Cluster analysis cluster analysis analysis

December 1, 2024 Data Mining: Concepts and Technique 1
Data Mining:
Concepts and
Techniques
— Slides for Textbook —
— Chapter 8 —
©Jiawei Han and Micheline Kamber
Intelligent Database Systems Research Lab
School of Computing Science
Simon Fraser University, Canada
http://www.cs.sfu.ca

Chapter 8. Cluster Analysis
 What is Cluster Analysis?
 Types of Data in Cluster Analysis
 A Categorization of Major Clustering Methods
 Partitioning Methods
 Hierarchical Methods
 Density-Based Methods
 Grid-Based Methods
 Model-Based Clustering Methods
 Outlier Analysis
 Summary

General Applications of Clustering
 Pattern Recognition
 Spatial Data Analysis
 create thematic maps in GIS by clustering feature
spaces
 detect spatial clusters and explain them in spatial
data mining
 Image Processing
 Economic Science (especially market research)
 WWW
 Document classification
 Cluster Weblog data to discover groups of similar
access patterns

Examples of Clustering Applications
 Marketing: Help marketers discover distinct groups in
their customer bases, and then use this knowledge to
develop targeted marketing programs
 Land use: Identification of areas of similar land use in
an earth observation database
 Insurance: Identifying groups of motor insurance
policy holders with a high average claim cost
 City-planning: Identifying groups of houses according
to their house type, value, and geographical location
 Earth-quake studies: Observed earth quake epicenters
should be clustered along continent faults

What Is Good Clustering?
 A good clustering method will produce high quality
clusters with
 high intra-class similarity
 low inter-class similarity
 The quality of a clustering result depends on both the
similarity measure used by the method and its
implementation.
 The quality of a clustering method is also measured
by its ability to discover some or all of the hidden
patterns.

Requirements of Clustering in Data
Mining
 Scalability
 Ability to deal with different types of attributes
 Discovery of clusters with arbitrary shape
 Minimal requirements for domain knowledge to
determine input parameters
 Able to deal with noise and outliers
 Insensitive to order of input records
 High dimensionality
 Incorporation of user-specified constraints
 Interpretability and usability

 Summary

Data Structures
 Data matrix
 (two modes)
 Dissimilarity matrix
 (one mode)


















np
x
...
nf
x
...
n1
x
...
...
...
...
...
ip
x
...
if
x
...
i1
x
...
...
...
...
...
1p
x
...
1f
x
...
11
x
















0
...
)
2
,
(
)
1
,
(
:
:
:
)
2
,
3
(
)
...
n
d
n
d
0
d
d(3,1
0
d(2,1)
0

Measure the Quality of Clustering
 Dissimilarity/Similarity metric: Similarity is expressed in
terms of a distance function, which is typically metric:
d(i, j)
 There is a separate “quality” function that measures
the “goodness” of a cluster.
 The definitions of distance functions are usually very
different for interval-scaled, boolean, categorical,
ordinal and ratio variables.
 Weights should be associated with different variables
based on applications and data semantics.
 It is hard to define “similar enough” or “good enough”
 the answer is typically highly subjective.

Type of data in clustering analysis
 Interval-scaled variables:
 Binary variables:
 Nominal, ordinal, and ratio variables:
 Variables of mixed types:

Interval-valued variables
 Standardize data
 Calculate the mean absolute deviation:
where
 Calculate the standardized measurement (z-score)
 Using mean absolute deviation is more robust than
using standard deviation
.
)
...
2
1
1
nf
f
f
f
x
x
(x
n
m 



|)
|
...
|
|
|
(|
1
2
1 f
nf
f
f
f
f
f
m
x
m
x
m
x
n
s 






f
f
if
if s
m
x
z



Similarity and Dissimilarity Between
Objects
 Distances are normally used to measure the similarity
or dissimilarity between two data objects
 Some popular ones include: Minkowski distance:
where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two
p-dimensional data objects, and q is a positive
integer
 If q = 1, d is Manhattan distance
q
q
p
p
q
q
j
x
i
x
j
x
i
x
j
x
i
x
j
i
d )
|
|
...
|
|
|
(|
)
,
(
2
2
1
1







|
|
...
|
|
|
|
)
,
(
2
2
1
1 p
p j
x
i
x
j
x
i
x
j
x
i
x
j
i
d 







Similarity and Dissimilarity Between
Objects (Cont.)
 If q = 2, d is Euclidean distance:
 Properties

d(i,j)  0

d(i,i) = 0

d(i,j) = d(j,i)

d(i,j)  d(i,k) + d(k,j)
 Also one can use weighted distance, parametric
Pearson product moment correlation, or other
disimilarity measures.
)
|
|
...
|
|
|
(|
)
,
( 2
2
2
2
2
1
1 p
p j
x
i
x
j
x
i
x
j
x
i
x
j
i
d 







Binary Variables
 A contingency table for binary data
 Simple matching coefficient (invariant, if the binary
variable is symmetric):
 Jaccard coefficient (noninvariant if the binary variable
is asymmetric):
d
c
b
a
c
b
j
i
d





)
,
(
p
d
b
c
a
sum
d
c
d
c
b
a
b
a
sum




0
1
0
1
c
b
a
c
b
j
i
d




)
,
(
Object i
Object j

Dissimilarity between Binary
Variables
 Example
 gender is a symmetric attribute
 the remaining attributes are asymmetric binary
 let the values Y and P be set to 1, and the value N be set to
0
Name Gender Fever Cough Test-1 Test-2 Test-3 Test-4
Jack M Y N P N N N
Mary F Y N P N P N
Jim M Y P N N N N
75
.
0
2
1
1
2
1
)
,
(
67
.
0
1
1
1
1
1
)
,
(
33
.
0
1
0
2
1
0
)
,
(















mary
jim
d
jim
jack
d
mary
jack
d

Nominal Variables
 A generalization of the binary variable in that it can
take more than 2 states, e.g., red, yellow, blue, green
 Method 1: Simple matching
 m: # of matches, p: total # of variables
 Method 2: use a large number of binary variables
 creating a new binary variable for each of the M
nominal states
p
m
p
j
i
d 

)
,
(

Ordinal Variables
 An ordinal variable can be discrete or continuous
 order is important, e.g., rank
 Can be treated like interval-scaled
 replacing xif by their rank
 map the range of each variable onto [0, 1] by
replacing i-th object in the f-th variable by
 compute the dissimilarity using methods for
interval-scaled variables
1
1



f
if
if M
r
z
}
,...,
1
{ f
if
M
r 

Ratio-Scaled Variables
 Ratio-scaled variable: a positive measurement on a
nonlinear scale, approximately at exponential scale,
such as AeBt
or Ae-Bt
 Methods:
 treat them like interval-scaled variables — not a
good choice! (why?)
 apply logarithmic transformation
yif = log(xif)
 treat them as continuous ordinal data treat their
rank as interval-scaled.

Variables of Mixed Types
 A database may contain all the six types of variables
 symmetric binary, asymmetric binary, nominal,
ordinal, interval and ratio.
 One may use a weighted formula to combine their
effects.
 f is binary or nominal:
dij
(f)
= 0 if xif = xjf , or dij
(f)
= 1 o.w.
 f is interval-based: use the normalized distance
 f is ordinal or ratio-scaled
 compute ranks rif and
 and treat zif as interval-scaled
)
(
1
)
(
)
(
1
)
,
( f
ij
p
f
f
ij
f
ij
p
f
d
j
i
d







1
1



f
if
M
r
zif

 Summary

Major Clustering Approaches
 Partitioning algorithms: Construct various partitions
and then evaluate them by some criterion
 Hierarchy algorithms: Create a hierarchical
decomposition of the set of data (or objects) using
some criterion
 Density-based: based on connectivity and density
functions
 Grid-based: based on a multiple-level granularity
structure
 Model-based: A model is hypothesized for each of the

 Summary

Partitioning Algorithms: Basic
Concept
 Partitioning method: Construct a partition of a
database D of n objects into a set of k clusters
 Given a k, find a partition of k clusters that optimizes
the chosen partitioning criterion
 Global optimal: exhaustively enumerate all
partitions
 Heuristic methods: k-means and k-medoids
algorithms
 k-means (MacQueen’67): Each cluster is represented
by the center of the cluster
 k-medoids or PAM (Partition around medoids)
(Kaufman & Rousseeuw’87): Each cluster is

The K-Means Clustering Method
 Given k, the k-means algorithm is implemented in 4
steps:
 Partition objects into k nonempty subsets
 Compute seed points as the centroids of the
clusters of the current partition. The centroid is
the center (mean point) of the cluster.
 Assign each object to the cluster with the
nearest seed point.
 Go back to Step 2, stop when no more new
assignment.

The K-Means Clustering Method
 Example
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10

Comments on the K-Means Method
 Strength
 Relatively efficient: O(tkn), where n is # objects, k is #
clusters, and t is # iterations. Normally, k, t << n.
 Often terminates at a local optimum. The global
optimum may be found using techniques such as:
deterministic annealing and genetic algorithms
 Weakness
 Applicable only when mean is defined, then what
about categorical data?
 Need to specify k, the number of clusters, in advance
 Unable to handle noisy data and outliers
 Not suitable to discover clusters with non-convex
shapes

Variations of the K-Means Method
 A few variants of the k-means which differ in
 Selection of the initial k means
 Dissimilarity calculations
 Strategies to calculate cluster means
 Handling categorical data: k-modes (Huang’98)
 Replacing means of clusters with modes
 Using new dissimilarity measures to deal with
categorical objects
 Using a frequency-based method to update modes
of clusters
 A mixture of categorical and numerical data: k-
prototype method

The K-Medoids Clustering Method
 Find representative objects, called medoids, in clusters
 PAM (Partitioning Around Medoids, 1987)
 starts from an initial set of medoids and iteratively
replaces one of the medoids by one of the non-
medoids if it improves the total distance of the
resulting clustering
 PAM works effectively for small data sets, but does
not scale well for large data sets
 CLARA (Kaufmann & Rousseeuw, 1990)
 CLARANS (Ng & Han, 1994): Randomized sampling
 Focusing + spatial data structure (Ester et al., 1995)

PAM (Partitioning Around Medoids)
(1987)
 PAM (Kaufman and Rousseeuw, 1987), built in Splus
 Use real object to represent the cluster
 Select k representative objects arbitrarily
 For each pair of non-selected object h and selected
object i, calculate the total swapping cost TCih
 For each pair of i and h,
 If TCih < 0, i is replaced by h

Then assign each non-selected object to the
most similar representative object
 repeat steps 2-3 until there is no change

PAM Clustering: Total swapping cost TCih=jCjih
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
j
i
h
t
Cjih = 0
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
t
i h
j
Cjih = d(j, h) - d(j, i)
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
h
i
t
j
Cjih = d(j, t) - d(j, i)
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
t
i
h j
Cjih = d(j, h) - d(j, t)

CLARA (Clustering Large Applications)
(1990)
 CLARA (Kaufmann and Rousseeuw in 1990)
 Built in statistical analysis packages, such as S+
 It draws multiple samples of the data set, applies PAM
on each sample, and gives the best clustering as the
output
 Strength: deals with larger data sets than PAM
 Weakness:
 Efficiency depends on the sample size
 A good clustering based on samples will not
necessarily represent a good clustering of the
whole data set if the sample is biased

CLARANS (“Randomized” CLARA) (1994)
 CLARANS (A Clustering Algorithm based on Randomized
Search) (Ng and Han’94)
 CLARANS draws sample of neighbors dynamically
 The clustering process can be presented as searching a
graph where every node is a potential solution, that is,
a set of k medoids
 If the local optimum is found, CLARANS starts with new
randomly selected node in search for a new local
optimum
 It is more efficient and scalable than both PAM and
CLARA
 Focusing techniques and spatial access structures may

 Summary

Hierarchical Clustering
 Use distance matrix as clustering criteria. This
method does not require the number of clusters k as
an input, but needs a termination condition
Step 0 Step 1 Step 2 Step 3 Step 4
b
d
c
e
a a b
d e
c d e
a b c d e
Step 4 Step 3 Step 2 Step 1 Step 0
agglomerative
(AGNES)
divisive
(DIANA)

AGNES (Agglomerative Nesting)
 Introduced in Kaufmann and Rousseeuw (1990)
 Implemented in statistical analysis packages, e.g.,
Splus
 Use the Single-Link method and the dissimilarity
matrix.
 Merge nodes that have the least dissimilarity
 Go on in a non-descending fashion
 Eventually all nodes belong to the same cluster
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10

A Dendrogram Shows How the
Clusters are Merged Hierarchically
Decompose data objects into a several levels of nested
partitioning (tree of clusters), called a dendrogram.
A clustering of the data objects is obtained by cutting the
dendrogram at the desired level, then each connected
component forms a cluster.

DIANA (Divisive Analysis)
 Introduced in Kaufmann and Rousseeuw (1990)
 Implemented in statistical analysis packages, e.g.,
Splus
 Inverse order of AGNES
 Eventually each node forms a cluster on its own
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10

More on Hierarchical Clustering
Methods
 Major weakness of agglomerative clustering methods
 do not scale well: time complexity of at least O(n2
),
where n is the number of total objects
 can never undo what was done previously
 Integration of hierarchical with distance-based
clustering
 BIRCH (1996): uses CF-tree and incrementally adjusts
the quality of sub-clusters
 CURE (1998): selects well-scattered points from the
cluster and then shrinks them towards the center of
the cluster by a specified fraction
 CHAMELEON (1999): hierarchical clustering using
dynamic modeling

BIRCH (1996)
 Birch: Balanced Iterative Reducing and Clustering using
Hierarchies, by Zhang, Ramakrishnan, Livny
(SIGMOD’96)
 Incrementally construct a CF (Clustering Feature) tree, a
hierarchical data structure for multiphase clustering
 Phase 1: scan DB to build an initial in-memory CF tree
(a multi-level compression of the data that tries to
preserve the inherent clustering structure of the
data)

Phase 2: use an arbitrary clustering algorithm to
cluster the leaf nodes of the CF-tree
 Scales linearly: finds a good clustering with a single scan
and improves the quality with a few additional scans
 Weakness: handles only numeric data, and sensitive to

Clustering Feature Vector
Clustering Feature: CF = (N, LS, SS)
N: Number of data points
LS: N
i=1=Xi
SS: N
i=1=Xi
2
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
CF = (5, (16,30),(54,190))
(3,4)
(2,6)
(4,5)
(4,7)
(3,8)

CF Tree
CF1
child1
CF3
child3
CF2
child2
CF6
child6
CF1
child1
CF3
child3
CF2
child2
CF5
child5
CF1 CF2 CF6
prev next CF1 CF2 CF4
prev next
B = 7
L = 6
Root
Non-leaf node
Leaf node Leaf node

CURE (Clustering Using
REpresentatives )
 CURE: proposed by Guha, Rastogi & Shim, 1998
 Stops the creation of a cluster hierarchy if a level
consists of k clusters
 Uses multiple representative points to evaluate the
distance between clusters, adjusts well to arbitrary
shaped clusters and avoids single-link effect

Drawbacks of Distance-Based
Method
 Drawbacks of square-error based clustering method

Consider only one point as representative of a
cluster

Good only for convex shaped, similar size and
density, and if k can be reasonably estimated

Cure: The Algorithm
 Draw random sample s.
 Partition sample to p partitions with size s/p
 Partially cluster partitions into s/pq clusters
 Eliminate outliers

By random sampling

If a cluster grows too slow, eliminate it.
 Cluster partial clusters.
 Label data in disk

Data Partitioning and Clustering

s = 50

p = 2

s/p = 25
x x
x
y
y y
y
x
y
x
 s/pq = 5

Cure: Shrinking Representative
Points
 Shrink the multiple representative points towards the
gravity center by a fraction of .
 Multiple representatives capture the shape of the
cluster
x
y
x
y

Clustering Categorical Data:
ROCK
 ROCK: Robust Clustering using linKs,
by S. Guha, R. Rastogi, K. Shim (ICDE’99).
 Use links to measure similarity/proximity
 Not distance based
 Computational complexity:
 Basic ideas:
 Similarity function and neighbors:
Let T1 = {1,2,3}, T2={3,4,5}
O n nm m n n
m a
( log )
2 2
 
Sim T T
T T
T T
( , )
1 2
1 2
1 2



Sim T T
( , )
{ }
{ , , , , }
.
1 2
3
1 2 3 4 5
1
5
0 2
  

Rock: Algorithm
 Links: The number of common neighbours for
the two points.
 Algorithm
 Draw random sample
 Cluster with links
 Label data in disk
{1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,3,5}
{1,4,5}, {2,3,4}, {2,3,5}, {2,4,5}, {3,4,5}
{1,2,3} {1,2,4}
3

CHAMELEON
 CHAMELEON: hierarchical clustering using dynamic
modeling, by G. Karypis, E.H. Han and V. Kumar’99
 Measures the similarity based on a dynamic model
 Two clusters are merged only if the
interconnectivity and closeness (proximity) between
two clusters are high relative to the internal
interconnectivity of the clusters and closeness of
items within the clusters
 A two phase algorithm
 1. Use a graph partitioning algorithm: cluster
objects into a large number of relatively small
sub-clusters
 2. Use an agglomerative hierarchical clustering
algorithm: find the genuine clusters by

Overall Framework of
CHAMELEON
Construct
Sparse Graph Partition the Graph
Merge Partition
Final Clusters
Data Set

 Summary

Density-Based Clustering
Methods
 Clustering based on density (local cluster criterion),
such as density-connected points

Major features:

Discover clusters of arbitrary shape

Handle noise

One scan

Need density parameters as termination
condition
 Several interesting studies:
 DBSCAN: Ester, et al. (KDD’96)
 OPTICS: Ankerst, et al (SIGMOD’99).
 DENCLUE: Hinneburg & D. Keim (KDD’98)
 CLIQUE: Agrawal, et al. (SIGMOD’98)

Density-Based Clustering: Background
 Two parameters:
 Eps: Maximum radius of the neighbourhood
 MinPts: Minimum number of points in an Eps-
neighbourhood of that point
 NEps(p): {q belongs to D | dist(p,q) <= Eps}
 Directly density-reachable: A point p is directly density-
reachable from a point q wrt. Eps, MinPts if

1) p belongs to NEps(q)
 2) core point condition:
|NEps (q)| >= MinPts
p
q
MinPts = 5
Eps = 1 cm

Density-Based Clustering: Background
(II)
 Density-reachable:
 A point p is density-reachable
from a point q wrt. Eps, MinPts if
there is a chain of points p1, …,
pn, p1 = q, pn = p such that pi+1 is
directly density-reachable from
pi
 Density-connected
 A point p is density-connected to
a point q wrt. Eps, MinPts if there
is a point o such that both, p and
q are density-reachable from o
wrt. Eps and MinPts.
p
q
p1
p q
o

DBSCAN: Density Based Spatial
Clustering of Applications with Noise
 Relies on a density-based notion of cluster: A cluster is
defined as a maximal set of density-connected points
 Discovers clusters of arbitrary shape in spatial
databases with noise
Core
Border
Outlier
Eps = 1cm
MinPts = 5

DBSCAN: The Algorithm
 Arbitrary select a point p
 Retrieve all points density-reachable from p wrt Eps
and MinPts.
 If p is a core point, a cluster is formed.
 If p is a border point, no points are density-
reachable from p and DBSCAN visits the next point
of the database.
 Continue the process until all of the points have
been processed.

OPTICS: A Cluster-Ordering Method (1999)
 OPTICS: Ordering Points To Identify the Clustering
Structure
 Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99)
 Produces a special order of the database wrt its
density-based clustering structure
 This cluster-ordering contains info equiv to the
density-based clusterings corresponding to a
broad range of parameter settings
 Good for both automatic and interactive cluster
analysis, including finding intrinsic clustering
structure
 Can be represented graphically or using
visualization techniques

OPTICS: Some Extension from
DBSCAN

Index-based:

k = number of dimensions

N = 20

p = 75%

M = N(1-p) = 5
 Complexity: O(kN2
)
 Core Distance
 Reachability Distance
D
p2
MinPts = 5
e = 3 cm
Max (core-distance (o), d (o, p))
r(p1, o) = 2.8cm. r(p2,o) = 4cm
o
o
p1



Reachability
-distance
Cluster-order
of the objects
undefined
 ‘

DENCLUE: using density functions
 DENsity-based CLUstEring by Hinneburg & Keim
(KDD’98)
 Major features
 Solid mathematical foundation
 Good for data sets with large amounts of noise
 Allows a compact mathematical description of
arbitrarily shaped clusters in high-dimensional data
sets
 Significant faster than existing algorithm (faster than
DBSCAN by a factor of up to 45)
 But needs a large number of parameters

 Uses grid cells but only keeps information about
grid cells that do actually contain data points and
manages these cells in a tree-based access
structure.
 Influence function: describes the impact of a data
point within its neighborhood.
 Overall density of the data space can be calculated
as the sum of the influence function of all data
points.
 Clusters can be determined mathematically by
identifying density attractors.
 Density attractors are local maximal of the overall
Denclue: Technical Essence

Gradient: The steepness of a slope
 Example
 


N
i
x
x
d
D
Gaussian
i
e
x
f 1
2
)
,
(
2
2
)
( 
 





N
i
x
x
d
i
i
D
Gaussian
i
e
x
x
x
x
f 1
2
)
,
(
2
2
)
(
)
,
( 
f x y e
Gaussian
d x y
( , )
( , )


2
2
2

Density Attractor

Center-Defined and Arbitrary

 Summary

Grid-Based Clustering Method
 Using multi-resolution grid data structure
 Several interesting methods
 STING (a STatistical INformation Grid approach)
by Wang, Yang and Muntz (1997)
 WaveCluster by Sheikholeslami, Chatterjee, and
Zhang (VLDB’98)

A multi-resolution clustering approach using
wavelet method
 CLIQUE: Agrawal, et al. (SIGMOD’98)

STING: A Statistical Information
Grid Approach
 Wang, Yang and Muntz (VLDB’97)
 The spatial area area is divided into rectangular cells
 There are several levels of cells corresponding to
different levels of resolution

Grid Approach (2)
 Each cell at a high level is partitioned into a number of
smaller cells in the next lower level
 Statistical info of each cell is calculated and stored
beforehand and is used to answer queries
 Parameters of higher level cells can be easily calculated
from parameters of lower level cell

count, mean, s, min, max

type of distribution—normal, uniform, etc.
 Use a top-down approach to answer spatial data queries

Start from a pre-selected layer—typically with a small
number of cells
 For each cell in the current level compute the confidence
interval

Grid Approach (3)
 Remove the irrelevant cells from further
consideration
 When finish examining the current layer, proceed
to the next lower level
 Repeat this process until the bottom layer is
reached
 Advantages:

Query-independent, easy to parallelize,
incremental update

O(K), where K is the number of grid cells at the
lowest level
 Disadvantages:

All the cluster boundaries are either horizontal
or vertical, and no diagonal boundary is

WaveCluster (1998)
 Sheikholeslami, Chatterjee, and Zhang (VLDB’98)
 A multi-resolution clustering approach which
applies wavelet transform to the feature space
 A wavelet transform is a signal processing
technique that decomposes a signal into
different frequency sub-band.
 Both grid-based and density-based
 Input parameters:
 # of grid cells for each dimension
 the wavelet, and the # of applications of wavelet
transform.

WaveCluster (1998)
 How to apply wavelet transform to find clusters
 Summaries the data by imposing a
multidimensional grid structure onto data space
 These multidimensional spatial data objects are
represented in a n-dimensional feature space
 Apply wavelet transform on feature space to find
the dense regions in the feature space
 Apply wavelet transform multiple times which
result in clusters at different scales from fine to
coarse

What Is Wavelet (2)?

Quantization

Transformation

WaveCluster (1998)
 Why is wavelet transformation useful for clustering
 Unsupervised clustering
It uses hat-shape filters to emphasize region
where points cluster, but simultaneously to
suppress weaker information in their boundary
 Effective removal of outliers
 Multi-resolution
 Cost efficiency
 Major features:
 Complexity O(N)
 Detect arbitrary shaped clusters at different scales
 Not sensitive to noise, not sensitive to input order
 Only applicable to low dimensional data

CLIQUE (Clustering In QUEst)
 Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98).
 Automatically identifying subspaces of a high
dimensional data space that allow better clustering than
original space
 CLIQUE can be considered as both density-based and
grid-based

It partitions each dimension into the same number of
equal length interval
 It partitions an m-dimensional data space into non-
overlapping rectangular units
 A unit is dense if the fraction of total data points
contained in the unit exceeds the input model
parameter

A cluster is a maximal set of connected dense units
within a subspace

CLIQUE: The Major Steps
 Partition the data space and find the number of points
that lie inside each cell of the partition.
 Identify the subspaces that contain clusters using the
Apriori principle
 Identify clusters:
 Determine dense units in all subspaces of interests
 Determine connected dense units in all subspaces
of interests.
 Generate minimal description for the clusters
 Determine maximal regions that cover a cluster of
connected dense units for each cluster
 Determination of minimal cover for each cluster

Salary
(10,000)
20 30 40 50 60
age
5
4
3
1
2
6
7
0
20 30 40 50 60
age
5
4
3
1
2
6
7
0
Vacation(
week)
age
Vacation
Salary 30 50
 = 3

Strength and Weakness of CLIQUE
 Strength
 It automatically finds subspaces of the highest
dimensionality such that high density clusters exist
in those subspaces
 It is insensitive to the order of records in input and
does not presume some canonical data
distribution
 It scales linearly with the size of input and has
good scalability as the number of dimensions in
the data increases
 Weakness
 The accuracy of the clustering result may be
degraded at the expense of simplicity of the

 Summary

Model-Based Clustering Methods
 Attempt to optimize the fit between the data and some
mathematical model
 Statistical and AI approach
 Conceptual clustering

A form of clustering in machine learning

Produces a classification scheme for a set of unlabeled
objects

Finds characteristic description for each concept (class)
 COBWEB (Fisher’87)

A popular a simple method of incremental conceptual
learning

Creates a hierarchical clustering in the form of a
classification tree

Each node refers to a concept and contains a probabilistic
description of that concept

COBWEB Clustering Method
A classification tree

More on Statistical-Based
Clustering
 Limitations of COBWEB

The assumption that the attributes are
independent of each other is often too strong
because correlation may exist
 Not suitable for clustering large database data –
skewed tree and expensive probability
distributions
 CLASSIT

an extension of COBWEB for incremental
clustering of continuous data
 suffers similar problems as COBWEB
 AutoClass (Cheeseman and Stutz, 1996)
 Uses Bayesian statistical analysis to estimate the
number of clusters

Popular in industry

Other Model-Based
Clustering Methods
 Neural network approaches
 Represent each cluster as an exemplar, acting as
a “prototype” of the cluster
 New objects are distributed to the cluster whose
exemplar is the most similar according to some
dostance measure
 Competitive learning
 Involves a hierarchical architecture of several
units (neurons)
 Neurons compete in a “winner-takes-all” fashion
for the object currently being presented

Model-Based Clustering Methods

Self-organizing feature maps
(SOMs)
 Clustering is also performed by having
several units competing for the current
object
 The unit whose weight vector is closest to
the current object wins
 The winner and its neighbors learn by having
their weights adjusted
 SOMs are believed to resemble processing
that can occur in the brain
 Useful for visualizing high-dimensional data
in 2- or 3-D space

 Summary

What Is Outlier Discovery?
 What are outliers?
 The set of objects are considerably dissimilar
from the remainder of the data
 Example: Sports: Michael Jordon, Wayne
Gretzky, ...
 Problem
 Find top n outlier points
 Applications:
 Credit card fraud detection
 Telecom fraud detection
 Customer segmentation
 Medical analysis

Outlier Discovery:
Statistical
Approaches
 Assume a model underlying distribution that
generates data set (e.g. normal distribution)
 Use discordancy tests depending on
 data distribution
 distribution parameter (e.g., mean, variance)
 number of expected outliers
 Drawbacks
 most tests are for single attribute
 In many cases, data distribution may not be
known

Outlier Discovery: Distance-
Based Approach
 Introduced to counter the main limitations imposed
by statistical methods
 We need multi-dimensional analysis without
knowing data distribution.
 Distance-based outlier: A DB(p, D)-outlier is an object
O in a dataset T such that at least a fraction p of the
objects in T lies at a distance greater than D from O
 Algorithms for mining distance-based outliers
 Index-based algorithm
 Nested-loop algorithm
 Cell-based algorithm

Outlier Discovery: Deviation-
Based Approach
 Identifies outliers by examining the main
characteristics of objects in a group
 Objects that “deviate” from this description are
considered outliers
 sequential exception technique
 simulates the way in which humans can
distinguish unusual objects from among a series
of supposedly like objects
 OLAP data cube technique
 uses data cubes to identify regions of anomalies
in large multidimensional data

 Summary

Problems and Challenges
 Considerable progress has been made in scalable
clustering methods
 Partitioning: k-means, k-medoids, CLARANS
 Hierarchical: BIRCH, CURE
 Density-based: DBSCAN, CLIQUE, OPTICS
 Grid-based: STING, WaveCluster
 Model-based: Autoclass, Denclue, Cobweb
 Current clustering techniques do not address all the
requirements adequately
 Constraint-based clustering analysis: Constraints exist
in data space (bridges and highways) or in user
queries

Constraint-Based Clustering Analysis
 Clustering analysis: less parameters but more user-
desired constraints, e.g., an ATM allocation problem

Summary
 Cluster analysis groups objects based on their
similarity and has wide applications
 Measure of similarity can be computed for various
types of data
 Clustering algorithms can be categorized into
partitioning methods, hierarchical methods, density-
based methods, grid-based methods, and model-
based methods
 Outlier detection and analysis are very useful for fraud
detection, etc. and can be performed by statistical,
distance-based or deviation-based approaches
 There are still lots of research issues on cluster
analysis, such as constraint-based clustering

References (1)
 R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering
of high dimensional data for data mining applications. SIGMOD'98
 M. R. Anderberg. Cluster Analysis for Applications. Academic Press, 1973.
 M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points to
identify the clustering structure, SIGMOD’99.
 P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World Scietific,
1996
 M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering
clusters in large spatial databases. KDD'96.
 M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial databases:
Focusing techniques for efficient class identification. SSD'95.
 D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine
Learning, 2:139-172, 1987.
 D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An approach
based on dynamic systems. In Proc. VLDB’98.
 S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for large
databases. SIGMOD'98.
 A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.

References (2)
 L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster
Analysis. John Wiley & Sons, 1990.
 E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets.
VLDB’98.
 G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to
Clustering. John Wiley and Sons, 1988.
 P. Michaud. Clustering techniques. Future Generation Computer systems, 13, 1997.
 R. Ng and J. Han. Efficient and effective clustering method for spatial data mining.
VLDB'94.
 E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very
large data sets. Proc. 1996 Int. Conf. on Pattern Recognition, 101-105.
 G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution
clustering approach for very large spatial databases. VLDB’98.
 W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to Spatial
Data Mining, VLDB’97.
 T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering
method for very large databases. SIGMOD'96.

http://www.cs.sfu.ca/~han
Thank you !!!

Cluster analysis cluster analysis analysis

More Related Content

Similar to Cluster analysis cluster analysis analysis (20)

Recently uploaded (20)

Cluster analysis cluster analysis analysis