only fragments of patterns that occur frequently are considered. The goal
of this work is to increase the number of customers. Noncustomer patterns with no
comparable customer patterns indicate that some changes to the link structure or Web
page designs may be in order. The project proposes rules and the use of a proxy server
to dynamically change the link structures of pages.
7.5 EXERCISES
1. (Research) Find and describe two different approaches used by Web sites to per
form personalization. Be as specific as possible.
2. (Research) Perform the same Web search using three different search engines.
Describe the results. Your description should include the number of documents
retrieved. Compare the differences of the five top pages found by each. Hypothesize
why these differences exist.
3. Construct a trie for the string (A, B, A , C).
4. Construct a suffix tree for the string (A, B, A , C) .
5. Given the following sessions, {(A, B, A, C), (C, B, D, F) , (A, B, A)}, indicate the
sequential patterns, forward sequences, and maximal frequent sequences assuming
a minimum support of 30%. Assume that each session occurs from a different user.
6. The use of a Web server through a proxy (such as an ISP) complicates the collecti�n
of frequent sequence statistics. Suppose that two users use one proxy and have t e
following sessions:
• User 1: ( 1 , 3, 1, 3, 4, 3, 6, 8, 2, 3, 6)
• User 2: (2, 3, 4, 3, 6, 8, 6, 3, 1)
When these are viewed together by the Web server (taking into account the time
stamps), one large session is generated:
( 1 , 2, 3, 3, 4, 1, 3, 6, 3, 8, 4, 3, 6, 3, 6, 1, 8, 2, 3, 6)
Identify the maximal frequent sequences assuming a rninimum support of 2. What
are the maximal frequent sequences if the two users could be separated?
1. (Research) Perform a literature survey concerning current research into solutions
to the proxy problem identified in Exercise 6.
7.6 BIBLIOGRAPH IC NOTES
A recent survey of Web mining has been published [KBOO] and contain� an �xcellent
bibliography and a survey of hundreds of Web mining publications. Th1s art1cl� �ro
vides both an information retrieval view and a database v1ew of Web content mmmg.
Both [Za199] and [CMS97] contain taxonomies of Web mining activities.
There have been many published articles exploring crawlers. Incremental crawlers
were examined in [CGMOO]. Focused crawlers are studied in [CDAR98], [CDI98],
[CvdBD99], and [DCL+oo]. The periodic crawler was investigated in [CGMOO]. The
context focused crawler was first investigated in [DCL+oo]. Harvest and Essence were
described in [BDH+95] and [HS93]. The Virtual Web View with the MLDB was pro-
posed in [Za199]. . . . . .
An excellent introduction to personalizatwn appeared m a spec1al 1ssue of the
Communications o
f the ACM [RieOO]. A special section within this same issue is aimed
specifically at personalization using Web usage mining [SpiOO]. Firefly's automatic per-
sonalization approach was examined in [SM95].
. . . .
A recent doctoral dissertation has examined many 1ssues associated w1th �eb mm
ing [Zai:99]. In addition to providing an excellent bibliography
.":ith compansons of
various Web mining activities, this work also proposes a Web rrurun
.
g query la�guage,
W
ebML. WebML accesses the Web data, which have been transformed mto a multllayered
database. Information about WordNet is available on the Web [WorOO].
The suffix tree data structure is actually a PATRICIA trie [Bay74] constructed for
the relevant suffixes. Efficient algorithms for suffix trees were shown in [�cC76].
Sequential patterns as applied to Web log were studied in [AS95]. MaXImal frequent
sequences were studied in [XD01a].
. .
The World Wide Web Consortium (W3C) 1s a consortiUm of over 400 member
organization.s whose purpose is to develop protocols needed to ensure the use and growth
of the Web. Various versions of HTML have been proposed. The most recent, XHTML
1.0 uses the syntax of XML. XML is the next-generation markup language to be used
by
'
web documents. The more structured nature of XML facilitates easier access and
querying for Web documents. Both XML and HTML are based on standard generalized
markup languaged (SGML), which is an ISO standard from 1986.

Recently there have been several proposals for query languages aimed at the Web.
Most of these are extensions of SQL (WebSQL, W3QL, WebOQL), while others are
based on deductive type rules (WebLog). Instead of using relations, WebSQL uses virtual
relations, which are viewed as abstractions of Web documents [MMM96]. When using
WebSQL, the actually work of accessing the Web is performed by a traditional search
engine. WebSQL queries are converted into search engine queries, and results of these
queries are compiled and returned to the user. Similarly, W3QL [KS95] takes advantage
of a search engine to access the Web. W3QL, however, allows the use of external code or
Unix commands to be embedded within the query. WebLog uses deductive rules rather
than an SQL-like syntax [LSS96]. It is considered to be a second-generation language
because it actually can generate new Web documents. Based on OQL, WebOQL views
data as consisting of trees, while groups of trees are called webs [GM98].
There are several ongoing research prototypes examining Web mining. The WEB
MINER system being developed at DePaul University [MobOO] consists of both Web log
preparation steps (cleaning, transaction identification, and integration) and mining func
tions. An ongoing researo
)l project at the University of Minnesota, W
ebSIFT, has produced
a comprehensive system design, including preprocessing, knowledge discovery, and pat
tern analysis steps [CTS97]. The data mining functions performed include classification
of Web pages, identification of sequential patterns of Web usage data, clustering of both
pages and users, generation of association rules, and creation of usage statistics.
IBM has a Web mining product called SurfAid Analytics [IBMOO]. SurfAid per
forms traversal pattern analysis, referral analysis, and otqer data mining activities. Refer
ral analysis determines where visitors came from when they entered a Web page.
8.1
C H A P T E R 8
Spatial M i n i ng
8.1 INTRODUCTION
8.2 SPATIAL DATA OVERVIEW
8.3 SPATIAL DATA MINING PRIMITIVES
8.4 GENERALIZATION AND SPECIALIZATION
8-5 SPATIAL RULES
8.6 SPATIAL CLASSIFICATION ALGORITHMS
8.7 SPATIAL CLUSTERING ALGORITHMS
8.8 EXERCISES
INTRODUCTION
Spatial data are data that have a spatial or location component. Spatial data can be
viewed as data about objects that themselves ar't located in a physical space. This may
be implemented with a specific location attribute(s) such as address or latitude/longitude
or may be more implicitly included such as by a partitioning of the database based
on location. In addition, spatial data may be adcessed using queries containing spatial
operators such as near, north, south, adjacent, and contained in. Spatial data are stored in
spatial databases that contain the spatial data and nonspatial data about objects. Because
of the inherent distance information associated with spatial data, spatial databases are
often stored using special data structures or indices built using distance or topological
information. As far as data mining is concerned, this distance information provides the
basis for needed similarity measures.
Spatial data are required for many current information technology systems. Geo
graphic inf
ormation systems (GIS) are used to store infomtation related to geographic
locations on the surface of the Earth. This includes applications related to weather,
community infrastructure needs, disaster management, and hazardous waste. Data min
ing activities include prediction of environmental catastrophes. Biomedical applications,
including medical imaging and illness diagnosis) also require spatial systems.
Spatial mining, often called spatial data mining or knowledge discovery in spatial
databases, is data mining as applied to spatial databases or spatial data. Some of the
applications for spatial data mining are in the areas of GIS systems, geology, environ
mental science, resource management, agriculture, medicine, and robotics. Many of the
techniques discussed in previous chapters are applied directly to spatial data, but there
also are new techniques and algorithms developed specifically for spatial data mining.
221

222 Chapter 8 Spatial Mining
We investigate these issues in this chapter. Before investigating spatial mining, we first
provide a brief introduction to spatial data and databases.
8.2 SPATIAL DATA OVERVIEW
Accessing spatial data can be more complicated than accessing nonspatial data. There
are specialized operations and data structures used to access spatial data.
8.2.1 Spatial Queries
Because of the complexity of spatial operations, much work has been performed to
examine spatial query processing and its optimization.
A traditional selection query accessing nonspatial data uses the standard comparison
operations: >, <, :::;, ::=:, :j=. A spatial selection is a selection on spatial data that may use
other selection comparison operations. The types of spatial comparators that could be
used include near, north, south, east, west, contained in, and overlap or intersect. The
following are examp�es of several spatial selection queries:
'
• Find all houses near Mohawk Elementary School.
• Find the nearest fire station to 963 1 Moss Haven Drive in Dallas.
A special join operation applied to two spatial relations is called a spatialjoin. In
some ways, a spatial join is like a regular relational join in that two records are joined
together if they have features in common. With a traditional join, two records must
have attributes in common that satisfy a predefined relationship (such as equality in an
equijoin). With a spatial join, the relationship is a spatial one. The type of relationship is
based on the type of spatial feature. For example, the nearest relationship may be used
for points, while the intersecting relationship is used for polygons.
In GIS applications, it is common to have different views of the same geographic
area. For example, city developers must be able to see where infrastructure facilities are
located, including streets, power lines, phone lines, and sewer lines. At another level,
they might be interested in actual elevations, building locations, and rivers. Each of these
types of information could be maintained in separate GIS files. Merging these disparate
data can be performed using a special operator called a map overlay.
A spatial object usually is described with both spatial and nonspatial attributes.
Some sort of location type attribute must be included. The location attribute could identify
a precise point, such as a latitude or longitude pair, or it may be more logical such as
a street address or zip code. Often, different spatial objects are identified by different
locations, and some sort of translation between one attribute and the other is needed to
perform spatial operations between the different objects. As in SAND, the nonspatial
attributes may be stored in a relational database, while each spatial attribute is stored in
some spatial data structure. Each tuple in the relationship represents the spatial object,
and a link to the spatial data structure is stored in the corresponding position in the
nonspatial tuple.
Many basic spatial queries can assist in data mining activities. Some of these
queries include:
• A region query or range query is a query asking for objects that intersect a given
region specified in the query.
Section 8.2 Spatial Data Overview 223
• A nearest neighbor query asks to find objects that are close to an identified object.
• A distance scan finds objects within a certain distance of an identified object, but
the distance is made increasingly larger.
All of these queries can be used to assist in clustering or classification.
8.2.2 Spatial Data Structures
Because of the unique features of spatial data, there are many data structures that have
been designed specifically to store or index spati:;tl data. In this section, we briefly exam
ine some of the more popular data structures. Many of these structures are based on
extensions to conventional indexing approaches, such as B-trees or binary search trees.
Nonspatial database queries using traditidmal indexing structures, such as a B
tree, access the data using an exact match query. However, spatial queries may use
proximity measures based on relative locations of spatial objects. To efficiently perform
these spatial queries, it is advisable that objects close in space be clustered on disk. To
this end, the geographic space under consideration may be partitioned into cells based
on proximity, and these cells would then be related to storage locations (blocks on disk).
The corresponding data structure would be constructed based on these cells.
A common technique used to represent a spatial object is by the smallest rectangle
that completely contains that object, minimum bounding rectangle (MBR). We illustrate
the use of MBRs by looking at a lake. Figure 8.1(a) shows the outline of a lake. If we
orient this lake in a traditional coordinate system with the horizontal axis representing
east-west and the perpendicular axis north-south, we can put this lake in a rectangle
(with sides parallel to the axes) that contains it. Thus, in Figure 8.1(b) we show an MBR
that can be used to represent this lake. Alternatively, in Figure 8. l(c) we could represent
it by a set of smaller rectangles. This option can provide a closer fit to the actual object,
but it requires multiple MBRs. An MBR can easily be represented by the coordinates for
two nonadjacent vertices. So we could represent the MBR in Figure 8.1(b) by the pair
{(XJ , YJ), (x2, Y2)}. There are other ways to store the MBR values, and the orientation of
the MBRs need not be with the axes.
We use the triangle shown in Figure 8.2(a) as a simple spatial object. In
Figure 8.2(b) we show an MBR for the triangle. Spatial indices can be used to assist
in spatial data mining activities. One benefit of the spatial data structures is that they
cluster objects based on location. This implies that objects that are close together in the
<XJ>Yt>
(a) Lake (b) MBR for lake (c) Smaller MBRsforlake
FIGURE 8. 1 : MBR example.

224 Chapter 8
�
Ll
L3l
�
�
L:C=::::J
�
�
Spatial Mi ning
(a)Triangle (a) MBR forTriangle
FIG U RE 8.2: Spatial object example.
2
10 9 6
11 1J. 7 8
18 17
15 16 19 20
3 4
(a) Representing triangle with quadrants (b) Quad tree
FIGURE 8.3: Quad tree example.
n-dimensional space tend to be stored close together in the data structure and on disk.
Thus, these structures could be used to reduce the processing overhead of an algorithm
by limiting its search space. In effect, filtering is performed as you traverse down a tree.
In addition, spatial queries can be more efficiently answered by use of these structures.
Quad Tree. One of the original data structures proposed for spatial data is that of
a quad tree. A quad tree represents a spatial object by a hierarchical decomposition of the
space into quadrants (cells). This process is illustrated in Figure 8.3(a) using the triangle
in Figure 8.2. Here the triangle is shown as three shaded squares. The spatial area has
been divided into two layers of quadrant divisions. The number of layers needed depends
on the precision desired. Obviously, the more layers, the more overhead is required for
the data structure. Each level in the quad tree corresponds to one of the hierarchical
layers. Each of the four quadrants at that layer has a related pointer to a node at the
next level if any of the lowest level quadrants are shaded. We label the quadrants at each
Section 8.2 Spatial Data Overview 225
level in a counterclockwise direction starting at the upper right quadrant (as shown in the
figure). Square 0 is the entire area. Square 1 is the upper right at level one. Square 15
is the square in the lower left comer at the second level. In this figure, the triangle is
represented by squares 12, 13, and 14 because it intersects these three regions. The quad
tree for this triangle is shown in Figure 8.3(b). Only nodes with nonempty quadrants are
shown. Thus, there are no nodes for quadrants 1 and 4 and 1their subquadrants.
MBRs are similar to the quadrants in the quad tree except that they do not have to
be of identical sizes. If hierarchies of MBRs exist, they do not have to be regular as in
the quadrant decompositions.
R-Tree. One approach to indexing spatial data represented as MBRs is an R-tree.
Each successive layer in the tree identifies smaller rectangles. In an R-tree, cells may
actually overlap. An object is represented by an MBR that is located within one cell.
Basically, a cell is the MBR that contains the related set of objects (or MBRs) at a lower
level of decomposition. Each level of decomposition is identified with a layer in the tree.
As spatial objects are added to the R-tree, it is created and maintained by algorithms
similar to those found for B-trees. The size of the tree is related to the number of objects.
Looking at a space with only the basic triangle, as seen in Figure 8.2, a tree with only a
root node would be created. We illustrate a more complicated R-tree in Figure 8.4. Here
there are five objects represented by the MBRs D, E, F, G, and H. The entire geographic
space is labeled A and is shown as the root of the tree in Figure 8.4(b). Three of the
objects (D, E, F) are contained in an MBR labeled B, while the remaining two (G, H)
are in MBR C.
Algorithms to perform spatial operators using an R-tree are relatively straight
forward. Suppose we wished to find all objects that intersected with a given object.
Representing the query object as an MBR, we can search the upper levels of the R-tree
to find only those cells that intersect the MBR query. Those subtrees that do not intersect
the query MBR can be discarded.
k-D Tree. A k-D tree was designed to index multiattribute data, not necessarily
spatial data. The k-D tree is a variation of a binary search tree where each level in the
A
D E
B
F c
(a) Partitioning with MBRs (b) R-tree
FIGURE 8.4: R-tree example.

226 Chapter 8
D
E
H
�
�
B
n
u
Spatial Mining
A
F
G
c
(a) Divide and conquer partitioning
FIGURE 8.5: k-D tree example.
A
(b) k-D tree
tree is used to index one of the attributes. We illustrate the use of the k-D tree assunring
a two-dimensional space. Each node in the tree represents a division of the space into
two subsets based on the division point used. in addition, the division alternates between
the two axes.
In Figure 8.5 we show a k-D tree using the same data we used for the R-tree. As
with the R-tree, each lowest level cell has only one object in it. However, the divisions
are not made using MBRs. Initially, the entire region is viewed as one cell and thus the
toot of the k-D tree. The area is divided first along one dimension and then along another
dimension until each cell has only one object in it. In this example, we see that the entire
region, A, is first divided into two cells (B, C) along the horizontal axis. Then, looking
at B, we see that it is divided into D and E. D is finally divided into H and I.
8.2.3 Thematic Maps
Thematic maps illustrate spatial objects by showing the distribution of attributes or
themes. Each map shows one (or more) of the thematic attributes. These attributes
describe the important nonspatial features of the associated spatial object. For exam
ple, one thematic map may show elevation, average rainfall, and average temperature.
Raster-based thematic maps represent the spatial data by relating pixels to attribute val
ues of the data. For example, in a map showing elevation, the color of the pixel can be
associated with the elevation of that location. A vector-based thematic map represents
objects by a geometric structure (such as their outline or MER). In addition, the object
then has the thematic attribute values.
8.2.4 Image Databases
In image databases the data are stored as pictures or images. These databases are used
in many applications, including medicine and remote sensing.
Some early classification work performed using large image databases looked at
ways to classify astrononrical objects. One of the applications of this work is to identify
Section 8.3 Spatial Data Mining Primitives 227
volcanos on Venus from images taken by the Magellan spacecraft [FWD93]. This system
consisted of three parts: data focusing, feature extraction, and classification. The first
component deternrines which of the areas of the images is the most likely to contain
volcanos. Here the intensity of a central point of a region is compared with that of the
background. The important features of these areas are extracted and stored in the second
part. The focusing portion compares the intensity of a central point of a region with
that of the background. During the second phase, interesting features are identified and
extracted. Finally, these features are classified based on classifiers built using training
data provided by domain experts. The third portion uses a decision tree to perform the
actual classification. The tree is created using ID3 and training examples provided by
experts. An accuracy of 80% was achieved.
A related work also used decision trees to classify stellar objects [FS93]. As with
the volcano work, the first two steps were to identify areas of the images of inter
est and then to extract information about these areas. Multiple trees were created, and
from these sets of rules were generated for classification. Accuracy was found to be
approximately 94%. When compared to several neural network approaches, the decision
tree/rules approach was found to be much more accurate. Both of these studies found the
need to normalize the extracted features to compensate for differences between different
images. For example, two images could differ based on the angle at which the image
was taken.
8.3 SPATIAL DATA MINING PRIMITIVES
Operations needed to support spatial data nrining involve those required for spatial
databases. We review some of these in this section. In these discussions, we assume
that A and B are spatial objects in a two-dimensional space. Each object can be viewed
as consisting of a set of points in the space: (xa , Ya) E A and (xb , Yb) E B.
As defined in [EFKSOO], there are several topological relationships that can exist
between two spatial objects. These relationships are based on the ways in which two
objects are placed in a geographic domain:
• Disjoint: A is disjoint from B if there are no points in A that are contained in B.
• Overlaps or intersects: A overlaps with B if there is at least one point in A that
is also in B.
• Equals: A equals B if all points in the two objects are in common.
• Covered by or inside or contained in: A is contained in B if all points in A are
in B. There may be points in B that are not in A.
• Covers or contains: A contains B iff B is contained in A.
While data nrining tasks may not specifically address these relationships, the similarity
between spatial objects certainly can be defined based partially on these relationships.
Based on the placement of the objects in the space, relationships with respect
to direction may be defined. These usually are defined by adding the traditional map
orientations to the space. Thus, we have the relationships such as north, south, east,
west, and so on. What makes these relationships difficult to identify is the irregular
shape of spatial objects and the fact that they may overlap.

228 Chapter 8 Spatial Min ing
As mentioned in Chapter 3, the Euclidean and Manhattan measures are often
to me su th d· b
·
used
a re e 1stance
.
etween two pomts. The distance between two spatial ob·ects
can be defined as extensiOns to these two traditional definitions:
�
• Minimum:
dis(A , B) = min dis((xa . Ya), (xb, Yb))
(xa ,Ya)EA, (xb,Yb)EB
(8. 1)
• Maximum:
dis(A , B) = max dis((xa , Ya) , (xb, Yb))
(Xa,ya)EA,(xb,Yb)EB
(8.2)
• Average:
• Center:
dis(A, B) = dis((Xea , Yea) , (Xeb, Yeb)) (8.4)
where (Xea , Yea) is a center point for object A and (Xeb, Yeb) for B.
Note th� si�larity to distance measures used in clustering. In fact, you can think of
t
�e spatial obJect as a clu�ter �f the points within it. The center points used for the last
distance formula can be Identified by finding the geometric center of the b' t p
1 'f MB
. . o �ec . or
exa�p e, I
.
an R IS used, the distance between objects could be found using the
Euclidean
.
dist�ce between the center of the MBRs for the two objects.
.
Spatial �bJects may be retrieved based on selection, aggregation, orjoin-type opera-
�I�ns. A selectiOn �ay b� performed based on the spatial or nonspatial attributes. Retriev
mo ?as�� on s�atlal attnbutes could be performed using one of the spatial operators. A
spatial JOin retneves based on the relationship between two spatial objects.
8.4 GENERALIZATION AND SPECIALIZATION
The us� of a concept hie
.
r�chy shows le"
�
·els of relationships among data. When applied
to s�atlal
.
data ch�actenstics, concept hierarchies allow the development of rules and
relatw�ships at differ�nt levels in the hierarchy. This is similar to the use of roll up
and �nl
.
l down oper�tl�ns i
.
n O�AP. "W_e have also seen this idea used in generalized
assocm
.
twn rule
.
s. A Siillllar Idea IS used m the generalization and specialization concepts
found m mac�ne learning
:
In these cases, however, the hierarchy is not necessarily
related t� s�ati�l data. Spatial data mining techniques have involved both generalization
and specializatiOn type approaches.
8.4. 1 Progressive Refinement
Because of the massive amounts of data found in spatial applications approximate
answers �ay be made before finding more accurate ones. The use of MBRs is a method
�o appro�Imate the shape of an object. Quad trees, R-trees, and most other spatial index
mg techmques use a type of progressive refinement. They estimate the shape of objects at
8.4.2
Section 8.4 Genera lization and Special ization 229
higher levels in the tree structure, and lower-level entries provide more precise descrip
tions of the spatial objects. Progressive re
finement can be viewed as filtering out data
that are not applicable to a problem.
With progressive refinement, the hierarchical levels are based on spatial relation
ships. Example 8.1 illustrates the idea of progressive refinement. Here spatial relationships
can be applied at a more coarse (move up the hierarchy) or more fine (move down the
hierarchy) level.
EXAMPLE 8.1
Suppose that a computer science student wishes to identify apartments close to the SMU
Computer Science and Engineering (CSE) Department. A given database listing available
apartments in the Dallas metroplex will contain many apartments nowhere near the
SMU campus. An initial filtering of the inappropriate elements can be made by finding
apartments that are "generalized close" to the CSE Department. This can be performed
at any of the levels in the concept hierarchy, Figure 8.6 shows the idea. The closest
apartments to SMU probably would be in the Park Cities. By filtering out all apartments
in all subtrees other than those for the Park Cities, apartments that are fairly close to
SMU would be found. Suppose that a lower level in the concept hierarchy existed that
included zip code. If apartments in the same zip code as the CSE Department were found,
an even finer estimate of close could be used. This process quickly filters out apartments
that could not possibly be used to answer the question. Here a coarser predicate is first
used to filter out potential answers. This predicate can be recursively refined until the
precise answers are found. Note that when looking at the concept hierarchy, the coarser
predicates can be applied to the MBRs at the higher levels, while the finer predicates are
applied at the lower levels.
Generalization
As with OLAP, generalization is driven by a concept hierarchy and can be viewed as
the process of deriving information at a high level based on information found at lower
levels. Concept hierarchies for spatial data can be both spatial and nonspatial. A spa
tial hierarchy is a concept hierarchy that shows the relationships between geographic
areas. Figure 8.6 shows a spatial hierarchy. In Chapter 6, Figure 6.7 illustrated a nonspatial
Dallas-Fort Worth Metroplex
Forth Worth Dallas Arlington Mid-cities Northern suburbs Park cities
�
Preston Hollow M Streets Lakewood East University Park Highland Park
FIGURE 8.6: lllustration of progressive refinement used in Example 8. 1 .

hierarchy. Generalization can bepetformed using either ofthese two hierarchies. When the
spatial data are generalized, the nonspatial data must be appropriately changed to reflect
the nonspatial data associated with the new spatial area. Similarly, when the nonspatial
data are generalized, the spatial data must be appropriately modified. Using these two types
of hierarchies, generalization as applied to spatial data can be divided into two subclasses:
spatialdata dominant and nonspatial data dominant [LH093]. Both ofthese subclassescan
be viewed as a type of clustering. Spatial data dominant does the clustering based on spatial
locations (so that objects close together are grouped), whereas nonspatial data dominant
clusters by similarity of nonspatial attribute values. These approaches are referred to as an
attribute-orientedinduction because the generalization process is based on attribute values.
With spatial data dominant generalization, generalization is first applied to the
spatial data, and then the related nonspatial attributes are modified accordingly. General
ization is petformed until a threshold number of regions is reached. For example, deter
mining the average rainfall in the southwestern United States could be done by finding
the mean average rainfall for all states shown to be in the Southwest by a spatial hier
archy. Thus, the spatial hie�archy determines which lower-level regions are found in the
higher-level region being queried. Determining how to apply the generalization to the
nonspatial data is, however, not always a straightforward aggregation operation. Deter
mining the average rainfall in this case actually treats each state the same. However, a
weighting by geographic area might be used to provide a more accurate average rainfall
for the higher-level region being queried.
An alternative approach is to generalize the nonspatial attribute values as well.
Generalization is based on grouping of data. Adjacent regions are merged if they have
the same generalized values for the nonspatial data. Suppose that instead of average
rainfall values, we simply returned values that represented the southwestern cluster. We
could assign values of heavy, medium, light, and so on to describe the rainfall rather than
providing actual numeric values. Algorithm 8.1 shows the spatial-dominant approach. A
threshold that indicates the maximum number of regions may be given. Based on this
threshold, the correct level in the hierarchy is chosen, and thus the number of regions is
determined.
ALGORITHM 8.1
Input :
D
H
c
q
/ / Spat ial database
/ / Spatial hierarchy
/ / Concept hierarchy
//Query
Output :
R I /Rule that states the general characteristics requested
SPATIAL - data- dominant algorithm:
d = set of data obtained from D based on selection criteria in q;
Following the structure of H, combine data into regions unt i l
e i ther the desired threshold number of regions is found
or the requested level in H i s obt aine d ;
for each region found do
perform an attribute- oriented induction on the
nonspat ial attributes ;
Generate and output a rule that summarizes the resul ts found ;
Section 8.4 Generalization and Specialization 231
Although not shown here, the nonspatial-data-dominant generalization technique
works in a similar fashion. The first step in this algorithm is to retrieve the data based on
the nonspatial selection criteria stated in the query. Needed attribute-oriented induction
is then petformed on the retrieved nonspatial data. The nonspatial concept hierarchies are
consulted to petform this. During this step, nonspatial attribute values are generalized
to higher-level values. These generalizations are higher-level summary values of the
lower-level specific values. For example, if average temperature were generalized, several
different average temperatures (or ranges) could be combined and labeled "hot." The third
step is to petform spatial-oriented generalization. Here neighboring regions with the same
(or similar) nonspatial generalized values are merged. This is done to reduce the number
of regions returned in response to the query.
A negative of these approaches is that the hierarchy must be predefined by domain
experts, and the quality of any data mining requests depends on the hierarchy provided.
The complexity to create the hierarchies is 0 (n log n).
8.4.3 Nearest Neighbor
We introduced the idea of a nearest neighbor in Chapter .5 with respect to clustering.
This idea of identifying objects that are close together is a common query type in spatial
databases. The nearest neighbor distance is the minimum distance between an object and
all other objects in the space.
8.4.4 STING
The ST
atistical INformation Grid-based method (STING) uses a hierarchical technique to
divide the spatial area into rectangular cells similar to a quad tree. The spatial database
is scanned once, and statistical parameters (mean, variance, distribution type) for each
cell are determined. Each node in the grid structure summarizes the information about
the items within it. By capturing this information, many data mining requests, includ
ing clustering, can be answered by examining the statistics created for the cells. Thus,
only clusters with vertical and horizontal boundaries are generated. However, the entire
database need not be scanned after this statistical information is captured. This can be
quite efficient when several data mining requests may be made against the data. Unlike
the generalization and progressive refinement techniques, no predefined concept hierarchy
must be provided.
The STING approached can be viewed as a type of hierarchical clustering tech
nique. The first step is to create a hierarchical representation (like a dendrogram). The
created tree successively divides the space into quadrants. The top level in the hierarchy
consists of the entire space. The lowest level has one leaf for each of the smallest cells.
The original proposal was for a cell to have four subcells (grids) at the next lowest level.
The division of cells is identical to that petformed for quad trees. In general, however,
the approach would work with any hierarchical decomposition of the space. Figure 8.7
illustrates the nodes at the first three levels of the constructed tree.
The process to create the tree is shown in Algorithm 8.2. Each cell in the space
corresponds to a node in the tree and is described with both attribute-independent (count)
data and attribute-dependent (mean, standard deviation, minimum, maximum, distribu
tion) data. As the data are loaded into the database, the hierarchy is created. Placement
of an item into a cell is completely determined by its physical position. Algorithm 8.2

(a) Level l (b) Level 2 (c) Level 3
FIGURE 8.7: Nodes in STING structure.
is divided into two parts. The first part creates the hierarchy and the second part fills in
the values. Since the number of nodes in the tree is less than the number of items in the
database, the complexity of STING BUILD is O(n).
ALGORITHM 8.2
Input :
D
k
Output :
T
/ /Data to be placed in the hi erarchi cal structure
/ /Number of de s i red cells at the l owest level
/ / Tree
STING BUILD algorithm:
/ / Create empty tree from top down .
T = root node with data values init ial i z e d ; / / Initially only
i = 1 ;
repeat
for each node in level i do
root node
create 4 chi ldren nodes with init ial values ;
i = i + 1 ;
unt i l 4
i
= k ;
I I Populate tree from bottom up .
for each i tem in D do
determine leaf node j associated with the pos i t i on of D;
update values of j based o n attribute values in item ·
i := log4 (k) ;
'
repeat
i := i - 1 ;
for each node J ln level i do
update values of j based on att ribute values
in i t s 4 chi ldren ;
unt i l i = 1 ;
I
The actual STING algorithm is shown in Algorithm 8.3. The algorithm assumes that
a query, q, that can be answered from the stored statistical information in the constructed
tree, T, is requested. Such a query might be to find the range of price of apartments
near SMU.
.
The statistics (minimum and maximum) of the apartment rental prices for
the appropnate cells should be determined. The cell that SMU is in would determine
the actual values for those closest to SMU. In addition, the query might retrieve the
Section 8.5 Spatial Rules 233
information for the cells surrounding this cell or perhaps at the next highest level in the
tree that contains the c�ll where SMU is located. Th� nearby cells could be determined
using some distance function. The crucial concept here is that the appropriate cells must
be determined and then the information from those cells, in the constructed tr�e must be
retrieved. A breadth-first tree traversal is used to examine the tree. However, a complete
traversal of the tree is not performed. Only children of relevant nodes are examined.
Here the concept of relevance is much like that with IR queries except that relevance is
determined by estimating the proportion of the objects in that cell that meet the query
conditions. The complexity of the STING algorithm is O (k) where k is the number of
cells at the lowest level. Obviously, this is the space taken up by the tree itself. When
used for clustering purposes, k would be the largest number of clusters created.
ALGORITHM 8.3
Input :
T
q
Output :
R
/ /Tree
//Query
//Regi ons of relevant cells
STING algorithm:
i = 1
repeat
for each node in level i do
det ermine if this cell i s rel evant to q and mark as such ;
i = i + 1
until all layers in the tree have been vis ited ;
ident ify neighboring cells of rel evant c e l l s to create
regions of cell s ;
Calculating the likelihood that a cell is relevant to a query is based on the percentage
of the objects in the cell that satisfy the query constraints. Using a predefined confidence
interval, if this proportion is high enough, then that cell is labeled as relevant. The
statistical information associated with these relevant cells is used to answer the query. If
this approximate answer is not good enough, then the associated relevant objects in the
database may have to be examined to provide a more exact response. The cells found by
STING approximate those found by DBSCAN. Cells that are found to be close enough
to relevant cells are included in the regions of cells that are found by the algorithm.
8.5 SPATIAL RU LES
Spatial rules can be generated that describe the relationship between and structure of
spatial objects. There are three types of rules that can be found during spatial data
mining [KAH96]. Spatial characteristic rules describe the data. Spatial discriminant
rules describe the differences between different classes of the data. They describe the
features that differentiate the different classes. Spatial association rules are implications
of one set of data by another. The following examples illustrate these three types of rules:
• Characteristic rule: In Dallas the average family income is $50,000.
• Discriminant rule: In Dallas the average family income is $50,000, while in Plano
the average family income is $75,000.

• Association rule: In Dallas the average family income for families living near
White Rock Lake is $100,000.
Characterization is the process of finding a description for a database or some
subset thereof. All of these rules can be thought of as spedal types of characterizations.
The characteristic rule is the simplest.
Another common approach to summarizing spatial data is that of performing a
trend detection, which is viewed as a regular change in one or more nonspatial attribute
values for spatial objects as you move away from another spatial object [EFKS98]. For
example, the average price per square foot of a house may increase as the proximity to
the ocean increases. Regression analysis may be used to identify a trend detection.
8.5.1 Spatial Association Rules
Spatial association rules are association rules about spatial data objects. Either the
antecedent or the consequent of the rule must contain some spatial predicates (such
as near):
• Nonspatial antecedent and spatial consequent: All elementary schools are located
close to single-family housing developments.
• Spatial antecedent and nonspatial consequent: If a house is located in Highland
Park, it is expensive.
• Spatial antecedent and spatial consequent: Any house that is near downtown is
south of Plano.
Support and confidence for spatial association rules is defined identically to that
forregular association rules. Unlike traditional association rules, however, the underlying
database being examined usually is not viewed as a set of transactions. Instead, it is a
set of spatial objects.
The simplest spatial association rule generation algorithm is found in [KH95]. The
approach is similar to that discussed earlier for classification in that a two-step approach is
used. As with traditional association rule algorithms, all assodation rules that satisfy the
minimum confidence and support are generated by this algorithm. Because of the large
number of possibilities for topological relationships, it is assumed that the data mining
request indicates what spatial predicate(s) is to be used. Once the relative subset of the
database is determined, relationships of this type are ide.ntified. It initially is assumed
that "generalized" versions of the topological relationships are used. The generalized
relationships are satisfied if some objects higher Up the concept hierarchy satisfy it.
For example, zip codes may be used instead of the exact structure of the house. At
this level, a filtering is performed to remove objects that could not possibly satisfy the
relationship.
To illustrate the concept of generalization with the spatial relationships, we follow
the example found in [Kop99]. Suppose that the topological relationship being examined
is "close_to." The GIS system would define precisely what this predicate means. For
example, it could define the relationship based on the Euclidean distance between the
two spatial objects. In addition, it might be defined differently based on the type of
objects in question. The generalization of "close_to" that is written as "g_close_to" may
Section 8.5 Spatial Rules 235
be defined by a hierarchy that shows that g_close_to contains close_to as well as other
predicates (such as contains and equal). A first step in d�termining satisfiability of the
close_to predicate would be to look at a coarse evaluation of g_close_to. The co�se
evaluation is used as a type of filter to efficiently rule out objects that could not posstbly
satisfy the true predicate. The coarse predicate coarse_g_close_to is satisfied by objects
if their MBRs satisfy g_close_to. Only those objects that satisfy coarse_g_close_to are
examined to see if they satisfy g_close_to.
The five-step algorithm is outlined in Algorithm 8.4. It is assumed that a �ata
mining query is input. The query contains selection informat�on that �s used to r�tneve
the objects from the database that are of interest. The topologtc�l predtcates �e��ng the
spatial relationships of interest are also input. Using these predtcates, P, an tmttal
.
table
is built Cp that identifies which pairs of objects satisfy P at a coarse level. The mput
minim�m s�pports are actually a set of support values to be used at different levels in
the processing. s[l] is the support level to be used at the coarse filtering level. Af�er th�s
filtering, the pairs of objects that satisfy the coarse predicates are counted to see tf therr
support is above the minimum. In effect, this frequent coarse predicate
.
(FCP) database
is the set of large one-itemsets. The predicates in FCP are then exammed to fin
.
d the
frequent predicates at a fine level (FFP). The last step expands these frequ�nt pre�t�ates
of size 1 to all arbitrary predicate sizes and then generates the rules as wtth tradttlonal
association rules. This is performed similarly to Apriori. By finding the FCRs first, the
number of objects to be examined is reduced at the last step.
ALGORITHM 8.4
Input :
D
c
s
Ci
q
p
/ /Data , inc luding spat ial and nonspatial attributes
/ / Concept hierarchies
/ /Minimum support for levels
/ /Conf idence
/ /Query to retri eve interested obj ects
/ / Topologi cal predi cate ( s ) of interest
Output :
R / /Spat ial assoc iation rules
SPATIAL association rule algorithm:
d = q(D) ;
CP is built by applying the coarse predicate version of P to d ;
1 1 CP cons i s t s of the set of coarse predicates sat i s f ied by
pairs of obj ects in d .
determine the set o f frequent coarse predicates FCP by finding
the coarse predicates that satisfy s ;
f ind the set of frequent fine predicates FFP from FCP;
f ind R by finding all frequent fine predicates and then
generating rules ;
This algorithm works in a similar manner to the Apriori algorithm in that
large "predicate sets" are determined. Here a predicate set is a �et of
.
predicates of
interest. A !-predicate might be {(close_to, park) }, so all spatial o�Jects that are
close_to a park will be counted as satisfying this predicate. A 2-predtcate could be
{(close_to, park), (south_of, Plano)}. Counts of 1-predicate sets are counted, then th?se
that are large are used to generate 2-predicate sets, and these are then counted. In actuality,

the algorithm can be used to generate multilevel association rules if desired or rules at a
coarse level rather than a fine leveL
8.6 SPATIAL CLASSIFICATION ALGORITHMS
Spatial classification problems are used to partition sets of spatial objects. Spatial objects
could be classified using nonspatial attributes, spatial predicates (spatial attributes), or
spatial and nonspatial attributes. Concept hierarchies may be used, as may sampling. As
with other types of spatial mining, generalization and progressive refinement techniques
may be used to improve efficiency.
8.6.1 103 Extension
The concept of neighborhood graphs has been applied to perform classification of spatial
objects using an ID3 extension [EKS97]. A neighborhood graph is a graph constructed
from the objects in the space. Each object becomes a node in the graph. The edges
are constructed from the neighbors; that is, two nodes are connected by an edge in the
neighborhood graph if 9ne is a neighbor of the other. "Neighbor" can be defined based
on any relationship between the spatial objects such as distance less than a particular
threshold, satisfiability of a topological relationship between the objects, or direction
relationship. Note that some of the relationships are order relationships and others are not.
The idea of the algorithm is to take into account the objects that are near a given
object. A max-length indicator is input that specifies the maximum length of a neighbor
hood path starting at the node. This then identifies a set of nodes that are associated with
the target hade. ID3 then considers for classification purposes not only the nonspatial
attributes of the target object, but also those in neighboring objects.
8.6.2 Spatial Decision Tree
One spatial classification technique builds decision trees using a two-step process similar
to that used for association rules [KHS98]. The basis of the approach is that spatial objects
can be described based on objects close to them. A description of the classes is then
assumed to be based on an aggregation of the most relevant predicates for objects nearby.
To construct the decision tree, the inost relevant predicates (spatial and nonspatial)
are first determined. It is hoped that this process will create smaller and more accurate
decision trees. These relevant predicates are the ones that will be used to build the
decision tree. It is assumed that a training sample is used to perform this step and that
weights are assigned to attributes and predicates. Initial weights are 0. Two corresponding
objects are examined for each object. The nearest miss is the spatial object closest to the
target object that is in a different class. The nearest hit is the closest target in the same
class. For each predicate value in the target object, if the nearest hit object has the same
value, then the weight of that predicate is increased. If it has a different value, then the
weight is decreased. Likewise, the weight is decreased (increased) if the nearest miss
has the same (different) value. Only predicates with positive weights above a predefined
threshold are then used to construct the tree. It is proposed that, because of the complexity
of finding the relevant predicates, relevant predicates be found first at a coarse level and
then at a finer leveL MBRs, instead of actual objects, and a generalized coarse close_to
relationship are first used to find the relevant predicates. Then these relevant predicates
and the true objects are used during the second pass.
8.7
Section 8.7 Spatial Cl ustering Algorithms 237
For each object in the sample, the area around it, called its buffer, is examined.
A description of this buffer is created by aggregating the values of the most relevant
predicates of the items in the buffer. Obviously, the size and shape of the buffer impact
the resulting classification algorithm. It is possible, although unrealistic, to perform an
exhaustive search around all possible buffer sizes and shapes. The objective would be to
choose the one that results in the best discrimination between classes in the training set.
This would be calculated using the information gain. Other approaches based on picking
a particular shape were examined, and the authors finally used circles (equidistance
buffers).
To construct the tree, it is assumed that each sample object has associated with it a
set of generalized predicates that it satisfies. Counts of the number of objects that satisfy
(do not satisfy) each predicate can then be determined. This is then used to calculate
information gain as is done in ID3. Instead of creating a multiway branching tree, a
binary decision tree is created. The resulting algorithm to construct the decision tree is
shown in Algorithm 8.5.
ALGORITHM 8.5
Input :
D
c
Output :
T
/ /Data , inc luding spatial and nonspatial attribut e s
/ / Concept hierarchi es
//B inary de c i s i on tree
SPATIAL deci s ion tree algorithm:
f ind a sample S of data from D with known classi f i cat ion;
ident i fy the be st predicates p to us e for clas s i f i c ation ;
determine the best buffer size and shape ;
us ing p and C, general i z e the predicates for each buf f e r ;
bui l d binary T us ing the generali zed predi cates and ID3 ;
SPATIAL CLUSTERING ALGORITHMS
Spatial clustering algorithms must be able to work efficiently with large multidimen
sional databases. In addition, they should be able to detect clusters of different shapes.
Figure 8.8 illustrates what we mean. This figure shows clusters in a two-dimensional
space. Obviously, by looking at this figure it is easy to see that there are four different
clusters, each of a fairly irregular shape. A good spatial clustering algorithm should be
able to detect these four clusters even though the shapes are not regular, and some points
in one cluster may actually be closer to some points of other clusters rather than to points
in its own cluster. An algorithm that works using centroids and simple distance measures
probably will not be able to identify the unusual shapes.
Other desirable features for spatial clustering are that the clusters found should be
independent of the order in which the points in the space are examined and that the
clusters should not be impacted by outliers. In Figure 8.8 the outliers in the lower right
part of the figure should not be added to the larger cluster close to them.
Many of the clustering algorithms discussed in Chapter 5 may be viewed as spa
tial. In the following sections, we evaluate additional algorithms specifically targeted to
spatial data.

238 Chapter 8 Spatial M ining
•
.
·
.
····=-- ··
·t
·
·
....,..,.. �.•.. ,.
··
·
. t��> ..., .. .. . . .
}/;�.'�
·:�
·')-:':.
'
�r I.
···":'· : ,
.
)
,:
•::.. ,:�
.
·.·..;�J
FIGURE 8.8: Different shapes for spatial clusters.
8.7.1 CLARANS Extensions
The main memory assumption of CLARANS is totally unacceptable for large spatial
databases. Two approaches to improve the performance of CLARANS by taking advan
tage of spatial indexing structures have been proposed [EKX95].
The first approach uses a type of sampling based on the structure of an R*-tree (an
R-tree variant). To ensure the quality of the sampling, the R*-tree is used to guarantee
that objects from all areas of the space are examined. The most central object found in
each page of the R*-tree is used to represent that page in the search. The most central
object is the object (of all objects stored on that page) with the smallest distance from
it to the center of the page. Remember that the page is actually the MBR that contains
all the objects in that page. So the center of that MBR can be defined as the geometric
center of the bounding rectangle. CLARANS is then used to find clusters for these central
objects. The k medoids found in this step represent the k clusters to be found for the
database as a whole. Since the R*-tree clusters objects that are spatially near on a node
in the tree (and thus page), it is reasonable to believe that this approach to sampling finds
good medoids.
The second technique improves on the manner in which the cost for a medoid
change is calculated (see Formula 5.10 in Chapter 5.). Instead of examining the entire
database, only the objects in the two affected clusters must be examined. A region
query can be used to retrieve the needed objects. An efficient technique to retrieve only
the objects in a given cluster is based on the construction of a polyhedron around the
cluster medoid. The constructed polyhedron is called the Voronoi polyhedron or V
oronoi
diagram. This polyhedron is created by constructing perpendicular bisectors between
pairs of medoids. This process is illustrated in Figure 8.9. This then defines the cluster.
The objectives within a Voronoi diagram are closer to the medoid of that polyhedron
than to any other.
8.7.2
Section 8.7 Spatial Clustering Algorithms 239
.';·�:
��·�W:
.
· ·
·
·
· · · ·
· '
· .
· · · · · · · · · · · ·
· ·
·
. . . . : ! � : n ; : :,: : : : . .
·
;:/:;��mmv�.�;·..:...
(a) Perpendicular bisector (b) Voronoi polyhedrons
FIGURE 8.9: Voronoi polyhedron.
SD(CLARANS)
Spatial dominant CLARANS [SD(CLARANS)] assumes that items to be clustered contain
both spatial and nonspatial components. It first clusters the spatial components using
CLARANS and then examines the nonspatial attributes within each cluster to derive a
description of that cluster. For example, clustering of vegetation in remote areas may
find that one area (cluster) is predominantly a forest of pine trees, while another contains
massive open plains and grassy areas. SD(CLARANS) assumes that some learning tool,
such as DBLEARN [HCC92], is used to derive the descliption of the cluster. This
description can be viewed as a generalized tuple; that is, by using a concept hierarchy,
the attribute values for the set of tuples in a cluster can be generalized to provide summary
values at a higher level in the hierarchy. The learning tool performs this task. Algorithm
8.6 outlines the SD(CLARANS) algorithm. Note that it is a combination of CLARANS,
DBLEARN, and the spatial-dominant algorithm discussed earlier in this chapter. It also
assumes that in the first step an initial filtering of the da1ta using a relevance based
on the nonspatial data is performed. Any clustering algorithm could be used in place
of CLARANS in this algorithm. In our algorithm we show that the number of desired
clusters is input. However, the authors of the original version propose an approach to
determine the "most natural number of clusters" [NH94].
ALGORITHM 8.6
Input :
D
k
Output :
/ /Data to be clustered
//Number of des ired cells at the lowes t l eve l
K / / Set of clusters
SD (CLARANS) algori thm :
I I Find set of tuples that s at i s fy select ion criteria .
d = s e lect tuples from D based on nonspat ial select ion criteria ;
/ /App ly CLARANS to d based on spatial att ributes .
K = CLARANS(d) ;
// Perform attribute general ization .
for each k E K do
apply DBLEARN to the nonspat i al attributes in k;

240 Chapter 8 Spatia l Mining
In contrast to SD(CLARANS), nonspatial dominant CLARANS [(NSD(CLARANS)j
first looks at the nonspatial attributes. By performing a generalization on these attributes,
a set of representative tuples, one representing each cluster, can be found. Then the
algorithm determines which spatial objects go with which representative tuple to finish
the clustering process.
8.7.3 DBCLASD
A recent spatial clustering algorithm based on DBSCAN has been proposed that is called
DBCLASD (Distribution Based Clustering of LArge Spatial Databases). DBCLASD
assumes that the items within a cluster are uniformly distributed and that points out
side the cluster probably do not satisfy this restriction. Based on this assumption, the
algorithm attempts to identify the distribution satisfied by the distances between nearest
neighbors. As with DBSCAN, a cluster is created around a target element. Elements are
added to a cluster as long as the nearest neighbor distance set fits the uniform distribution
assumption. Candidate elements are determined and then are added to the current cluster
if they satisfy a menbership criteria. Candidate elements are determined by executing a
region query ,Ising a circle of radius m centered around a point p that was just added to
the cluster; m is chosen based on the following formula:
A
m > (8.5)
Here N is the number of points in the cluster and A is its area. The added points then
become new candidates.
The area of the cluster is estimated by using grids that enclose the cluster with a
polygon. When a point is added to a cluster, the grid containing that point is added to the
polygon. The closeness of the polygon to the real shape of the cluster depends on the size
of the grids. If the grids are too large, the shape may not approximate the cluster well.
If they are too small, the cluster could actually be estimated by disconnected polygons.
The grid length is chosen to be the largest value in the nearest neighbor distance set.
The algorithm DBCLASD is shown in Algorithm 8.7. Since the x2 test usually
requires at least 30 elements, the authors assumed that 29 neighboring points are initially
added to each cluster [XEKS98]. The last step expands a cluster based on the expected
distribution of the nearest neighbor distance set of C using the candidates found in c. Each
candidate is added one at a time to C, and the distribution of the nearest neighbor distance
set is estimated. If it still has the desired distribution, the points in the neighborhood of
this candidate are added to the set of candidates; otherwise the candidate is removed
from C. This process continues until c is empty. The points in the neighborhood of a
given point are determined based on the radius value stated above.
ALGORITHM 8.7
Input :
D / / Spat ial obj ects to be clustered
Output :
K / / Set of clusters
DBCLASD algori thm:
k = 0 ; II Ini tially there are no clusters .
Section 8.7 Spatial Clustering Algorithms 241
c = 0; I / Initial ize the set of candidat e s ' to be empty .
for each point p in D do
i f p is not in a cluster , then
create a new cluster C and put p in C;
add neighboring point s of p to C;
for each point q in C do
add the points in the neighborhood of q that have not
been processed to c ;
expand C;
Performance studies show that DBCLASD successfully finds clusters of arbitrary
shapes. Only points on the boundary of clusters are assigned to the wrong cluster.
8.7.4 BANG
The BANG approach uses a grid structure similar to a k-D tree. The structure adapts to
the distribution of the items so that more dense areas have a larger number of smaller
grids, while less dense areas have a few large ones. The grids (blocks) are then sorted
based on their density, which is the number of items in the grid divided by its area. Based
on the number of desired clusters, those grids with the greatest densities are chosen as
the centers of the clusters. For each chosen grid, adjacents grids are added as long as
their densities are less than or equal to that of the current cluster center.
8.7.5 WaveCiuster
The W
aveCluster approach to generating spatial clusters looks at the data as if they were
signals like STING, WaveCluster uses grids. The complexity of generating clusters is
O(n) and is not impacted by outliers. Unlike some approaches, WaveCluster can find
arbitrarily shaped clusters and does not need to know the dtesired number of clusters. A
set of spatial objects in an n-dimensional space are viewed! as a signal. The boundaries
of the clusters correspond to the high frequencies. Clusters themselves are low-frequency
with high amplitude. Signal processing techniques can be used to find the low-frequency
portions of the space. The authors propose that a wavelet transf
orm be used to find the
clusters. A wavelet transform is used as a filter to determine: the frequency content of the
signal. A wavelet transform of a spatial object decomposes it into a hierarchy of spatial
images. They can be used to scale an image to different sizes.
8.7.6 Approximation
Once spatial clusters are found, it is beneficial to determine why the clusters exist; that
is, what are the unique features of the clusters? Approximation can be used to identify
the characteristics of clusters. This is done by determining the features that are close
to the clusters. Clusters can be distinguished based on features unique to them or that
are common across several clusters. Here, features are spatial objects such as rivers,
oceans, schools, and so on. For example, some clusters may be unique partly because
they are close to the ocean or close to good schools. It usually is assumed that features
and clusters are represented by more complex closed polygons than by simple MBRs.

Aggregate proximity is defined as a measure of how close a cluster (or
f 1 )
. ., ( b' .
th
. . group
o e ements IS to a 1eature or to an o �ect m e space). This IS not a measu
. re �
distance from the cluster boundary, but rather to the points in the cluster. Traditional
data structures, such as R-trees and k-D trees, cannot be used to efficiently find th
aggregate proximity relationships because they focus on a cluster (object) boundary
e
::
opposed to the objects in the cluster. The aggregate proximity distance may be measured
by the sum of distances to all points in the cluster.
The aggregate proximity relationship finds the k closest features to a cluster. The
CRH algorithm has been proposed to identify these relationships [KN96]. C stands for
encompassing circle, R for isothetic rectangle, and H for convex hull. These are defined
as follows:
• Isothetic rectangle: MBR containing a set of points where the sides ofthe rectangle
are parallel to the coordinate axes.
• Encompassing circle: Circle that contains a set of points; found by using the
diagonal of the ti.sothetic rectangle as its diameter.
• Convex hull: Minimum bounding convex shape containing a set of points.
What makes these shapes efficient is that given a set of n points, the first two points
can be found in 0 (n) time and the last in 0 (n lg n) time. Each type of geometric shape
is viewed as a bounding structure around a feature. These three types of encompassing
geometric shapes are used as multiple levels of filtering of the possible close features.
These are used in order of increasing accuracy and decreasing efficiency. The concept of
using these three types of bounding polygons is shown in Figure 8.10, which illustrates
a school. The school is fairly accurately represented by a convex hull, but less accurately
represented by a rectangle and a circle. The objective is to obtain a balance between
accuracy and efficiency in identifying the relationships.
The first step in CRH is to apply the encompassing circle. The features (using
the circular approximation) that are ranked the highest (those that are viewed to be the
Circle
Rectangle
Convex hull
School
FIGURE 8. 1 0: CRH polygons.
...
I ' ,
I ' ,
I ',
�------:��
closest) to a given cluster are then sent to the filter at the next level. At this level the
isothetic rectangle is used to represent the features, and the features are again ranked
based on proximity to the cluster. The highest ranking features at this level are examined
at the final level, where a convex hull bounding polygon is used to estimate each feature.
This approach is used for each cluster. The desired number of features identified at each
level is indicated as input to the algorithm. Although different techniques can be used to
rank the features, intersection may be used or actual distances may be calculated. The
CRH algorithm uses various optimization features to reduce the overall complexity and
to eliminate redundant computation of distances.
8.8 EXERCISES
1. (�esearch) Compare the R-tree to the R*-tree.
2. (Research) Another commonly used spatial index is the grid file. Define a grid
file. Compare it to a k-D tree and a quad tree. Show the grid file that would be
used to index the data found in Figure 8.5.
Most spatial data structures were proposed many years ago. Quad trees were first intro
duced to handle queries on composite keys [FB74]. The k-D tree was proposed in
[Ben75]. There have been many variation of the k-D tree for use with spatial data
data [OSDH93]. The grid file was proposed in [NH84].
There have been many excellent surveys examining spatial data and spatial data
structures. A survey of spatial and multimedia data, including indexing, can be found in
[ZCF+97]. One unpublished survey of spatial indexing techniques not only provides a
taxonomy of the approaches, but also identifies the strengths and weaknesses of the vari
ous techniques [OSDH93]. Nievergelt and Widmayer have written an extremely easy-to
read yet thorough survey of spatial data structures with an excellent bibliography [NW97].
Other surveys of spatial data structures are available [Sam95a, GG98]. This last survey
[GG98] is an extensive examination of multidimensional indexing techniques, including
spatial and nonspatial. It includes a comparison of the various techniques. Additional sur
veys look at query processing of spatial data [Sarn95b]. A more general survey [Gtit94]
covered spatial data modeling, querying spatial· databases, spatial indexing, and archi
tectural approaches. Spatial relationships based on direction are examined in [EFKSOO].
The original proposal for R-trees can be found in [Gut84]. The R*-tree is a more efficient
improvement on the R-tree [BKSS90]. Many extensions to the basic R-tree have been
proposed [OSDH93]. The STING approach was proposed in [WYM97].
There also exist some surveys of spatial data mining. [EFKSOO] contains a survey
of the algorithms, relationships, and operations needed to support spatial data mining.
The concept of progressive refinement has been studied extensively in a recent doctoral
dissertation [Kop99]. A recent book [MHOl ] is a collection of many different spatial data
mining articles.
Articles that provide an overview of clustering in spatial databases can be found
in [EKSX98], [HKTOl], and [NH94]. In fact, many of the clustering techniques intro
duced in Chapter 5 can be viewed as spatial: K-means and K-medoids and CLARANS.
DBCLASD was proposed in [XEKS98]. WaveCluster was examined in [SCZ98]. Aggre
gate proximity is defined in [KN96]. However, the authors of the original version

proposed an approach to determine the "most natural number of clusters" [NH94] based
on a concept called silhouette coefficients [KR90].
Some of the earliest work on spatial classification was found in [FWD93]. Here
decision tree techniques were used to categorize objects in the sky. Specifically, stars
and galaxies were identified.
Many traditional clustering and classification algorithms have been modified to
perform well for spatial databases. DBSCAN has been generalized into generalized
DBSCAN (GDBSCAN), which clusters objects using both spatial and nonspatial attributes
[SEKX98]. It was examined in astronomy, biology, earth science, and geography appli
cations.
A spatial data mining query language based on SQL, GMQL (geo-mining query
language), has been proposed [Kop99]. This is based on DMQL and is used in DBMiner
and GeoMiner.
The SAND approach was initially examined in [AS91].
C H A P T E R 9
Temporal M i n i ng
9.1 INTRODUCTION
9.2 MODELING TEMPORAL EVENTS
9.3 TIME SERIES
9.4 PATIERN DETECTION
9.5 SEQUENCES
9.6 TEMPORAL ASSOCIATION RULES
9.7 EXERCISES
9.1 INTRODUCTION
Databases traditionally do not contain temporal data. Instead, the data that are stored
reflect data at a single point in time. Thus, it may be called a snapshot database. For
example, an employee database normally contains only the current company employ
ees rather than all employees who have ever worked for the company. However, many
questions cannot be answered by this snapshot data. A company CEO might wish to
determine trends in the hiring and firing of etiJ.ployees, or he might wish to obtain infor
mation about the ethnic diversity of employees and how it has changed over time. These
types of data mining questions require temporal data. In a temporal database, data are
maintained for multiple time points, not just one time point. Example 9. 1 illustrates the
use of a temporal database that stores employee data. Obviously, storing three separate
tuples for one employee with so much redundant information is not efficient, and tech
niques can be used to eliminate this redundancy. However, this illustrates the concept.
Each tuple contains the information that is current from the date stored with that tuple
to the date stored with the next tuple in temporal order.
EXAMPLE 9.1
XYZ Corp. uses a temporal database to store employee information. It maintains the
Social Security number (SSN), employee name, address, and salary for each employee.
When a new tuple is stored in the database, the current date is added to this infor
mation. Joe Smith is hired on 2/12/02 at a salary of $50,000. On his six-month per
formance evaluation, he is given a $2,000 raise and a promotion. On 12/10/02, he
moves to a new address. At the end of 2002, there are three tuples in the database
for Joe Smith:
245

246 Chapter 9
Date
2/12/02
8112/02
12/10/02
Temporal Mining
Name
Joe Smith
Joe Smith
Joe Smith
SSN
123456789
123456789
123456789
Address
10 Moss Haven
10 Moss Haven
13 Chesterton
Salary
$50,000
$52,000
$52,000
Analysis of temporal (or time-varying) data presents many interesting challenges.
For example, there may be many different interpretations for time. In Example 9.1 the
date stored in the record is the date representing when that information becomes current.
This is often called the valid time. The valid time for information is the time during
which the information is true in the modeled world. This usually consists of a start time
and an end time. The end time in the example is implied by the start time of the next
temporal record for the same employee. Another time that could have been used is the
transaction time. The transaction time is the timestamp associated with the transaction
that inserted this record. This could be different from the start time for the valid time
interval. The transaction time interval is the time the tuple actually existed in the database.
For example, Joe Smith may have indicated on 11/15/02 that he would have the new
address effective 12/10/02. The start valid time for the new address was 12/10/02, but
the transaction time was 1 1/15/02. Other types of times may be used as well. When the
employee information changes, a new tuple is inserted. Changes and deletes may occur
only to change data that was incorrectly inserted.
So far we have seen that temporal data often involve a duration of time; that
is, a start time and an end time. In this interpretation, the range of values [ts, te] is
associated with each record. Here ts is the start time and te is the end time. Different
temporal interpretations may be used. A timestamp of a specific time instance may be
used instead of a range. This is common with time series data where specific values
are associated with times. For example, a common time series is to show the price of a
specific stock at the stock market close each day. This stock quote is the price at that
specific point in time.
Many different examples for temporal data exist. Satellites continually collect
images and sensory data. This information is temporal and is associated with specific
points in time (when the data were obtained). In a hospital, printouts of heartbeats may
be kept for patients. This represents a continuous view of temporal data. When an EEG
is taken for a patient, several different brain waves are measured in parallel. Each wave
represents a continuous set of data over time.
Temporal databases usually do not accept the same types of updates and queries
as traditional snapshot databases. The only updates that are allowed are corrections and
versions. Actual modifications of tuples usually are not allowed. Instead, a new tuple with
a different valid time would be added. Temporal queries may involve fairly complicated
temporal selection criteria. For example, it would not make sense to ask for the salaries
of all employees. Instead, a temporal value would be needed such as: Find the salaries
for all employees on 7/9/01. Or a more complicated range query could be asked: Find
the names of all employees who had a salary greater than $100,000 between 1/1/01
and 12/3 1/01. A temporal query q involves a valid time range yq = [t.i , t%] in the
request. Here t.i is the start time and ti is the end time of the query's time range. In
Section 9. 1 Introduction . 247
the last example, these values were 1/1/01 and 12/31/01, Jrespectively. The time range
then exists for the query as well as for the data. Suppose that yd = [tf, t1J is the valid
time range for a tuple. Special temporal queries then can involve various combinations
of these two ranges:
• Intersection query: A tuple is retrieved only if its valid time range intersects that
of the query: yd n yq f= 0.
• Inclusion query: A tuple is retrieved only if its valid time range is completely
contained in the time range for the query: t.i :S tf :S t1 :S t% .
• Containment query: A tuple is retrieved only if its valid time range contains that
of the query: tf :S t.i :S t% :S t1.
• Point query: A tuple is retrieved only if it is valid at a particular point in time:
tf = ti = t% = t1.
When considering time, there are at least four types of databases:
• Snapshot: The database system provides no support for any temporal attribute.
The stored data usually are assumed to represent data that are valid at the current
time.
• Transaction time: The only temporal data supported by the database is the time
associated with the transaction that inserted the data. This could be a timestamp
for when the transaction was committed (or perhaps was requested) or it could be
a range.
• Valid time: This database supports a valid time range for the data. It may be
stored, as with the transaction time, using a unique value or a range. If it is a
unique value, this is the start of the time range, and the end is the beginning of
the next time range for the data with the same key.
• Bitemporal: A bitemporal databases supports both transaction time and valid time.
With temporal data, the concept of a key is complicated as well. In the salary
database, the employee SSN can no longer determine a unique tuple. Temporal informa
tion is needed as well.
As with spatial mining, several specialized data structures have been proposed
to assist in temporal mining. There are many specialized data structures that have been
proposed to index temporal databases that we do not discuss here. These usually are gen
eralizations of B+-trees and are similar to those structures we saw with spatial databases.
One difference, of course, is that time is usually one dimension where as space may
be two or three dimensions. These structures usually assume that a valid time range is
associated with each tuple. One complicating factor is the use of the current time. Unlike
spatial data, the temporal dimension keeps expanding. Thus, for items that are currently
valid, it is not possible to have the current end time for the range. One solution to this
problem is to use a special time value called now, which is the current time. Thus, a
range that ends in "now" means that it is valid up to the current time. The resulting effect
is that time ranges that end in "now" actually keep expanding in size.

248 Chapter 9 Temporal Mining
Mining of temporal data involves many of the conventional data mining activities
but, of course, is complicated by the temporal aspect and the more complicated types
of queries. For example, time series data may be clustered based on similarities found.
However, determining the similarity between two different sets of time series data is
difficult, as was shown in Chapter 1. Given a time series, a future value also may be
predicted. Association rules may involve temporal aspects and relationships. Web usage
mining discussed in Chapter 7 involved temporal data. The combination of spatial and
temporal mining also is used.
9.2 MODELING TEMPORAL EVENTS
There have been many different techniques used to model a sequence of temporal events.
We briefly examine three of them: Markov models (MMs), hidden Markov models, and
recurrent neural networks (NN). Suppose that you are given the problem of recognizing
the string of characters "the." This can be viewed as a temporal sequence of events. Each
event recognizes one character. One of the earliest techniques used to model a sequence
of events was a finite s;ate recognizer (FSR) or finite state machine (FSM). Figure 9. 1
illustrates an FSR for the sequence "the." The temporal aspect is implied by the arcs. It
can be viewed that the individual events (or characters) occur at specific time intervals.
While FSRs can be used to recognize a known sequence, they do not scale well
when the vocabulary is large. They also do not work well to model transitions between
states that are not precisely defined. Mmkov models and their variant, hidden Markov
models, extend the basic idea of an FSR but scale well and are more general. Figure 9.2
shows a simple Markov model. Notice the similarity and differences between it and the
FSR in Figure 9. 1 . One of the major differences is that the transitions (arcs) are not
associated with specific input values. Just as with an FSM, a Markov model (MM) is a
directed graph that can be used to recognize a pattern. Each node is associated with a
state in recognizing a sequence of events (or pattern). Although our example shows a
start and end node, these nodes need not be present. One of the major differences is that
a transition (arc) is associated with a probability, transition probability. The probability,
PiJ• on an arc (i, }} is the probability that a transition will be made from state i to state
j. In Figure 9.2, the probability of transitioning from state 1 to state 2 is 0.3, while that
of staying in state 1 is 0.7. The sum of the weights on the edges coming out of a node
is 1. Any arcs not shown are assumed to have a probability of 0. The probabilities can
be combined to determine the probability that a pattern will be produced by an MM.
For example, with the MM in Figure 9.2, the probability that the transitions along the
FIGURE 9. 1 : FSR for sequence "the."
Section 9.2 Modeling Temporal Events 249
0.7
0.5 0.3 1
0.3
�
0.1
FIGURE 9.2: Simple Markov model.
major horizontal path are all taken is 0.3 x 0.5 x 0.6 = 0.09. As with an FSR, there is
always one state that is designated as the current state. A major property of a Markov
model is the Markov property, which states that, given the current state, the transition
probability is independent of any previous states. Thus, an MM is memoryless. A more
formal definition of an MM is found in Definition 9. 1 .
DEFINITION 9.1. A Markov model (MM) is a directed graph ( V, A } with ver
tices representing states V = {v, , v2, . . . , v11} and arcs, A = {(i, j} I v;, VJ E V},
showing transitions between states. Each arc (i.j} is labeled with a probability PiJ
of transitioning from Vi to v1 . At any time t, one state is designated as the current
state v1 • At time t, the probability of any future transitions depends only on v1 and
no other earlier states.
The transition probabilities in an MM are learned during a training phase, where
counts are kept for each transition.
Markov models have been used in many different applications. Speech recognition
and natural language processing are very common applications for MMs. Suppose that
an MM is created to model certain phrases. The individual nodes could represent sounds
or words. A sequence of these would be a phrase. Given a phrase, the probability that
that phrase occurs is the product of the probabilities from the start state to the end
state using the transitions associated with each word in sequence. In this manner, the
most likely sequences can be found, and the most likely sequence is the one that is
"recognized." Given a model, the probability of the occurrence of a sequence of events
can be determined. You also could determine the probability of being in a particular state
at a particular time. Another application is in the area of system reliability. Here an MM
is used to model the system operation. The transition probabilities can be determined by
domain experts or learned from training data. The resulting model can be used to do
such things as determine system availability and predict the mean time between failures.
An extension to the MM that still satisfies the Markov property, is the hidden
Markov model (HMM). A major difference between the MM and HMM is the fact that the
states in an HMM need not correspond to observable states. An HMM models a process
that produces as output a sequence of observable symbois. The HMM will actually output
these symbols. Given a sequence of symbols, the HMM can be constructed to produce
these symbols. What is hidden is the state sequence that produced these symbols. There is
no relationship between states and real-world observable values. An observation sequence
can be produced by more than one state sequence.
As with the MM, the HMM consists of a set of states with transition probabilities. In
addition, the HMM has associated with each state an observation probability ·distribution.

0.5 0.5
y
P(H) = 0.5 P(H) = 0.3
P(T) = 0.5 P(T) = 0.7
0.5
FIGURE 9.3: Simple hidden Markov model (modified from [RJ86]).
An example of an HMM, modified from [RJ86], is found in Figure 9.3. One of the
most obvious differences between the HMM and the MM is the presence of the extra
probabilities. This represents the hidden part of the model and is associated with the
observable output from each state. This example models results that can be found from
tossing two coins. The first state is associated with one coin and the second state is
associated with the second coin. The first coin is a fair coin in that the probability of
obtaining a head or tail is each 0.5. The second coin is biased toward obtaining a tail
as that probability is 0.7.1The state transitions are all 0.5, which means that after tossing
a coin it is equally likely that the next toss will occur with either coin. The hidden
probabilities are used to determine what the output from that state will be, while the
public (or transition) probabilities are used to determine the next state that will occur.
Notice a major difference between an HMM/MM and an FSR is that the HMMIMM
models a system. It is not built simply to recognize a sequence of events. Thus, they
have many more applications than an FSR. Not only can they recognize, but also they
can forecast (predict). They are much more general, but at the same time they are more
complex. Many problems exist when determining what the HMM model should actually
look like:
• Size: Determining the number of states is not obvious. They need not be associated
with a real-world observable event.
• Transition probabilities: Determining what the transition probabilities are is dif
ficult. Domain experts and/or learning algorithms can be used to determine the
probabilities, much as is done with NNs.
• Hidden observation probabilities: As with the transition probabilities, these prob
abilities can be learned.
A more formal definition of an HMM is found in Definition 9.2. The transition
probabilities and observation probabilities are fixed for a given state.
DEFINITION 9.2. A hidden Markov model (HMM) is a directed graph (V, A)
with vertices representing states V = {v1 , vz, . . . , vn} and arcs, A = {(i, j)
1 vi , v1 E V}, showing transitions between states. Each HMM has the follow
ing additional components:
1. Initial state distribution used to determine the starting state at time 0, vo.
2. Each arc (i.j) is labeled with a probability PiJ of transitioning from Vi to vJ.
This value is fixed.
3. Given a set of possible observations, O{o1 , oz, . . . , ok}, each state, Vi , con
tains a set of probabilities for each observation, {Pi l , Pi2• . . . , Pik}.
Section 9.2 Modeling Temporal Events 251
Given an HMM, an observation sequence is generated based on the algorithm
shown in Algorithm 9. 1. This algorithm assumes that a sequence of m observations is
produced. The variable t represents time.
ALGORITHM 9.1
Input :
H I IHMM
Output :
S = (so, s1, . . . , sm-1) I I Output sequence
HMM observation sequence algorithm :
t = O
Based on initial s t at e di stribution, determine Vt i
repeat
Output St based on observation probabi l i t ies {Ptl , Pt2 , . . . , Ptkl ;
Choos e Vt+l based on t rans iti on probabi l ities at Vt i
t = t + l ;
until t = k;
There are three basic HMM problems [RJ86]:
1. Given a sequence of observed elements and an HMM, what is the probability that
the HMM actually produced the sequence? Note that this is associated with the
recognition problem. If the probability is low, then this model probably did not
produce it. As a result, the system that is modeled by the HMM probably did not
produce it.
2. Given a sequence of observed values and an HMM, what is the most likely state
sequence that produced this sequence?
3. How can the model parameters (transition probabilities, observation probabilities,
and starting state distribution) be improved? This problem is similar to that of how
learning is accomplished for a NN.
Relatively efficient algorithms have been proposed to solve all three of these problems.
Traditional feedforward neural networks cannot easily be used to model tempo
ral events because there is no mechanism of time. However, there are advanced neural
network (NN) architectures that can be used for both recognition problems and predic
tion problems. In a recurrent neural network (RNN) a neuron can obtain input from any
other neuron, including those in the output layer. Specifically, the outputs from nodes
in the hidden or output layers are fed back as input to an earlier layer. As RNNs store
information about time, they can be used for temporal prediction applications. How
ever, they are quite difficult to use and to train. Unlike traditional feedforward NNs, the
time that it takes for a recurrent NN to produce output is not known. This is because
the hidden and output layer nodes will continually be activated until the model sta
bilizes. Recurrence implies that the current state of the network depends not only on
the current input values but also on those of the previous cycle (from the previous
outputs).
Figure 9.4 shows the basic structure of an RNN. In Part (a), the structure for a
feedforward NN is shown. Part (b) shows an RNN. In (b), output from the hidden layer
is fed not only to the output layer but also into a new input layer referred to as a context

Input Hidden Output Input Hidden Output
, - -I , - -I , - -I ,- -I , - -I , - -I
I I I I I I __..I I I I I I
I I I I I I I
�:
I I I
I I I I I I I I I I
I I I I I I
I I I I I I
� � I � I � I
�
I I I I I I I
r-----:
I
I I I I I I ----+1 I
I I I I I I I I I I
I I I I I I I � I
I_ _ _
I 1_ _ _1 I_ _ _
I I_ _ _
I 1_ _ _1
Context
(a) Feedforward NN (b) RNN
FIGURE 9.4: Recurrent neural network.
I I
I I
I I
I I
1_ _ _1
layer. In this structure the input to the hidden layer then comes from nodes in the input
and context layers.
9.3 TIME SERIES
A time series is a set of attllibute values over a period of time. Alternative definitions
exist. Some investigators view a time series as consisting only of numeric values. Some
investigators assume that the values are at specific, evenly spaced time intervals. Time
series data may be continuous or discrete. In this text we take a general view that
encompasses all of these. As seen in Definition 9.3, a time series is a set of attribute
values over a period of time.
DEFINITION 9.3. Given an attribute, A, a time series is a set of n values:
{(t1 , a1) , (t2 , a2) , . . . , (t11 , a11)}. Here there are n time values and for each a corre
sponding value of A. Often the values are identified for specific well-defined points
in time, in which case the values may be viewed as a vector (a1 , az, . . . , an) .
DEFINITION 9.4. One time series Y' = (y;1 , • • • , Yim ) is a subseries of another
Y = (y1, . . . , y11 ) if, 'v'l :::; j :::; m - 1 , ij < ij+t and 'v'1 :::; j :::: m, 31 :::: k :::: n
such that Yij = Yk.
Typical data mining applications for time series include determining the similarity
between two different time series and predicting future values for an attribute, given a
time series of known values. Obviously, the prediction is a type of classification, while
the similarity can be thought of as either clustering or classification. Given several time
series, we may want to determine which time series are like each other (clustering).
Alternatively, we may be given a time series to find which time series from a set are
like this one (classification). A special type of similarity analysis is that of identifying
patterns within time series.
9.3.1 Time Series Analysis
Time series analysis may be viewed as finding patterns in the data and predicting future
values. Detected patterns may include:
• 'fiends: A trend can be viewed as systematic nonrepetitive changes (linear or
nonlinear) to the attribute values over time. An example would be that the value
of a stock may continually rise.
• Cycles: Here the observed behavior is cyclic.
Section 9.3 Time Series 253
• Seasonal: Here the detected patterns may be based on time of year or month or
day. As an example, the sales volumes from department stores always jump around
Christmas.
• Outliers: To assist in pattern detection, techniques may be needed to remove or
reduce the impact of outliers.
·
Identifying patterns with real-world data may be difficult because of noise, outliers, errors,
and missing data. Multiple patterns may be observed in the same data. Looking at gross
sales made by retail companies, there usually are large increases around Christmas each
year. This seasonal change is independent of the general trend for cost of goods sold to
increase (perhaps based on population increase and inflation).
9.3.2 Trend Analysis
Many straightforward techniques can be used to detect trends in time series. Smoothing
is an approach that is used to remove the nonsystematic behaviors found in a time
series. For example, the general trend may be that a time series is increasing in value.
However, when the specific attribute values are examined, there are many decreases in
value. Smoothing usually takes the form of finding moving averages of attribute values.
Given a window in time around a particular time point, the local average of all attribute
values is used instead of the specific value actually found at this point. Median value, as
opposed to mean value, normally is used because it is less sensitive to outliers. Figure 9.5
illustrates the process. Smoothing is used to filter out noise and outliers. It also can be
used to predict future values because the resulting data are easier to fit to a known
function (linear, logarithmic, exponential, etc.).
Detecting seasonal patterns in time series data is more difficult. One approach
is to detect correlations between attributes at evenly spaced intervals. For example, a
- Values - o- - Moving average
10
4
2
FIGURE 9.5: Smoothing using a moving average.

correlation may be found between every twelfth value (in monthly sales data). The time
difference between the related items is referred to as the lag. With the sales data, the lag
is 12. Autocorrelation functions can be generated to determine the correlations between
data values at different lag intervals. A correlogram graphically shows the autocorrelation
values for different lag values.
The covariance measures how two variables change together. It can be used as the
basis for determining the relationship between either two time series or seasonal trends
in one time series. An autocorrelation coefficient, rk. measures the correlations between
time series values a certain distance, lag k, apart. Several different approaches have been
used for autocorrelation. A correlation coefficient, introduced in Chapter 3, measures the
linear relationship between two variables (or that between the same variable at a given
time lag). Positive values indicate that both variables increase together, while negative
values indicate that as one increases the other decreases. A value close to zero indicates
that there is little correlation between the two variables. One standard formula to measure
correlation is the correlation coefficient r, sometimes called Pearson 's r. Given two time
series, X and Y, with IIeans X and Y, each with n elements, the formula for r is
1
L (x; - X)(y; - Y)
(9.1)
Applying this to find the correlation coefficient with lag of k, rk. on a time series X =
(xi, x2, . . . , Xn) is straightforward. The first time series is X' = (x1 , x2, . . . , Xn-k), while
the second time series is X" = (xk+1 , Xk+2, . . . , Xn). Example 9.2 illustrates the use of
autocorrelation coefficients.
EXAMPLE 9.2
By looking at the graph in Figure 9.6, it is obvious that several patterns exist. One is the
fact that the values rise linearly for two time units and then drop and restart. Thus, there
is an obvious autocorrelation with a lag of 3. In this case we find that rk = 1 because
there is a perfect positive relationship.
FIGURE 9.6: Correlation with lag = 3.
Sectio n 9.3 Time Series 255
9.3.3 Transformation
To assist in pattern detection, the actual time series data may be transformed in some
manner. A logarithmic transformation can be used to stabilize the variance and to make
seasonal effects constant over years. Transformation is also used to solve the dimension
ality curse. The dimensionality curse is the fact that many problems are caused by data
sets with many dimensions. Data mining on time series data with many variables is not
only difficult but also expensive. Data structures to store high-dimensional data are not
very efficient. Transformation can be used to reduce the number of dimensions. Note that
feature extraction also reduces the number of dimensions.
9.3.4 Similarity
We saw examples for examining the similarity between patterns in Web usage min
ing. Indeed, these applications are temporal data mining. Given a target pattern X =
(XJ , x2, . . . , Xn) and a sequence Y = (y, , y2, . . . , Ym). the problem is that of determin
ing sim(X, Y) . Here n may or may not be the same as m. Matching may be based on
matching the two series completely, matching subseries thereof, or more advanced types
of matching. One series may be scaled or shifted to match the other. Gaps or "don't
care" values may have to be added to one series to match the second.
Some common distance measures that may be used are Euclidean, linear correlation,
and discrete Fourier transform. We have already seen the Euclidean distance metric.
There are problems with these common distance measures:
• Length: X and Y may be of different lengths but may still be quite similar.
• Scale: While the general shape of X and Y may be identical, the scale may be
somewhat different. For example, one may use a log scale. Different metrics may
be used (Fahrenheit vs. Centigrade).
• Gaps: One of the series may be missing some of the values that exist in the other
series.
• Outliers: This is similar to the problem to gaps, except that it is assumed that the
extra values in one series may be due to erroneous readings.
• Baseline: The actual baseline values may differ. This means the time between
successive values in X and Y may differ.
One similarity approach is to look at the longest common subseries between two
series [BDGM97]. When looking at X and Y, the idea is to find the longest subseries
that they have in common. For example, suppose X = (10, 5, 6, 9, 22, 15, 4, 2) and
Y = (6, 9, 10, 5, 6, 22, 15, 4, 2). The longest common subseries is (22, 15, 4, 2) . The
sim(X, Y) = ljn = 4/9, where l is the length of largest common subseries and n is the
length of the largest of the two given series. While this approach handles some of the
issues stated above, it does not handle the scale or baseline problem.
A recent similarity measure has been proposed to solve these issues [BDGM97].
The basic idea is to convert one series to the other using a linear transformation function
f to convert a value from one series to a value in the next. This function, along with
an allowed tolerated difference, E in the results, compensates for the scale and baseline

256 Chapter 9 Tempora l Mining
issues. The baseline issue is also addressed by allowing a slight difference, up to 8, in the
time values used. The resulting similarity function sim<,o (X, Y) is shown in Definition
9.5 (modified from [BDGM97]). The maximum is taken over all possible values for f.
The closer si�.o (X, Y) is to 1, the more similar X and Y are.
DEFINITION 9.5. Given integer value 8 > 0, real number E < 1, and linear
function function f, and two time series X, Y with the longest one of length n.
Let X' = (xi1 , Xi2 , . . • , Xi, ) and Y' = (yj1 , yh, . . . , yj11. ) be the longest subseries
in X and Y, respectively, where:
• V1 :=: k :=: m - 1, I ik - }k 1:=: 8 and
• V1 ::: k ::: m, ci�E) ::: f(xik) ::: Yjk (l + E)
Then, simE,o (X, Y) = maxf(mjn)
The longest common subseries between two given series can be found in O(n2).
Thus, the most difficulf part of finding simE,o(X, Y) is finding f. Several algorithms
have been proposed to find the function. An exact algorithm is O(n3), while approximate
algorithms with better behavior are also proposed [BDGM97].
9.3.5 Prediction
The prediction (or f
orecasting) of time senes data can use some of the techniques dis
cussed earlier, such as regression. However, in practice, time series data are replete with
errors and noise. Using simple regression often is not sufficient. Given a discrete time
series over equally spaced time intervals, the forecasting problem is to predict a value
at time t, x1 (l), and a lead time of l. It is assumed that previous time series values,
(x1 , x2, . . . , x1 ), are known. The objective is to minimize the mean square of the devia
tions x1+t - x1 (l). Various models may be used to represent the time series values and
thus predict future values. We briefly review some of these models here.
Studies of time seties prediction often assume that the time series is stationary.
This means that the values come from a model with a constant mean. More complex
prediction techniques may assume that the time series is nonstationary. A time series
usually represents values that are dependent on each other, but they may be viewed
as being generated from a series of independent values called shocks. The shocks are
randomly drawn from a normal distribution with a zero mean. A sequence of these
random values is thought of as representing a white noise process. This white noise
process is transformed into the time seties by a linearfilter, which may be viewed as a
simple weighted sum of previous shocks.
A special case of the linear filter model is one that assumes that time series values
are dependent on earlier ones. Autoregression, then, is a method of predicting a future
time seties value by looking at previous values. Given a time series X = (x1 , x2 , . . . , Xn ).
a future value, Xn+ 1 , can be found using
(9.2)
Here en+I represents a random error, at time n + 1. In addition, each element in the time
series can be viewed as a combination of a random error and a linear combination of
previous values. Here the <Pi are the autoregressive parameters. Alternatively, the value
Section 9.4 Pattern Detection 257
may be viewed as a weighted sum of previous deviations from the mean. Autoregression
models may be stationary or nonstationary.
Another dependency that may exist between values in a time series is that of
a moving average. Here a future value, Xn+1 , can be found using a moving average
model applied to a set of previous consecutive •talues. There are many different mov
ing average models, and any model could be used. In addition, there may be a lag
between the point where the moving average is applied and the prediction value. For
example, a seasonal forecast for sales could be based on an average of the sales for
the prior season 12 months earlier. A future time seties value, Xn+ l . can be predicted
using:
(9.3)
where ai is a shock. For a moving average, then, the time series value is predicted based
on a weighted average of a set of previous shock values.
Autoregression and moving average can be combined to created a model of a time
series that is called ARMA (Autoregressive Moving Average). In practice, this combined
model is sufficient to represent many real-world time series. When the model is not
stationary, an extension of ARMA, ARIMA (Autoregressive Integrated Moving Average)
may be used. The ARIMA model has become quite popular, but it is relative complex
and requires an expert to use it effectively.
9.4 PATTERN DETECTION
Given a set of data values (d1 , d2, . . . , dn ) where di is collected at time ti and ti < tj
iff i < j, the pattern detection problem is to deterrnine a given pattern that occurs in
this sequence. This can be viewed as a type of classification problem where the pattern
to be predicted is one found in a given set of patterns. Typical pattern detection appli
cations include speech recognition and signal proceS;sing. Spelling correctors and word
processors also use simple pattern detection algorithms. Although these simpler cousins
of the true data mining pattern detection problems are precise, the more general pattern
detection problems are fuzzy with no exact matches. Approximations are needed. While
humans are good at detecting such patterns, machines are not.
9.4.1 String Matching
The string matching problem assumes that both a long text document and a short pattern
are given. The problem is to determine where the pattern is found in the text. Example 9.3
illustrates the pattern detection problem when it is applied to string matching. This prob
lem is a common one, with many applications in word processing.
EXAMPLE 9.3
Martha Holder is editing her resume using a popular worq processor. She has just gotten
married and wishes to change the name Holder to her new last name of Laros, where
approptiate. Not all occurrences of Holder, however, should be changed. For example,
she does not want to change the author's names of previous publications that were made
under her maiden name. Using the word processor, she repeatedly finds all occurrences
of Holder in the vita. She then must examine the context to determine whether it should
be changed to Laros. In this case, the pattern being matched is (H, o, l, d, e, r). Only

258 Chapter 9 Temporal Mini ng
words that are an exact match to this pattern should be found. Note that here each letter
is viewed as if it occurred at a later point in time. In actuality, it is a later point in the
document.
One of the earliest string matching algorithms is the Knuth-Morris-Pratt or KMP
algorithm. KMP creates a finite state machine (FSM), which is used to recognize the
given pattern. The FSM represents all possible states that exist when scanning a string to
match the given pattern. Each node in the FSM relates to one of these states. Figure 9.7
shows an FSM created to recognize the pattern "ABAABA." Here there are seven states.
State i represents the fact that the first i characters in the pattern match the most recent i
characters in the string. State six is designated as the recognizer state with two concentric
circles. The arcs in the graph are labeled with the character from the pattern that causes a
transition between the two states as indicated. Transitions labeled with "*" indicate that
this transition is taken with any other character found in the string. The KMP algorithm
creates the FSM for a given pattern. The FSM can then be applied to the string by
starting at the first charadter in the string. From a given state, the next character in the
string determines which transition is taken. The accepting state of the FSM is reached
only when the pattern is found in the string. The worst-case behavior of the application
of the FSM is O(m + n), where m is the length of the pattern and n is the length of the
string. The preprocessing phase to create the FSM is O (m) in space and time.
Another algorithm that builds on the KMP approach is called the Boyer-Moore, or
BM, algorithm. The same FSM is constructed to recognize the pattern, but the pattern is
applied to the string in a right-to-left pattern. For example, when looking for the string
"ABAABA," if the sixth character in the string is not A, then we know that the pattern
is not found in the string starting at the first character in the string. We also know that
if the sixth character is neither an "A" nor a "B," then the pattern does not exist in the
string starting at any of the first six characters. The BM needs only one comparison to
determine this, while the KMP would have to examine all of the first six characters.
Again, the BM is O(m + n) in the worst-case scenario, but the expected and best cases
A
FIGURE 9.7: FSM for string "ABAABA."
Section 9.4 Pattern Detection 259
are better than this. The actual performance depends (of course) on both the pattern and
the string.
Even though KMP and BM are pattern recognition algorithms, they usually are
not thought of as data mining applications. The identification of patterns in these earlier
techniques is precise. Most data mining pattern matching applications are fuzzy; that
is, the pattern being compared to (i.e., the class representative) and the object being
classified will not match precisely. However, as we will see, there are more advanced
pattern recognition algorithms that are similar in that graphical structures are built to
specifically recognize a pattern. In effect, these true data inining applications build on
these earlier non-data mining algorithms.
When examining text strings, it often is beneficial to determine the "distance"
between one string and another. For example, spelling checkers use this concept to
recommend corrections for misspelled words. Again, these usually are not thought of
as data mining activities, but the distance measure technique we discuss here is often
the basis for more advanced distance measure approaches. Suppose that we wish to
convert A = (a1 , az, . . . , an} to B = (b1 , bz, . . . , bm}. The basic idea is to determine the
minimum cost of steps that are needed to convert one string to another. There are three
operations that can be performed to convert string A to string B. Starting at the first
character in each string, each operation identifies what operation should be performed
on A and B to change A to B. Each operation not only indicates specific functions to be
performed but also associates a cost for it. The following assume that we are currently
examining ai in A and bj in B:
• Match: Leave ai and bj as they are. New character in A is ai+I and in B is bj+l ·
The cost of this operation is 0 if ai = bj ; otherwise the cost is oo.
• Delete: Drop ai from A. The new length of A is n - 1. The cost of this operation
is 1.
• Insert: Insert bj into A at position ai . All characters in A following the previous
ai are shifted down one, and the new length of A is n + 1. Next character in A is
ai+1 and in B is bj+I · The cost of this operation is 1.
The distance between string A and B is then determined by the minimum total cost for
all operations needed to convert A to B. For example; tHe distance from catch to cat is
2 because the c and h have to be deleted. Similarly, the distance from cat to hat is 2
because c must be deleted and h must be inserted. Example 9.4 illustrates the process.
EXAMPLE 9.4
Suppose that we wish to determine the distance between a string "apron" and "crayon."
By looking at the strings, we see that we can match at most three characters: either
a, o, n or r, o, n. Figure 9.8 illustrates the use of the first matching. Here the cost is 5
because we have to insert c, r, y and delete p, r. The figure shows that we can view the
problem as a shortest path between two points: the top left comer and the bottom right
corner.

D
Insert
iM;
e �
t
e
a
•
p
•
•
0
•
n
•
c a y 0
• • •
• • • •
• • • •
• •
• •
• •
FIGURE 9.8: Convert apron to crayon.
n
•
•
•
9.5 SEQU ENCES
A sequence is an ordered list of itemsets [AS95]. Definition 9.6 gives the definition of a
sequence.
DEFINITION 9.6. Let I = {h , h . . . , Im } be a set of items. A sequence, S, is:
S = (sJ , sz, . . . , sn) , where Si � I.
As with a time series, we find that there are many different definitions for a
sequence. In Chapter 7 the sequence was a list of web pages. A sequence is some
times viewed as an ordered list of attribute values from any domain. The individual
members of the sequence are sometimes viewed to be sets of items from some under
lying domains (alphabets). One common difference is that the sequence may not have
explicit relationships with time. The only requirement is that the entries be totally ordered.
As a matter of fact, the terms sequence and time series are often used interchangeably.
In this text, we use the two definitions as shown in Definition 9.3 and Definition 9.6.
The basic difference between the two concepts, then, is that a series is an ordered list
of values, while a sequence is an ordered list of sets of items or values. The length
of a sequence is the sum of the cardinalities of all itemsets in the sequence. A subse
quence of a given sequence is one that can be obtained by removing some items and any
resulting empty itemsets from the original sequence. We briefly examined the concept of
sequential patterns in Chapter 7. These are specific type� of subsequences in that they
are maximal.
DEFINITION 9.7. Let I = {h , h. . . . , Im } be a set of items. One sequence
T = (ti1 , • • • , ti111 ) is a subsequence of another S = (sJ , . . . , sn) if Vl :S j :S
m - 1 , ij < ij+! and V1 ::::; j ::::; m, 31 ::::; k :S n such that tij � Sk. In this case, S
contains T.
We assume that items are grouped together into transactions. The temporal feature
is added by assuming that a customer may obtain different items at different times.
Each set of items purchased at one time by a customer is a transaction. Example 9.5
Section 9.5 Sequences 261
illustrates the concept of a sequence. The sequence of itemsets purchased by a customer
is referred to as the customer-sequence. Note that customer Ct has the customer-sequence
({A, B}, {B, C}, {C}) . ({A}, {C}) is a subsequence of this, but ({A, C}, {B}) is not. To
be a subsequence, each itemset must be a subset of an item in the larger sequence. In
addition, the larger itemsets must satisfy the ordering indicated.
The use of support and confidence in sequences is defined in Definition 9.8 and
Definition 9.9, respectively. Given a minimum support threshold, a sequence is said to
be large or frequent if its support exceeds this threshold.
DEFINITION 9.8. Given a set of customers and transactions for each customer, the
support of a sequence s(S) is the percentage of total customers whose customer
sequence contains S.
DEFINITION 9.9. The confidence (a) for a sequence association rule S =} T is the
ratio of the number of customers (customer-sequences) that contain both sequences
S and T to the number that contain S.
An example of sequences is found in Example 9.5. As with traditional itemsets,
a lattice can be constructed to illustrate the sequences. The sequence lattice, originally
proposed in [PZOD99], uses the subsequence (as opposed to subset) relationship. The
data at one level in the lattice are obtained from that at the next lower level by adding
one item. This is done either by adding the item to one of the itemsets or by inserting it
as a singleton itemset somewhere in the sequence.
EXAMPLE 9.5
Let I = A, B, C, D, suppose that there are three customers, Ct . Cz, and C3, who purchase
these items at different times. The following table shows purchases made by these three
customers:
Customer Time Itemset
Ct 10 AB
Ct 20 BC
c1 30 D
Cz 15 ABC
Cz 20 D
C3 15 ACD
(In this table we have removed commas and set notation.) Given S = ({A}, {C}), we
see that the support is s(S) = 1/3 because it is contained only in the sequential pattern
found for customer CJ . A second sequence T = ({A}, {D}) has a support of s(T) = 2/3,
while U = ({B, C}, {D}) has a support of s(U) = 2/3. Figure 9.9 shows the lattice with
frequent sequences, assuming a minimum support of 2/3, only for this data.
As with frequent itemsets, a frequent sequence follows a large sequence property.
This means that any subsequence of a large (frequent) sequence is also frequent.

FIGURE 9.9: Frequent sequence lattice for Example 9.5.
9.5.1 AprioriAII
Algorithm AprioriAll in Chapter 7 contained a simple algorithm
.
for finding sequen�ial
patterns. AprioriAll works in three parts by first finding all frequent Items
.
ets, then relaci�g
original transactions with frequent itemsets, and finally finding s�quenttal patterns. This
algorithm does not scale well, partly because of the transformatwn step. It also would
be difficult to incorporate extensions such as sliding windows.
9.5.2 SPADE
The algorithm that we now introduce, SP
ADE (Sequential P
Attern Discovery
.
using Equiv
alence classes), identifies patterns by traversing the lattice top-down. To Improve pro
cessing, SPADE uses an id-list that associates the customers and transactions associated
with each item. Table 9.1 illustrates this concept for the data in Example 9.5. Here we
see the id-lists for sequences of length 1. These can be viewed as the atoms to construct
support counts for larger sequences. The support for a k-sequence can be dete�ned �y
looking at the intersection of any two of its (k - I)-subsequences. To accomplish this,
temporary id-lists are generated from the starting id-lists. To illustrate this process, look
at the sequence T = ({A}, {D}). Looking at Table 9.1, we see that the count for ({�})
is 3, as is that for ({D}). As seen in Example 9.5, T = ({A}, {D}) count of 2. To denve
this, an id-list for T is created by determining the intersection for the two subsequences:
Customer Time
Note that intersection must take times into account. Thus, its count is 2 and its support
is 2/3. This observation is used in SPADE to count the sequences and determine their
support. The lattice can be traversed to construct id-lists for higher-�evel
.
s�quences by
intersecting two subsequences at the next lower level. The problem with this IS that there
may not be enough memory to do this all in memory.
TABLE 9. 1 : ID-Lists for Sequences of Length 1
-------
A B c D
Customer Time Customer Time Customer Time Customer Time
Cr 10 Cr 10 C1 20 C1 30
c2 15 Ct 20 c2 15 c2 20
c3 15 c2 15 c3 15 C3 15
To address the space issue, th� lattice is divided into partitions and these partitions
are traversed independently. This reduces the memory requirements by reducing the
number of id-lists that must be kept at one time. An equivalence class concept is used
to accomplish this. A k length prefix for a sequence is determined by looking at the
first k items (and associated ordering) for the sequence. Given a sequence S, the k
length prefix of S is denoted by p(S, k), In Example 9.5, we looked at the sequence
U = ({B, C}, {D}). This sequence is a 3-sequence because it is of length 3. It has a
length 2 prefix of ({B, C}), as does another 3-sequence W = ({B, C, D}). ek is an
equivalence relation. As Set<n in Definition 9. 10, two sequences are ek equivalent if they
have identical prefixes of length k. Thus, we see that U is equivalent to W, written
as U = W(mod 82). If we had the id-lists for U and W with their counts, we could
determine the count.
DEFINITION 9.10. Two sequences S and T are equivalent, S = T(mod ek) iff
p(S, k) = p(T, k).
To partition the frequent sequence lattice, we look at those sequences in a 8k
equivalence class. In Figure 9.10 we have identified the equivalence classes [A]e1 , [B]e1 ,
(C]e1 , and [D]e1 • The supports for the sequences in each sublattice can be determined
by intersecting the id-lists for two sequences at the lower level. The partitioning of the
lattice can be accomplished by any of the ek equivalence classes. If the number of classes
for 8r is too large (i.e., the number of id-lists will not fit into memory), then a larger
value of k can be used. Figure 9.10 shows the lattice from Figure 9.9 with equivalence
classes for e[ .
Algorithm 9.2 shows the steps in SPADE. A breadth-first search or depth-first
search of the lattice can be performed to enumerate the large sequences within each
class. The first step is to find the frequent 1-sequences. This is performed by read
ing the id-lists into memory and counting the support for each customer. The fre
quent 2-sequences can then be found by intersecting the id-lists for the frequent 1-
sequences. A straightforward approach for this is to look at all (mover2) possible com
binations. The authors of the algorithm have proposed improvements to this naive tech
nique, which we do not discuss here [Zak98]. The equivalence classes, E, for 81 can
then be determined. SPADE successfully finds all frequent sequences in only three
database scans and has been shown to outperform other algorithms for identifying fre
quent sequences.

FIGURE 9. 1 0: e1 Equivalence classes for lattice in Figure 9.9.
t
ALGORITHM 9.2
Input :
D / / ID - l i s t s for customer transact i ons
s / / Support
output :
F I I Frequent sequences
SPADE algoritlun :
Determine frequent items , F1 ;
Determine frequent 2 - sequences , F2 ;
Find equivalence clas ses E for all 1 - sequence s [S]e1 ;
for each [S] E E do
Find frequent sequences F;
9.5.3 Generalization
The concept of subsequence has been generalized to include concept hierarchies and
more temporal information. These generalizations can make the conce?t of sequences
applicable to a wider range of applications. For example, one constramt would b� to
include a maximum time between elements in the sequence. For example, you nught
want to see customers who purchase a digital camera and then purchase a printer within
three months. This also illustrates the concept hierarchy problem. There are many types
of digital cameras. This sequence should be for any brand and type. This requires the
use of taxonomies as in generalized association rules. . .
Adding concept hierarchies to sequences is relatively straightforward. The tdea 1s
to change the definition of subsequence as seen in Definition 9.1 1 .
DEFINITION 9.11. One sequence T = (t;1 , • • • , t;111 ) is a subsequence of another
s = (s1 , . . . , sn) if Vl � j � m - l , ij < ij+l and Vl � j � m, 31 � k_ � n
_
such
that t;j £; sk or Vx E t;j x is an ancestor in a concept hierarchy for some 1tem m Sk.
Another technique that has been proposed is to look at a sliding window aroun�the
data [SA96b]. A sliding window is a maximum time difference used to group transactiOns
together. When transactions are grouped together, a sequence is said to exist in a customer
sequence if it exists in any of the transactions in a window. The effect of this is to increase
the support of sequences.
One last extension proposed in [SA96b] is to aPd a time constraint that indicates
the allowed ti�e between successive elements in the sequence. The time constraint is
a pair Umin. tmax) that indicates the minimum and maximum distances that are allowed
to exist in a sequence. These are allowable time gaps. The time qifference between
transactions with consecutive elements in the sequence must be greater than tmin but no
greater than tmax.
The three extensions also may be combined. With no concept hierarchy, a window
size of 0, and time constraints of (0, 09), there is the regular concept of sequences.
One algorithm has been proposed specifically to handle generalized sequential
patterns. This algorithm generalized sequential pattern (GSP) has been shown to outper
form an extended version of AprioriAll by up to 20 times [SA96b]. As with Apriori,
GSP scans the database several times. The support for all items
·
is determined during
the first scan. The input to the next scan is the frequent items (sequences of length one)
found during the first traversal. The algorithm works iteratively in this fashion. Dur
ing each scan, candidate sequences are generated from the frequent sequences of the
prior scanned and then counted. As with Apriori, the size of each candidate during a
database scan is the same. GSP terminates when no candidates at that pass are found to
be frequent.
To assist with the time constraint issue, the concept of contiguous subsequences is
used. A sequence will always contain all contiguous subsequences, but with time con
straints added, it may not contain noncontiguous subsequences. The definition of con
tiguous subsequence is found in Definition 9.12 [SA96b]. For example, (A , C, DE, D),
(AB , C, DE), and (AB , C, D, D) are contiguous subsequences of S = (AB , C, DE, D),
while (AB , DE, D) and (AB , C, D) are not. Notice that any time constraints that a
sequence satisfy will also always be satisfied by any contiguous sequence.
DEFINITION 9.12. One sequence T = (tt , . . . , tm) is a contiguous subsequence
of another S = (s1 , . . . , sn) if T is a subsequence of S and one of the following
applies:
• n = m; V2 � i � n - 1t; = s; ; and either ft = S] and I tm 1=1 Sn I - 1 or tm = Sn
and I ft 1=1 s1 I -
1. Note that if the cardinality of either St or sn is one, when
one item is dropped from it then that particular itemset in the subsequence is
dropped effectively making the lfmgth of T n - 1.
• n = m ; 32 � i � n - 1 such that I t; 1=1 s; I - 1 and for all other V1 � j �
nj j6 i ; ti = Si.
• T is a contiguous subsequence of another sequence U, which is a subsequence
of S.
The generation of candidate sequences must be handled differently than with can
didate itemsets in Apriori. For example, suppose it is found that ({A}) and ({B}) are
frequent during the first scan. These two sequences can be used to generate three can
didates: ({AB}), ({A}, {B}), and ({B}, {A}). In addition, the generalization constraints

266 Chapter 9 Tempora l Mining
must be satisfied. As with candidate generation in Apriori, candidates are generated by
joining frequent sequences from the prior scan. Here, however, joining is defined slightly
differently. Two sequences T = (tJ , . . . , t111) and S = (SJ , . . . , sn ) are joined if the sub
sequence of T obtained by dropping the first item in t1 is the same as a subsequence of
S obtained by dropping the last item in s,. When T and S are joined, the new sequence
obtained is either U = (tJ , . . . , tm , x) or U = (tJ , . . . , tm U x). The first sequence is
obtained if x = sn ; otherwise the second sequence is obtained.
9.5.4 Feature Extraction
The feature extraction problem is to extract k features from every sequence ai:J.d to
represent that sequence by those features. This approach may make it easier to perform
sequence analysis, and each sequence can be represented as a point in k-dimensional
space. R-trees or other multidimensional data structures can then be used to store and
search the time series data. The problem, of course, is how to extract the features.
As with time series clustering, identifying features that describe classes of sequences
is beneficial. One algoqthm, FEATUREMINE, has been proposed to extract features for
sequences [LZ099]. The approach is to use a preprocessor sequence mining algorithm
to extract features. Then classification can be performed on these features. A sequence
classifier maps each sequence into a class based on features. There are four goals for
features [LZ099]:
• Features should occur often in the database.
• Features should be usable to distinguish between classes.
• There should be no redundant features.
• Avoid searching the entire database to find the features.
The lastitemindicates how the feature extraction should not be performed. FEATUREMINE
uses SPADE and integrates a pruning technique into the algorithm. The approach is to
traverse the sequence latticein adepth-first manner to find the frequency sequences. Observe
that sequences at the root of the lattice are more general than those beneath it. As with
SPADE, to ensure that processing inmainmemory, the lattice is partitionedinto equivalence
class sections and the traversal actually is performed in each partition separately.
9.6 TE MPORAL ASSOCIATION RULES
With traditional association rules, a transaction can be viewed as the following:
(TID , CID , l] , In , . . . , Im )
where TID is the ID for the transaction, CID is the ID for the customer, and I1 , . . . , Im
are the items. When considered part of a temporal database, a transaction could be
viewed as
(TID , CID , /1 , In• . . . , I111 , ts , te)
where [ts, te] is the valid time range for the transaction. If this were a grocery store
transaction, fs = te could be the point in time that the transaction was completed. Alter
natively, if the transaction represented products ordered over the Web, ts might be the
time the order was placed and te might be the time the actual delivery was made. Thus,
[ts, te] would be the range of time the transaction was active. Once time is added to
Section 9.6 Temporal Association Rules 267
the database, different association rules can be found for different times or time ranges.
This is similar to the idea of combining clustering with association rules in the spatial
mining area. The analogy in temporal mining is to cluster the data based on time and then
determine the association rules. This can be done to examine the change in association
rules over time, to detect seasonal association rules, and to identify rules that may not
be found if looking at larger sets of data. For example, a grocery store could look at
association rules for an entire year. However, this would not identify frequent items sold
at particular times of the year. This would not allow the store to take advantage of some
of the most frequently sold items over short petiods of time. The importance of this
concept is demonstrated by the fact that many supermarket:. now have aisles dedicated
to the sale of seasonal products.
When time is added to the concept of association rules, different types of rules
may be generated. We observe several of these different interpretations in this section.
9.6.1 lntertransaction Rules
The basic association rule approaches look only at items occurring together within one
transaction. These may be viewed as intratransaction association rules. However, there
certainly are situations in wp_ich rules generated across transactions would be of interest.
For example, an electronics store manager might want to lmow if customers purchase
computer software after they purchase a computer. These purchases could occur in trans
actions at two differeqt · times. To define these new rules, the concept of a window
is applied to the transaction database. Recall that the basic association rule problem
assumes the existence of a set of items I = {/J , [z, . . . , I111 } and a database of trans
actions JJ = {tJ , tz, . . . , tn} where t; = In , li2, . . . , l;k and Iij E I. Assume that each
transaction t; has associated with it a value d; which could be time, location, or other
information desctibing the t�ansaction. We assume here th�t the value is time, so that d;
is the time that t; · �x�cuted: Although the original proposal in [TLHF99] viewed that d
could be any ordinal ·attributes, to simplify discussion here we look at specific integers
representing time. .A sliding window is viewed to be placed on top of D. Th� interval
between two transaction ti and fk is I di - d; 1. The number of transactions to 'be· included
in the window, w, is an input parameter.
9.6.2 Episode Rules
An episode rule is a generalization of association rules applied to sequences of events.
An event sequence S is an ordered list of events, each one occurring at a particular time.
Thus, it can be viewed as a special type of time series. An episode is a set of event
predicates, A, and a partial order, ::;:, on the events in A: {A, ::;:}. An event predicate
is a predicate that can be evaluated as true or false when applied to an actual event
occurrence. It could be as simple as to check the type or severity of an event. An episode
can be viewed as a directed graph where the vertices are the events and the arcs represent
the partial order. An episode B is a subepisode of an episode A if the graph of B is
a subgraph of the graph of A. A sequence of events, S, contains an episode if all the
alarm predicates are satisfied in S and these events satisfy the partial order. A formal
definition for episode rule is found in Definition 9.13. As with association rules, we may
also define support and confidence.

268 Chapter 9 Temporal M ining
DEFINITION 9.13. An episode rule is an implication of the form B ::::} A where
B and A are episodes and B is a subepisode of A.
An important application in networks is to predict the failure of a switching node.
Episode rules can be used to help solve this problem. If a failure can be accurately
predicted, then the node can be taken offline and replaced before the occurrence. When
viewed as a temporal data mining problem, it becomes one of predicting an event (failure)
based on a sequence of earlier events. These events can be viewed as the amount of
traffic passing through a node or alarms (messages generated by a network entity usually
describing a problem). All alarm can be viewed as a triple a = (t, s, m), where t
is the time the alarm occurred, s is where this alarm message came from, and m is
the alarm message itself [Kle99b]. The sequence of alarms could be viewed as a time
series.
The following preprocessing techniques may be used to perform some of the fol
lowing functions [Kle99b]:
• Remove redundan� alarms.
1
• Remove lower priority alarms if higher alarms also exist.
• Replace some alarms by either new information or perhaps higher-level alarms.
A correlation pattern is then used to match to the sequences that have been found in the
alarm data. This pattern may be compared to alarms that have occurred in a recent time
window. If the sequence of alarms that have occurred matches a correlation pattern, then
the associated correlation action is taken.
Two different approaches have been proposed to find episode rules. One approach,
WINEPI, applies a window to the events. Given an event sequence, S, the window is a
time span (ts , te) that defines a subseries of S, namely, those events (in order) that occur
in the window. Given an episode B, the subseries of B that occur in all windows of size
W is referred to as Bw . The window can be used to define support and confidence as
seen in Definition 9.14 and Definition 9.15. The support is the percentage of windows
in which the target episode occur.
DEFINITION 9.14. Given a set of subseries, Sw , of an episode, S, as defined by a
window W the support of a episode B, s(B) is the percentage of total subseries
in Bw that have S as a subepisode.
DEFINITION 9.15. The confidence (a) of an episode rule B ::::} A is the ratio of
the support of A to the support of B: �i�j.
9.6.3 Trend Dependencies
Trend dependencies are like association rules in that they compare attribute values, but
they do so over time [WM97]. For example, we might observe that an employee's
salary always increases over time. A formal definition (as found in [WM97]) is found in
Definition 9. 18. Note that the definition does not explicitly indicate that the two database
states mtist differ in time. Of course, this is our assumption here, but in general it is not
Section 9.6 · · Temporal Association Rules 269
necessary. To add this temporal aspect to it, we assume that the pattern on the left-hand
side is from a relation state valid at an earlier time than the pattern on the right-hand
side of the trend dependency.
DEFINITION 9.16. Let R be a schema containing atttibutes A1, Az, . . . , Am . The
domain for each attribute A; must be a totally ordered set. A pattern over R
is a set {(A 1 , e1 ), (A2, e2), . . . , (Am , Bm)} where V1 ::: i, j ::: m, A; I= Aj and
e; E {<, =, > , :::, ::::, :f.}.
DEFINITION 9.17. Let R be a schema contammg attributes At, Az, . . . , Am .
A pair of tuples tt , tz satisfy a pattern {(A J , Bt), (Az, Bz), . . . , (Am , em)} iff
tr (A; )e; tz(A;)V1 ::: i ::: m.
DEFINITION 9.18. A trend dependency is an implication of the form X ::::} Y
where X and Y are patterns over schema R.
A trend dependency, just as an association rule, is also subject to a support and a
confidence. For example, we would not be interested in the trend dependency concerning
salaries if it rarely were true.
DEFINITION 9.19. Given two relations, fr , [z over schema R, the support (s) for
a trend dependency X ::::} Y is the percentage of tuple pairs in It x [z that satisfy
both patterns X and Y. If I /1 x /z I= 0, then s = 0.
DEFINITION 9.20. Given two relations, I1 , /z over schema R, the confidence (a)
for a trend dependency X ::::} Y is the ratio of the number tuple pairs in It x /z
that satisfy both patterns X and Y to the number that satisfy X. If the number that
satisfy X is 0, then a = 0.
Example 9.6, which is adapted from [WM97], illustrates a trend dependency. In
this example, there are two database states: I It I= 6 and I /z I= 6. Thus, I h x lz I= 36.
Here X = (SSN, =) AND Y = (Salary, :::). The number of tuple pairs in h x /z that
satisfy both patterns is 4. The number that satisfies X is 5. Thus, the a = 4/5 = 80%
and s = 4/36 = 1 1 %.
EXAMPLE 9.6
Imagine having the data in Example 9. 1 for all employees at XYZ. Instead of viewing
it as one table, however, we want to look at it as three different instances: It , /z, and
h It contains the valid data at time 2/12/02, /z has the valid data for 8/12/02, and h
has the valid data for 12/10/02. A trend that can be observed (at least for Joe Smith)
is that an employee's salary always increases with time. This trend detection can be
stated as
(SSN, =) =::::} (Salary, :S)
Given two tuples t1 E /r and tz E h if ft (SSN) = tz(SSN) then ft (Salary) ::: tz(Salary) .
This holds for any two database states where the second state is at a later time. The
following tables show It and h

Name SSN Address Salary
Joe Smith 123456789 10 Moss Haven 50,000
Mary Jones 1 1 1 1 1 1 1 1 1 1 0 Main 75,000
Bill Adams 222222222 215 North 100,000
Selena Shepherd 876543298 25 Georgetown 15,000
Paul Williams 908734124 13 East 250,000
Martha Laros 873659365 1010 Fox 150,000
Name SSN Address Salary
Mary Jones 1 1 1 1 1 1 1 1 1 1 0 Main 85,000
Joe Smith 123456789 10 Moss Haven 52,000
Bill Adams 222222222 215 North 90,000
Selena Shepherd 876543298 25 Georgetown 15,000
Paul Williatbs 908734124 13 East 270,000
Bob Holder 838383838 22 South 20,000
Given these two states, the confidence and support of X ==> Y is ct = 4/5 = 80% and
s - 4/36 = 1 1%.
Trend dependencies are defined over only two database states. They are not easily
generalized to more states. As with association rules, we c�n state
.
a
.
trend dependency
problem as that of finding all trend dependencies with a gt�en rmru�um suppo� and
confidence over two identified database states. The complexity of this problem m the
worst-case scenario is quite high. There are 1 e ID possible combinations of attributes
and operations. Here e is the set of operators (we assume there are six in this case), and
D is the number of possible attribute pairs. In the example this becomes 616. Obviously,
an exhaustive search is not advisable. In fact, it has been shown that the general problem
is NP-complete [WM97]. When the set of operators is restricted to { < , = , >}, it becomes
polynomial and an efficient algmithm has been proposed [WM97].
9.6.4 Sequence Association Rules
We can use sequences in rules, which we call sequence association rules.
DEFINITION 9.21. Given a set of items I = {h , h . . . , Im } and a set of trans
actions grouped by customer in customer-sequences, a sequence association rule
is an implication of the form S ==> T, where S and T are sequences.
DEFINITION 9.22. The support for a sequence association rule S ==> T is the
percentage of customers (customer-sequences) that contain S and T.
DEFINITION 9.23. The confidence (ct) for a sequence association rule S ==> T,
is the ratio of the number of customers (customer-sequences) that contain both
sequences S and T to the number that contain S.
9.6.5
Section 9.6 Temporal Association Rules 271
As with conventional association rules, we can state the sequence association rule
problem to be that of finding sequence association rules with minimum support and
confidence. There are many applications that could use sequence association rules. In
the market basket area, the buying behavior over time can be used to predict future
buying behavior. This could be used to do targeted advertising to customers with the
first type of buying behavior. Note that we are predicting buying patterns over time,
not just within one transaction. An example of sequence association rules is found
in Example 9.7. The SPADE algorithm discussed earlier ca:n be used to find frequent
sequences, and these sequences can then be used to solve the sequence association
rule problem.
EXAMPLE 9.7
Using the data introduced for Example 9.5, we can construct the following sequence
association rules:
Rule
({A}, {C}) ==> ({A}, {D})
({B, C}, {D}) ==> ({A}, {C})
Calendric Association Rules
Support Confidence
113 1
113 112
Calendric·association rules, as defined in Definition 9.24, assume that each transaction, t; ,
is associated with a timestamp, t;s, when it was executed. In addition, time is assumed to
be divided into predefined units, t. A time interval, k, is defined by the range [kt, (k+ l)t).
A transaction occurs in a time interval k, which is defined by [kt, (k + l)t), if kt :=: t;s <
(k + l)t. D[k] is the subset of transactions that occur in time interval k. The support
of an itemset, X in D [k], is the percentage of transactions in D[k] that contain X. The
confidence of X ==> Y in D[k], is the ratio of the number of transactions in D [k] that
contain X U Y to the number that contain X. Example 9.8 illustrates use of a calendric
association rule. Here the time unit is a day. Notice, however, that the same data could
have been evaluated for calendric association rules where the time unit was a different
granule: hour, month, year, etc.
DEFINITION 9.24. Given a set of items I = Ut . h . . . , Im}, a set of transac
tions D = {tJ , t2, tn}, a time unit k, and a calendar C = {(s1 , et) , . . . , (sk. ek)}, a
calendric association rule is an association rule, X �· Y that occurs in D[k].
EXAMPLE 9.8
Suppose that a grocery store wishes to obtain information about purchases for a particular
day. In this case, the time unit is a day (24-hour period). The manager is interested in
finding all association rules in this time frame that satisfy a given minimum support and
confidence. The manager also is interested in association rules that satisfy the support and
confidence for all but five days in a given season. During the year 2001, the manager
defines two time intervals by looking at days in winter as defined by the calendar:

{(1, 79) , (355, 365)}. There are 90 time units, or days, in this calend�. With a mismatch
threshold of 5, he is then interested in only association rules that sattsfy the support and
confidence on at least 85 of the days.
One could imagine a regular association rule algorithm, such as Apriori, applied
to a subset of D created by finding all transactions that occur in the given
.
time int�r�al.
However, the problem may be more general, such as finding all calendric assoctatwn
rules that occur over any time interval (or some set of intervals). This could be used
to determine important association rules over any day or period of time (n
.
ot just o?e).
It is assumed that a particular calendar is defined with potentially many dtfferent ttme
granularities. The more general problem, then, is to find all calendric
.
associ�tion �les
that hold given this calendar. Given a calendar of time intervals and a t�me u�t•
.
vanous
occurrences of the time unit can be defined in each. In Example 9.8, the ttme urut ts a day,
but the intervals are the seasons consisting of days in the four seasons. An association rule
may satisfy the support and confidence for so�e of t�e t�e u�its. Thus, an additional
threshold, m, is used td indicate the number of ttme uruts m the mtervals of the c�lendar
in which the association rule does not hold. A calendar belongs to a rule X => Y tf there
are at most m mismatches. A calendric association rule algorithm has been proposed
that is given as input a set of possible calendars and a time unit [RMS98]. It first fi�ds
large itemsets over all time units and then determines which calendars belong to which
association rules.
9.7 EXERCISES
1. Using the time series data in Example 9.2, determine the autocorrelation with a lag
of 4. Explain what this value indicates.
2. Assume that you are given the following temperature values, Zt , taken at 5-minute
time intervals: {50, 52, 55, 58, 60, 57, 66, 62, 60}. Plot both Zt+2 and Zt · Does there
appear to be an autocorrelation? Calculate the correlation coefficient.
3. Plot the following time series values as well as the moving average found by
replacing a given value with the average of it and the ones preceding and following
it: {5, 15, 7, 20, 13, 5, 8, 10, 12, 1 1 , 9, 15}. For the first and last values, you are to
use only the two values available to calculate the average.
4. Using the MM in Figure 9.2, determine the probability that the model is in the
state labeled 't' after 3 characters.
5. Determine the probability that the sequence "ththe" is recognized by the MM in
Figure 9.2.
6. (Research) Investigate and describe two techniques which have been used to predict
future stock prices.
The original string matching algorithms, KMP and BM, were proposed �ver 20 years
ago [BM77] [KMP77]. A variation on KMP proposed by Aho and Corastck constructs
an FSM that can recognize multiple patterns [AC75].
Section 9.8 Bibl iographic Notes 273
One recent text has examined the impaCt of time on databases, logic, and data min
ing [BJW98]. There are several excellent surveys and tutorials concerning temporal data
mining, including a recent textbook that examines sequential patterns [AdaOO] [HLP01].1
Markov models and hidden Markov models have been extensively studied. An
excellent introduction to the topic of HMM can be found [RJ86].
Time series have been extensively studied in the literature with several introduc
tory books and surveys. One online statistics textbook contains a survey, [Sta01] that is
quite simple to understand and very complete. Many time series textbooks are available,
including [And71], [BJR94], and [BD96]. A recent dissertation examined the telecom
munication network alarm issues in detail [Kle99b]. One of the earliest proposals for
a recurrent network was by Jordan in 1986 [Jor86]. Jordan proposed feedback from
output units to special context input nodes. Note that this allows conventional NN back
propagation learning techniques. Elman proposed that R.NlTs allow a feedback form the
hidden layer to a separate context input layer [Elm90]. Recurrent neural networks have
been proposed to detect automobile emission problems that occur over time. One recent
dissertation has studied the temporal NNs [1198].
The various applications of time to association rules is becoming quite popular in
the research community. Intertransaction association rules were introduced in [TLHF99].
Calendric association rules were proposed in [RMS98] as an extension to the earlier
proposed cyclic association rules [ORS98]. The authors pmpose a calendar algebra to
manipulate time intervals using a predefined calendric system.
SPADE was first introduced in 1998 [Zak98]. The concept of subsequence gen
eralization was examined in [SA96b] The approach of applying windows to events in
WINEPI was proposed in [MTV95].
1Eamonn Keogh, "A Tutorial on Indexing and Mining T
ime Series Data," The 2001 IEEE International
Conference on Data Mining.

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Index
Ac
2
, 275
IR, 1 16
4Thought, 274
JDA Intellect, 281
Linear Discriminant Analysis, 1 23
NeoVista Decision Series, 281
ACC, 120
activation function, 64
bipolar, 65
Gaussian, 66
hyperbolic tangent, 66
linear, 65
sigmoid, 66
step, 65
threshold, 65
unipolar, 65
adaptive, 1 63
adaptive classifier combination, 120
additive, 3 1
affinity, 145
affinity analysis, 8
Affinium Model, 274
agglomerative clustering, 128
aggregate proximity, 241
aggregate proximity relationship, 242
aggregation hierarchy, 29
AGNES, 162
AI, 12, 42
AI Trilogy, 274
algorithms
l R, 1 1 7
Agglomerative, 133
Apriori, 1 73
Apriori-Gen, 1 72
ARGen, 1 69
Averagelink:, 137
Backpropagation, 1 09
BIRCH, 1 5 1
Count Distribution, 1 80
CURE, 1 57
Data D:istribution, 1 8 1
DBCLASD, 241
DBSCAN, 154
Decision Tree Build, 94
Decision Tree Processing, 60
EM, 49
GA Clustering, 147
Generate Rules, 1 15
Genetic Algorithm, 69
Gradient Descent, 1 10
HITS, 206
hrnm, 251
K Nearest Neighbors, 92
K-Means, 141
Maximal Reference Sequences, 216
MST, 136
Nearest Neighbor, 1 42
OAT, 217
PAM, 144
Partition, 178
Partitional MST, 1 39
PRISM, 1 19
Propagation, 1 06
Quantitative Association Rule, 1 86
Rule Extraction, 1 16
Sampling, 1 75
SD(CLARANS), 240
Sequential Patterns, 213
Simple Distance Based, 90
SPADE, 264
SPATIAL ASSOCIATION
RULE, 235
SPATIAL DECISION TREE, 237
SP
ATIJI.L-DATA-DOMINANT, 23 1
Squared Error, 140
STING, 233
STING BUILD, 232
Supervised Learning, 1 07
Alice, 275
Alice d'ISoft, 275
Alice/Server, 275
305

306 Index
alleles, 67
alternative hypothesis, 54
Amadea, 275
ANN, 61
AnswerTree, 275
approximation, 13, 241
Apriori, 169-170
Apriori-Gen, 170
AprioriAll, 262
ARIMA, 257
ARMA, 257
artificial intelligence, 12, 42
artificial neural networks, 61
association, 8
association rule, 8, 166, 212
association rule problem, 167
association rules, 164-192
Apriori, 169-173
CDA, 179-180
DDA, 180-181
inter-transaction, 267
intra-transaction, 267
parallel algorithms, 178-181
partitioning, 177-178
sampling, 173-177
temporal, 266-272
attribute-oriented induction, 230
authoritative, 205
authority, 205
Auto-Regressive Integrated Moving Average,
257
autocorrelation, 254
autocorrelation coefficient, 254
autoregression, 256
average, 130
average link, 133, 137
B-tree, 223
b-tree, 34
backlink, 205
back-percolation, 276
backpropagation, 108, 109
backward crawling, 200
backward traversal, 211
BANG, 241
batch gradient descent, 1 10
Bayes, 86, 199
Bayes Rule, 52
Bayes Theorem, 52
Bayesian classification, 86-89
BEA, 145
bias, 47
binary search tree, 223
bipolar, 65
bipolar activation function, 65
BIRCH, 150
bitmap index, 34
bivariate regression, 55
BM, 258
Bond Energy algorithm, 145
boosting, 101
border point, 153
box plot, 51
Boyer-Moore, 258
Braincel, 276
BrainMaker, 276
broker, 201
C4.5, 100-101
C5, 100
C5.0, 101, 286
CACTUS, 162
calendric association rule, 27 1
candidate, 168
Capri, 286
CARMA, 191
CART, 102-103, 276
CCPD, 191
CDA, 179, 191
cell, 223
center, 90
centroid, 90, 129, 1 30, 140
CF tree, 151
CFC, 200
CHAID, 123
CHAMELEON, 163
characterization, 8, 234
chi squared, 54, 123, 1 89
chi squared automatic interaction
detection, 123
chi squared statistic, 54
children, 67
chromosome, 67
CLARA, 144
CLARANS, 144
class, 76, 89
classification, 5, 75-124
Bayesian classification, 86-89
C4.5, 100-101
CART, 102-103
distance based simple, 89-90
generating rules, 1 16-1 19
generating rules from DT, 114-1 15
generating rules from NN, 115-1 16
ID3, 97-100
Issues, 77-80
KNN, 90-92
RainForest, 103
regression, 80-86
SPRINT, 103
classification and regression trees, 102
classification rule, 1 14
classification tree, 93
Clementine, 277
Clever, 205
click, 208
clickstrearn, 206
clique, 1 37
cluster mean, 140
clustering, 7, 125-163
agglomerative, 132-137
average link, 137
BEA, 145-146
BIRCH, 150-152
categorical, 157-159
complete link, 137
CURE, 154-157
DBSCAN, 152-154
divisive, 138
genetic algorithms, 146-147
hierarchical, 13 1-138
K-Means, 140-141
Minimum Spanning Tree, 138-139
Nearest Neighbors, 142
neural networks, 147-149
P
AM, 142-145
partitional, 138-149
single link, 134-136
Squared Error, 139-140
clustering feature, 150, 151
clustering problem, 127
clustering problem, alternative definition, 129
clusters, 125
CMC, 1 19
collaborative filtering, 203, 204
combination of multiple classifiers, 1 19
competitive layer, 148
complete link, 130, 133
compression, 12
concept, 27
concept hierarchy, 27, 184
confidence, 166, 188, 234, 261 , 268-271
confidence interval, 47
confusion matrlx, 80
connected component, 134
contains, 267
Context Focused Crawler, 200
context graph, 200
context layer, 251
contiguous subsequence, 265
contingency table, 189
continuous data, 100
convex hull, 242
conviction, 188
cookie, 209
core points, 153
Index 307
correlation, 55, 56, 1 87, 188, 253
correlation coefficient r, 56, 254
correlation pattern, 268
correlation rule, 187
correlogram, 254
cosine, 58
Count Distribution, 179
covariance, 254
covering, 1 16
crawler, 198
CRH, 242
CRISP-DM, 18, 20
cross, 67
crossover, 67, 68
Cubist, 277
CURE, 154
customer-sequence, 261
cyclic association rules, 273
DAG, 28, 215
Darwin; 277
data bubbles, 162
Data Distribution, 1 80
data mart, 38
data mining, 3, 9
Data Mining Query Language, 1 8
data model, 2 1
data parallelism, 178
data scrubbing, 38
data staging, 38
data warehouse, 35, 36
database, 12, 21
Database Management System, 21
database segmentation, 125
DataEngine, 278
DataMite, 278
DB, 12
DBCLASD, 240

308 Index
DBMiner, 279
DBMS, 17, 21
DBSCAN, 152
DCS, 120
DDA, 1 80
Decider, 279
Decider-Online, 279
decision support systems, 28
decision tree, 58, 59, 93
decision tree induction, 93
decision tree model, 60
decision trees, 58-61, 92-103
DecisionT
ime, 279
delta rule, 108
DENCLUE, 163
dendrogram, 131, 231
density-reachable, 153
descriptive model, 5
diameter, 129, 1 5 1
DIANA, 162
Dice, 58
dice, 40
dimension, 29
dimension table, 3 1
dimensional modeling, 29
dimensionality curse, 15, 255
dimensionality reduction, 1 5
dimensions, 1 5
directed acyclic graph, 28, 2 1 5
direction relationship, 227
directly density-reachable, 153
discordancy test, 130
dissimilarity, 58
dissimilarity measure, 58
distance, 58, 227
distance measure, 58, 90, 129
Euclidean, 58
manhattan, 58
distance scan, 223
distiller, 198
distributed, 178
division, 82
divisive clustering, 128
DMA, 191
DMG, 20
DMQL, 18, 202
domain, 22
downward closed, 170
drill down, 29, 40, 229
DSS, 28
DT model, 60
dynamic classifier selection, 1 20
EIS, 28
EM, 49
encompassing circle, 242
Enterprise Miner, 280
entity, 2 1
entity-relationship data model, 2 1
entropy, 97, 98
episode, 214, 267
episode rule, 267
Eps, 152
Bps-neighborhood, 152
equivalence classes, 76
equivalent, 263
ER data model, 2 1
ER diagram, 21
ESS, 28
Essence, 201
Euclidean, 58
Euclidean distance, 58, 158
evaluation, 10
event sequence, 267
evolutionary computing, 67
executive information systems, 28
executive support systems, 28
Exhaustive CHAID, 123
expanded dimension table, 34
expectation-maximization, 49
exploratory data analysis, 42
Extensible Markup Language, 198
extrinsic, 1 28
fact, 29
fact table, 3 1
fallout, 80
false negative, 79
false positive, 79
farthest neighbor, 1 37
FEATUREMINE, 266
feedback, 63
feedforward, 63
finite state machine, 248, 258
finite state recognizer, 248
Firefly, 204
fires, 65
firing rule, 65
fitness, 68
fitness function, 68
flattened, 33
flattened dimension table, 32
FN, 79
focused crawler, 198
forecasting, 256
forward references, 215
FP, 79
frequency distribution, 5 1
frequent, 261
frequent itemset, 168
FSM, 248, 258
fuzzy association rule, 1 92
fuzzy logic, 24
fuzzy set, 23
g-sequence, 2 1 8
GA, 68
Gain, 98
GainRatio, 101
GainSmarts, 280
gatherer, 201
Gaussian, 66
Gaussian activation function, 66
GDBSCAN, 244
gene, 67
generalization, 8, 202, 229
generalized association rules, 1 84, 229
generalized suffix tree, 21 1
genetic algorithms, 67-70
Geographic Information Systems, 221
gini, 103
GIS, 221
Google, 205
gradient descent, 109
batch, 1 10
incremental, 1 10
offline, 1 10
online, 1 10
GSP, 265
GST, 2 1 1
hard focus, 199
Harvest, 201
harvest rate, 1 99
hash tree, 182
HD, 1 9 1
heapify, 156
hebb rule, 108
hebbian learning, 147
hidden layer, 61
Hidden Markov Model, 249
hidden node, 61
hierarchical classifier, 1 99
hierarchical clustering, 128
high dimensionality, 15
histogram, 5 1
HITS, 205
HMM, 249
HNC Risk Suite, 280
HOLAP, 40
HPA, 1 9 1
HTML, 198
hub, 198, 205
Hybrid Distribution, 1 9 1
Hybrid OLAP, 40
hyperbolic tangent, 66
Index 309
hyperbolic tangent activation function, 66
HyperText Markup Language, 198
hypothesis testing, 54
ID3, 97-100, 237
IDD, 191
IDF, 26, 201
image databases, 226
incremental, 1 28
incremental crawler, 198
incremental gradient descent, 1 10
incremental rules, 184
incremental updating, 184
individual, 67
induction, 1 2
information, 97
information ain, 97
information gain, 98
information retrieval, 12, 26, 79
informational data, 35
input layer, 61.
input node, 61
integration, 15
Intelligent Miner, 281
inter-transaction association rules, 267
interconnections, 61
interest, 1 88
Internet Service Provider, 209
interpretation, 14
intra-transaction association rules, 267
intrinsic, 128
introduction, 3-20
inverse document frequency, 26, 201
IR, 12, 26

310 Index
ISO/IEC, 20
isothetic rectangle, 242
ISP, 209
itemset, 166
iterative, 163
Jaccard, 58
Jaccard's coefficient, 158
jackknife estimate, 47
Java Data Mining, 285
IDA Intellect, 281
JDBCMine, 282
JDM, 285
join index, 34
K Nearest Neighbors, 90-92
k-D tree, 156, 225
K-Means, 140
K-Medoids, 142
K-Modes, 141
k-sequence, 213
Kaidara Advisor, 282
KATE, 282
KDl , 285
KDD, 9
KDD object, 23
KDD process, 10
KDDMS, 18
KDnuggets, 274
key, 21
KlvlP, 258
KNN, 90
knowledge and data discovery management
system, 18
knowledge discovery in databases, 9
knowledge discovery in spatial databases, 221
KnowledgeSEEKER, 282
KnowledgeSTUDIO, 282
Knuth-Morris-Pratt algorithm, 258
Kohonen, 148
Kohonen self organizing map, 148
lag, 254
large, 261
large itemset, 168
large itemset property, 169, 177
large reference sequence, 216
large sequence property, 261
LDA, 123
learning, 106
learning parameter, 1 10
learning rate, 108
learning rule, 108
least squares estimates, 82
levelized dimension table, 34
lift, 188
likelihood, 49
linear, 65
linear activation function, 65
linear filter, 256
linear regression, 55, 80
link analysis, 8
location, 222
logistic regression, 85
LOGIT, 283
longest common subseries, 255
machine learning, 43
Magnify, 283
Magnum Opus, 283
major table, 3 1
manhattan, 58
manhattan distance, 58
Mantas, 283
map overlay, 222
market basket, 167
market basket analysis, 8, 1 67
MarketMiner, 284
Markov Model, 248
Markov Property, 249
MARS, 284
maximal forward reference, 216
maximal frequent forward
sequences, 215, 216
maximal reference sequences, 216
maximum likelihood estimate, 49
MBR, 223
MDD, 39
mean, 51
mean squared, 129
mean squared error, 47, 82, 108
median, 51
medoid, 129, 130
merged context graph, 200
method of least squares, 82
metric, 129
minimum bounded rectangle, 223
minimum item supports, 1 87
Minimum Spanning Tree algorithm, 135
minor table, 31
Minotaur, 284
Minotaur Transcure, 284
MinPts, 152
MINT, 218
mismatch, 272
missing data, 15, 77, 80, 100
MLDB, 201
MLE, 49
MLP, 1 13
MM, 248
mode, 5 1
MOLAP, 39
momentum, 1 12
monothetic, 128
moving average, 253, 257
MSapriori, 1 87
MSE, 47, 1 08
MST, 135
Multidimensional Database, 39
Multidimensional OLAP, 39
multilayer perceptron, 1 1 3
multimedia data, 1 5
Multiple Layered DataBase, 201
multiple linear regression, 55
multiple-level association rules, 1 85
mutation, 68
My Y
ahoo, 203
Naive Bayes, 86, 199
nearest hit, 236
nearest miss, 236
nearest neighbor, 134, 142, 23 1, 240
Nearest Neighbor algorithm, 142
nearest neighbor query, 222
negative border, 173
neighborhood, 152
neighborhood graph, 236
Net Perceptions Retail Analyst, 285
neural network, 63
neural network model, 63
neural networks, 61-66, 103-1 14
perceptron, 1 12
propagation, 105
RBF, 1 12
SOFM, 148
supervised learning, 106-1 1 2
unsupervised learning, 106, 147-149
News Dude, 204
NN, 61, 63, 1 03
NN model, 63
noise, 82
Noisy data, 15
noncompetitive learning, 147
nonhierarchical, 138
nonlinear regression, 85
nonparametric model, 46
Index 3 1 1
nonspatial data dominant generalization, 230
nonspatial hierarchy, 229
nonstationary, 256
normalized dimension table, 34
now, 247
NSD CLARANS, 240
null hypothesis, 54
OAT, 217
observation probability, 250
OC curve, 79
Ockham's razor, 5 1
offline gradient descent, 1 10
offspring, 67
OLAP, 18, 39, 229
OLTP, 23
online, 150, 163
OnLine Analytic Processing, 39
online gradient descent, 1 10
online transaction processing, 23
operational data, 35
operative characteristic curve, 79
OPTICS, 163
OPUS, 191
Opus, 283
Oracle Data Mining Suite, 277
Oracle9i Database, 285
outlier, 14, 82, 127, 130
outlier detection, 130
output layer, 61
output node, 61
overfitting, 14, 77
overlap, 58
page, 208
PageRank, 205
PAM, 142
PAR, 1 9 1
parallel, 178
parallelization, 1 80
parametric model, 46
parents, 67
Partek, 285
partial-completeness, 192
Partition, 177
partitional, 138
partitional clustering, 128
Partitioning, 177
Partitioning Around Medoids, 142

312 Index
path completion, 209
pattern, 9, 269
pattern detection, 257-266
pattern discovery, 2 1 1
pattern matching, 44
Pattern Query Language (PQL), 278
pattern recognition, 5, 44, 75
PDM, 191
Pearson's r, 254
perceptron, 1 12, 1 13
performance, 79
performance measures, 78
periodic crawler, 198
personalization, 202
point estimation, 47
PolyAnalyst, 285
polythetic, 128
population, 67
posterior probability, 52, 76
potentially large, 173
precision, 26
predicate set, 235
prediction, 7, 75, 82, 256
predictive model, 4
Predictive Modeling Mark-Up Language
(PMML), 286
predictor, 55
prefix, 263
preprocessing, 10
prior probability, 52, 87
PRISM, 1 17
privacy, 16
processing element function, 64
processing elements, 61
profile association rule, 191
profiling, 16
progressive refinement, 229
propagation, 105, 108
pruning, 1 00
quad tree, 224
Quadstone System, 286
quantitative association rule, 185
quartiles, 51
query language, 23
querying, 12
QUEST, 123
r correlation coefficient, 56
R-tree, 225
radial basis function, 1 12
radial function, 1 12
radius, 129
RainForest, 103
range, 5 1
range query, 222
rank sink, 205
rare item problem, 187
raster, 226
ratio rule, 192
RBF, 1 1 2
RBF network, 1 12
Re:order, 286
recall, 26
receiver operating characteristic
curve, 79
recurrent neural network, 25 1
referral analysis, 220
region query, 222
regression, 6, 55, 80-86
linear, 80
logistic, 85
nonlinear, 85
regression coefficients, 8 1
regressor, 55
related concepts, 21-45
relation, 22
relational algebra, 22
relational calculus, 22
relational model, 22
Relational OLAP, 39
relationship, 21
relative operating characteristic
curve, 79
relevance, 233
relevant, 26
reproduction, 67
response, 55
return on investment, 15, 36
rms, 47
RMSE, 47
RNN, 251
robot, 198
ROC curve, 79
ROCK algorithm, 158
ROI, 1 5, 36
ROLAP, 39
roll up, 29, 40, 229
root mean square, 47
root mean square error, 47
roulette wheel selection, 68
rules, 100, 1 14-1 19
S-Plus, 287
Sampling, 173
SAND, 222
satisfy, 269
scalability, 14, 17, 103
scalable parallelizable induction of decision
trees, 103
scatter diagram, 52
Scenario, 286
schema, 21
SD(CLARANS), 239
search, 13
search engine, 198
SeeS, 286
seed URLs, 198
segmentation, 8, 125
segments, 8
selection, 10
self organizing, 148
self organizing feature map, 148
self organizing map, 148
self organizing neural networks, 147
semantic index, 201
sequence, 260
sequence association rule, 270
sequence association rule problem, 271
sequence association rules, 270
sequence classifier, 266
sequence discovery, 9
sequential analysis, 9
sequential pattern, 9, 213, 260
serial, 128
session, 208, 209
sessionize, 208
set, 23
SGML, 219
shock, 256
SIGKDD, 20
sigmoid, 66
sigmoid activation function, 66
silhouette coefficient, 244
similarity, 26, 57
similarity measure, 129, 158
cosine, 58
Dice, 58
Jaccard, 58
overlap, 58
similarity measures, 26, 57-58
simultaneous, 128
single link, 130, 133
slice, 40
sliding window, 264
SLIQ, 124
smoothing, 253
snapshot, 245
snowflake schema, 34
SOFM, 148
soft focus, 199
SOM, 148
SPADE, 262
Index 313
spatial association rules, 233, 234
spatial characteristic rules, 233
spatial clustering, 237
spatial data, 221
spatial data dominant generalization, 230
spatial data mining, 221
spatial database, 221
spatial discriminant rule, 233
spatial hierarchy, 229
spatial join, 222
spatial mining, 221-244
spatial operator, 221
spatial selection, 222
spider, 198
splitting, 94, 100
splitting attributes, 94
splitting predicates, 94
SPRINT, 103, 124
SQL, 17, 23
squared error, 47, 138, 139
Squared Error algorithm, 139
squashing function, 64
tServ, 274
standard deviation, 5 1
star schema, 3 1 , 32
stationary, 256
STATISTICA Data Miner, 287
statistical inference, 42
statistical sigrilficance, 54
statistics, 12
step, 65
step activation function, 65
STING, 23 1
strength, 166
String to String Conversion, 259
subepisode, 267
subsequence, 260
subseries, 252
subtree raising, 100
subtree replacement, 100
suffix tree, 210
summarization, 8

314 I ndex
SuperQuery, 287
supervised learning, 43, 63, 106-1 12
support, 165, 166, 188, 216, 234, 261, 268,
269, 27 1
SurfAid Analytics, 288
surprise, 1 89
targeting, 196
task parallelism, 178, 180
temporal association rules, 266-272
temporal database, 246
temporal mining, 245-273
term frequency, 201
TF, 201
the 5As, 1 9
The Data Mining Suite, 278
thematic map, 226
threshold, 65
threshold activation function, 65
time constraint, 265
time line, 1 2
time series, 6 , 252-257
time series analysis, 6
TN, 79
topological relationship, 227
TP, 79
training data, 76
transaction, 261
transaction time, 246
transformation, 10, 255
transition probability, 248
traversal patterns, 207, 21 1
trend dependency, 268
trend detection, 234
trie, 209
true negative, 79
true positive, 79
unbiased, 47
unipolar, 65
unipolar activation function, 65
unsupervised learning, 43, 63, 106,
147-149
valid time, 246
variance, 51
vector, 226
vertical fragment, 145
virtual warehouse, 38
Virtual Web View, 201
visualization, 1 1 , 14
Visualizer Workstation, 288
Voronoi diagram, 238
Voronoi polyhedron, 238
VWV, 201
W
AP-tree, 214. 219
WaveCluster, 241
wavelet transform, 241
Web access patterns, 207
Web content mining, 197
Web log, 206
Web mining, 1 95-220
Web usage mining, 206
Web Watcher, 204
WebAnalyst, 288
WEBMINER, 220
WebMiner ASP, 288
WebML, 202, 219
WebSIFf, 220
Whatlf, 279
white noise, 256
WizWhy, 289
WordNet, 202
WordNet Semantic Network, 202
XML, 198
XpertRule Miner, 289
About the Author
Margaret H. Dunham received the B.A. and the M.S. in mathematics from Miami Uni
versity in Oxford, Ohio. She earned the Ph.D. degree in computer science from Southern
Methodist University. Professor Dunham's research interests encompass main memory
databases, data mining, temporal databases, and mobile computing. She is currently an
Associate Editor for IEEE Transactions on Knowledge and Data Engineering. She has
published numerous technical papers in such research areas as database concurrency con
trol and recovery, database machines, main memory databases, and mobile computing.
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