Getting to Know Data presentation basics

1
Data Mining:
Concepts and Techniques
— Chapter 2 —
Courtesy: Jiawei Han, Micheline Kamber, and Jian
Pei
University of Illinois at Urbana-Champaign
Simon Fraser University
©2013 Han, Kamber, and Pei. All rights

2
Chapter 2: Getting to Know Your Data
 Data Objects and Attribute Types
 Basic Statistical Descriptions of Data
 Data Visualization
 Measuring Data Similarity and Dissimilarity
 Summary

3
Types of Data Sets
 Record
 Relational records
 Data matrix, e.g., numerical matrix,
crosstabs
 Document data: text documents: term-
frequency vector
 Transaction data
 Graph and network
 World Wide Web
 Social or information networks
 Molecular Structures
 Ordered
 Video data: sequence of images
 Temporal data: time-series
 Sequential Data: transaction sequences
 Genetic sequence data
 Spatial, image and multimedia:
 Spatial data: maps
 Image data:
 Video data:
Document 1
season
timeout
lost
wi
n
game
score
ball
pla
y
coach
team
Document 2
Document 3
3 0 5 0 2 6 0 2 0 2
0
0
7 0 2 1 0 0 3 0 0
1 0 0 1 2 2 0 3 0
TID Items
1 Bread, Coke, Milk
2 Beer, Bread
3 Beer, Coke, Diaper, Milk
4 Beer, Bread, Diaper, Milk
5 Coke, Diaper, Milk

4
Data Objects
 Data sets are made up of data objects.
 A data object represents an entity.
 Examples:
 sales database: customers, store items, sales
 medical database: patients, treatments
 university database: students, professors, courses
 Also called samples , examples, instances, data points,
objects, tuples.
 Data objects are described by attributes.
 Database rows -> data objects; columns ->attributes.

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Attributes
 Attribute (or dimensions, features, variables):
a data field, representing a characteristic or
feature of a data object.
 E.g., customer _ID, name, address
 Types:
 Nominal
 Binary
 Numeric: quantitative
 Ordinal

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Attribute Types
 Nominal: categories, states, or “names of things”
 Hair_color = {auburn, black, blond, brown, grey, red, white}
 marital status, occupation, ID numbers, zip codes
 Binary
 Nominal attribute with only 2 states (0 and 1)
 Symmetric binary: both outcomes equally important e.g.,
gender
 Asymmetric binary: outcomes not equally important.

e.g., medical test (positive vs. negative)

Convention: assign 1 to most important outcome (e.g.,
HIV positive)
 Ordinal
 Values have a meaningful order (ranking) but magnitude
between successive values is not known.
 Size = {small, medium, large}, grades, army rankings
 Numeric
 Quantity (integer or real-valued)

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 Summary

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Basic Statistical Descriptions of Data
 Motivation
 To better understand the data: central tendency,
variation and spread
 Data dispersion characteristics
 median, max, min, quantiles, outliers, variance, etc.
 Graphical analysis
 Boxplot
 Histograms
 Scatter Plots

9
Measuring the Central Tendency
 Mean (algebraic measure) (sample vs. population):
Note: n is sample size and N is population size.
 Weighted arithmetic mean:
 Trimmed mean: chopping extreme values
 Median:
 Middle value if odd number of values, or average of the middle two values
otherwise
 Note: sort the given data before computing median
 Mode
 Value that occurs most frequently in the data
 Unimodal, bimodal, trimodal
 Note: sort the given data before computing model



n
i
i
x
n
x
1
1

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

 n
i
i
n
i
i
i
w
x
w
x
1
1
N
x




June 3, 2025
Data Mining: Concepts and
Techniques 10
Symmetric vs. Skewed Data
 Median, mean and mode of
symmetric, positively and negatively
skewed data
positively skewed negatively skewed
symmetric
Mode < median Mode > median

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Measuring the Dispersion of Data
 Quartiles, outliers and boxplots
 Quartiles: Q1 (25th
percentile), Q3 (75th
percentile)
 Inter-quartile range: IQR = Q3 –Q1
 Five number summary: min, Q1, median,Q3, max
 Boxplot: ends of the box are the quartiles; median is marked; add whiskers,
and plot outliers individually
 Outlier: usually, a value higher/lower than 1.5 x IQR
 Variance and standard deviation (sample: s, population: σ)
 Variance: (algebraic, scalable computation)
 Standard deviation s (or σ) is the square root of variance s2 (
orσ2)
 
  


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



n
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i
n
i
i x
N
x
N 1
2
2
1
2
2 1
)
(
1




Example
 Data: 30, 36, 47, 50, 52, 52, 56, 60, 63, 70, 70, 110
 Solution
 Data already in sorted order
 Total data = 12
 Q1 = 25th
percentile = 3rd
number here = 47
 Q3 = 9th
number = 63
 Q2 = 6th
number = median = 52
 IQR = 63-47 = 16
12

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Boxplot Analysis
 Five-number summary of a distribution
 Minimum, Q1, Median, Q3, Maximum
 Boxplot
 Data is represented with a box
 The ends of the box are at the first and third
quartiles, i.e., the height of the box is IQR
 The median is marked by a line within the
box
 Whiskers: two lines outside the box
extended to Minimum and Maximum
 Outliers: points beyond a specified outlier
threshold, plotted individually

Example
 Students marks in course-1: 30, 36, 47, 50, 52, 52, 56,
60, 63, 70, 70, 100
 Draw horizontal or vertical box plot
 Q1
 Q2
 Q3
 IQR
 Min
 Max
15

16
Graphic Displays of Basic Statistical Descriptions
 Boxplot: graphic display of five-number summary
 Histogram: x-axis are values, y-axis repres.
frequencies
 Scatter plot: each pair of values is a pair of
coordinates and plotted as points in the plane

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Histogram Analysis
 Histogram: Graph display of tabulated frequencies, shown as bars
 It shows what proportion of cases fall into each of several categories
 The categories are usually specified as non-overlapping intervals of
some variable. The categories (bars) must be adjacent
0
5
10
15
20
25
30
35
40
10000 30000 50000 70000 90000

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Histograms Often Tell More than Boxplots
 The two histograms
shown in the left may
have the same boxplot
representation
 The same values for:
min, Q1, median, Q3,
max
 But they have rather
different data
distributions

19
Scatter plot
 Provides a first look at bivariate data to see clusters of
points, outliers, etc
 Each pair of values is treated as a pair of coordinates
and plotted as points in the plane

20
Positively and Negatively Correlated Data
 The left half fragment is positively
correlated
 The right half is negative
correlated

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 Summary

23
Similarity and Dissimilarity
Can you find the 5 dissimilarities?

24

25
 Similarity
 Numerical measure of how alike two data objects are
 Value is higher when objects are more alike
 Often falls in the range [0,1]
 Dissimilarity (e.g., distance)
 Numerical measure of how different two data
objects are
 Lower when objects are more alike
 Minimum dissimilarity is often 0
 Upper limit varies
 Proximity refers to a similarity or dissimilarity

26
Data Matrix and Dissimilarity Matrix
 Data matrix
 n data points with p
dimensions
 Dissimilarity matrix
 n data points, but
registers only the
distance
 A triangular matrix

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0

27
Proximity Measure for Nominal Attributes
 Can take 2 or more states, e.g., red, yellow,
blue, green (generalization of a binary
attribute)
 Method 1: Simple matching
 m: # of matches, p: total # of variables
 Method 2: Use a large number of binary
attributes
 creating a new binary attribute for each of
p
m
p
j
i
d 

)
,
(

28
For attribute test-1, assume p = 1
So, d(2,1) = 1
d(4,1) = 0
similarly we can find other dissimilarity
values
So, obj i and j are dissimilar if d(i,j) = 1

29
Proximity Measure for Binary Attributes
 A contingency table for binary data
 Distance measure for symmetric binary
variables:
 Distance measure for asymmetric binary
variables:
 Jaccard coefficient (similarity measure for
asymmetric binary variables):
Object i
Object j

Example
 Assume: name is ID, gender is symmetric and others
are asymmetric
 Assume Y = P = 1. N = 0 for asymmetric attributes
 Consider only asymmetric attributes and find
dissimilarity or distance d(Jack, Jim)
30

31
Dissimilarity between Binary Variables
 Example
 Gender is a symmetric attribute. The remaining attributes are
asymmetric binary. Let the values Y and P be 1, and the value
N be 0
 Consider all asymmetric attributes
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
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0
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mary
jim
d
jim
jack
d
mary
jack
d

32
Distance on Numeric Data: Minkowski Distance
 Minkowski distance: A popular distance measure
where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-
dimensional data objects, and h is the order (the
distance so defined is also called L-h norm)
 Properties
 d(i, j) > 0 if i ≠ j, and d(i, i) = 0 (Positive definiteness)
 d(i, j) = d(j, i) (Symmetry)
 d(i, j)  d(i, k) + d(k, j) (Triangle Inequality)
 A distance that satisfies these properties is a metric

33
Special Cases of Minkowski Distance
 h = 1: Manhattan (city block, L1
norm) distance
 E.g., the Hamming distance: the number of bits that are different
between two binary vectors
 h = 2: (L2 norm) Euclidean distance
 h  . “supremum” (Lmax
norm, L
norm) distance.
 This is the maximum difference between any component (attribute)
of the vectors
)
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35
Example: Minkowski Distance
Dissimilarity Matrices
point attribute 1 attribute 2
x1 1 2
x2 3 5
x3 2 0
x4 4 5
L x1 x2 x3 x4
x1 0
x2 5 0
x3 3 6 0
x4 6 1 7 0
L2 x1 x2 x3 x4
x1 0
x2 3.61 0
x3 2.24 5.1 0
x4 4.24 1 5.39 0
L x1 x2 x3 x4
x1 0
x2 3 0
x3 2 5 0
x4 3 1 5 0
Manhattan (L1)
Euclidean (L2)
Supremum
0 2 4
2
4
x1
x2
x3
x4

36
Ordinal Variables
 An ordinal variable can be discrete or continuous
 Order is important, e.g., rank
 Can be treated like interval-scaled
 replace 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 

 3 states for test-2, so M = 3
 So the ranks are 3,1,2,3 for objects 1,2,3,4
 Normalizing ranks: 0.0, 0.5, 1.0 for ranks 1,2,3
respectively
 Replace these ranks in the table values and apply
dissimilarity metrics such as Euclidean
37

Euclidean distance will result into this
dissimilarity matrix
 If d(i,j) = 1, then i and j are most dissimilar
38

39
Attributes of Mixed Type
 A database may contain all attribute types
 Nominal, symmetric binary, asymmetric binary, numeric,
ordinal
 One may use a weighted formula to combine their effects
 if
 f is binary or nominal:
dij
(f)
= 0 if xif = xjf , or dij
(f)
= 1 otherwise
 f is numeric: use the normalized distance
 f is ordinal

Compute ranks rif and

Treat zif as interval-scaled
)
(
1
)
(
)
(
1
)
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ij
p
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f
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M
r
zif
hf
x
h
hf
x
h
jf
x
if
x
dij min
max 



40
Cosine Similarity
 A document can be represented by thousands of attributes, each
recording the frequency of a particular word (such as keywords) or
phrase in the document.
 Other vector objects: gene features in micro-arrays, …
 Applications: information retrieval, biologic taxonomy, gene feature
mapping, ...
 Cosine measure: If d1
and d2
are two vectors (e.g., term-frequency
vectors), then
cos(d1
, d2
) = (d1
 d2
) /||d1
|| ||d2
|| ,
where  indicates vector dot product, ||d||: the length of vector
d

41
Example: Cosine Similarity
 cos(d1
, d2
) = (d1
 d2
) /||d1
|| ||d2
|| ,
where  indicates vector dot product, ||d|: the length of vector d
 Ex: Find the similarity between documents 1 and 2.
d1
= (5, 0, 3, 0, 2, 0, 0, 2, 0, 0)
d2
= (3, 0, 2, 0, 1, 1, 0, 1, 0, 1)
d1
d2
= 5*3+0*0+3*2+0*0+2*1+0*1+0*1+2*1+0*0+0*1 = 25
||d1
||= (5*5+0*0+3*3+0*0+2*2+0*0+0*0+2*2+0*0+0*0)0.5
=(42)0.5
=
6.481
||d2
||= (3*3+0*0+2*2+0*0+1*1+1*1+0*0+1*1+0*0+1*1)0.5
=(17)0.5
= 4.12
cos(d1
, d2
) = 25 / (6.481 * 4.12) = 25/26.7= 0.94

Summary
 Data attribute types: nominal, binary, ordinal, interval-
scaled, ratio-scaled
 Many types of data sets, e.g., numerical, text, graph,
Web, image.
 Gain insight into the data by:
 Basic statistical data description: central tendency,
dispersion, graphical displays
 Data visualization: map data onto graphical
primitives
 Measure data similarity
 Above steps are the beginning of data preprocessing
 Many methods have been developed but still an active
area of research

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