Transcript What is Data?

Data Mining:
Concepts and Techniques
— Chapter 2 —
March 21, 2017
Data Mining: Concepts and Techniques
1
What is about Data?

General data characteristics

Basic data description and exploration

Measuring data similarity
March 21, 2017
Data Mining: Concepts and Techniques
2
What is Data?



Attributes
Collection of data objects and their
attributes
An attribute is a property or
characteristic of an object
 Examples: eye color of a
person, temperature, etc.
 Attribute is also known as
variable, field, characteristic, or
Objects
feature
A collection of attributes describe
an object
 Object is also known as record,
point, case, sample, entity, or
instance
March 21, 2017
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
10
Data Mining: Concepts and Techniques
3
Important Characteristics of Structured Data

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Dimensionality
 Curse of dimensionality
Sparsity
 Only presence counts
Resolution
Patterns depend on the scale
Similarity
 Distance measure


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Data Mining: Concepts and Techniques
4
Attribute Values

Attribute values are numbers or symbols assigned to an attribute

Distinction between attributes and attribute values


Same attribute can be mapped to different attribute
values
 Example: height can be measured in feet or meters
Different attributes can be mapped to the same set of
values
 Example: Attribute values for ID and age are integers
 But properties of attribute values can be different
 ID has no limit but age has a maximum and
minimum value
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Data Mining: Concepts and Techniques
5
Types of Attribute Values

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Nominal
 E.g., profession, ID numbers, eye color, zip codes
Ordinal
 E.g., rankings (e.g., army, professions), grades, height
in {tall, medium, short}
Binary
 E.g., medical test (positive vs. negative)
Interval
 E.g., calendar dates, body temperatures
Ratio

E.g., temperature in Kelvin, length, time, counts
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Data Mining: Concepts and Techniques
6
Properties of Attribute Values

The type of an attribute depends on which of the following properties it
possesses:


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


Distinctness:
= 
Order:
< >
Addition:
+ Multiplication:
*/
Nominal attribute: distinctness
Ordinal attribute: distinctness & order
Interval attribute: distinctness, order & addition
Ratio attribute: all 4 properties
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Data Mining: Concepts and Techniques
7
Attribute
Type
Description
Examples
Nominal
The values of a nominal attribute are
just different names, i.e., nominal
attributes provide only enough
information to distinguish one object
from another. (=, )
zip codes, employee
ID numbers, eye color,
sex: {male, female}
mode, entropy,
contingency
correlation, 2 test
Ordinal
The values of an ordinal attribute
provide enough information to order
objects. (<, >)
hardness of minerals,
{good, better, best},
grades, street numbers
median, percentiles,
rank correlation,
run tests, sign tests
Interval
For interval attributes, the
differences between values are
meaningful, i.e., a unit of
measurement exists.
(+, - )
calendar dates,
temperature in Celsius
or Fahrenheit
mean, standard
deviation, Pearson's
correlation, t and F
tests
For ratio variables, both differences
and ratios are meaningful. (*, /)
temperature in Kelvin,
monetary quantities,
counts, age, mass,
length, electrical
current
geometric mean,
harmonic mean,
percent variation
Ratio
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Data Mining: Concepts and Techniques
Operations
8
Discrete vs. Continuous Attributes

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Discrete Attribute
 Has only a finite or countably infinite set of values
 E.g., zip codes, profession, or the set of words in a
collection of documents
 Sometimes, represented as integer variables
 Note: Binary attributes are a special case of discrete
attributes
Continuous Attribute
 Has real numbers as attribute values
 Examples: temperature, height, or weight
 Practically, real values can only be measured and
represented using a finite number of digits
 Continuous attributes are typically represented as
floating-point variables
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Data Mining: Concepts and Techniques
9
Types of data sets

Record
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Graph
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Data Matrix
Document Data
Transaction Data
World Wide Web
Molecular Structures
Ordered
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Spatial Data
Temporal Data
Sequential Data
Genetic Sequence Data
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Data Mining: Concepts and Techniques
10
Important Characteristics of Structured Data

Dimensionality


Sparsity


Only presence counts
Resolution

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Curse of Dimensionality
Patterns depend on the scale
Data Mining: Concepts and Techniques
11
Record Data

Data that consists of a collection of records, each
of which consists of a fixed set of attributes
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
10
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Data Mining: Concepts and Techniques
12
Data Matrix


If data objects have the same fixed set of numeric attributes, then the data
objects can be thought of as points in a multi-dimensional space, where each
dimension represents a distinct attribute
Such data set can be represented by an m by n matrix, where there are m
rows, one for each object, and n columns, one for each attribute
Projection
of x Load
Projection
of y load
Distance
Load
Thickness
10.23
5.27
15.22
2.7
1.2
12.65
6.25
16.22
2.2
1.1
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Data Mining: Concepts and Techniques
13
Document Data

Each document becomes a `term' vector,
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each term is a component (attribute) of the vector,
the value of each component is the number of times
the corresponding term occurs in the document.
team
coach
pla
y
ball
score
game
wi
n
lost
timeout
season
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Document 1
3
0
5
0
2
6
0
2
0
2
Document 2
0
7
0
2
1
0
0
3
0
0
Document 3
0
1
0
0
1
2
2
0
3
0
Data Mining: Concepts and Techniques
14
Transaction Data
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A special type of record data, where
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each record (transaction) involves a set of items.
For example, consider a grocery store. The set of
products purchased by a customer during one
shopping trip constitute a transaction, while the
individual products that were purchased are the items.
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TID
Items
1
Bread, Coke, Milk
2
3
4
5
Beer, Bread
Beer, Coke, Diaper, Milk
Beer, Bread, Diaper, Milk
Coke, Diaper, Milk
Data Mining: Concepts and Techniques
15
Graph Data

Examples: Generic graph and HTML Links
2
1
5
2
<a href="papers/papers.html#bbbb">
Data Mining </a>
<li>
<a href="papers/papers.html#aaaa">
Graph Partitioning </a>
<li>
<a href="papers/papers.html#aaaa">
Parallel Solution of Sparse Linear System of Equations </a>
<li>
<a href="papers/papers.html#ffff">
N-Body Computation and Dense Linear System Solvers
5
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Data Mining: Concepts and Techniques
16
Chemical Data
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Benzene Molecule: C6H6
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Data Mining: Concepts and Techniques
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Ordered Data
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Sequences of transactions
Items/Events
An element of
the sequence
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Data Mining: Concepts and Techniques
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Ordered Data
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Genomic sequence data
GGTTCCGCCTTCAGCCCCGCGCC
CGCAGGGCCCGCCCCGCGCCGTC
GAGAAGGGCCCGCCTGGCGGGCG
GGGGGAGGCGGGGCCGCCCGAGC
CCAACCGAGTCCGACCAGGTGCC
CCCTCTGCTCGGCCTAGACCTGA
GCTCATTAGGCGGCAGCGGACAG
GCCAAGTAGAACACGCGAAGCGC
TGGGCTGCCTGCTGCGACCAGGG
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Data Mining: Concepts and Techniques
19
Ordered Data

Spatio-Temporal Data
Average Monthly
Temperature of
land and ocean
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Data Mining: Concepts and Techniques
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
General data characteristics

Basic data description and exploration

Measuring data similarity
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Data Mining: Concepts and Techniques
21
Mining Data Descriptive Characteristics
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Motivation
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Data dispersion characteristics
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To better understand the data: central tendency, variation
and spread
median, max, min, quantiles, outliers, variance, etc.
Numerical dimensions correspond to sorted intervals

Data dispersion: analyzed with multiple granularities of
precision
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Boxplot or quantile analysis on sorted intervals
Dispersion analysis on computed measures

Folding measures into numerical dimensions
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Boxplot or quantile analysis on the transformed cube
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Data Mining: Concepts and Techniques
22
Measuring the Central Tendency
1 n
 Mean (algebraic measure) (sample vs. population): x 
 xi
n i 1



Weighted arithmetic mean:
x
N
n
Trimmed mean: chopping extreme values
x
Median: A holistic measure


w x
i 1
n
i
i
w
i 1
i
Middle value if odd number of values, or average of the middle two
values otherwise
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Estimated by interpolation (for grouped data):
median  L1  (
Mode
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Value that occurs most frequently in the data
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Unimodal, bimodal, trimodal
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Empirical formula:
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N / 2  ( freq)l
freqmedian
) width
mean  mode  3  (mean  median)
Data Mining: Concepts and Techniques
23
Symmetric vs. Skewed Data

Median, mean and mode of
symmetric, positively and
negatively skewed data
positively skewed
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symmetric
negatively skewed
Data Mining: Concepts and Techniques
24
Measuring the Dispersion of Data
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Quartiles, outliers and boxplots
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Quartiles: Q1 (25th percentile), Q3 (75th percentile)
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Inter-quartile range: IQR = Q3 – Q1
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Five number summary: min, Q1, M, Q3, max
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Boxplot: ends of the box are the quartiles, median is marked, whiskers, and
plot outlier individually
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Outlier: usually, a value higher/lower than 1.5 x IQR
Variance and standard deviation (sample: s, population: σ)

Variance: (algebraic, scalable computation)
1 n
1 n 2 1 n
2
s 
( xi  x ) 
[ xi  ( xi ) 2 ]

n  1 i 1
n  1 i 1
n i 1
2

1
 
N
2
n
1
(
x


)


i
N
i 1
2
n
 xi   2
2
i 1
Standard deviation s (or σ) is the square root of variance s2 (or σ2)
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25
Boxplot Analysis

Five-number summary of a distribution:
Minimum, Q1, M, Q3, Maximum

Boxplot
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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 extend to
Minimum and Maximum
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Data Mining: Concepts and Techniques
26
Histogram Analysis

Graph displays of basic statistical class descriptions
 Frequency histograms
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A univariate graphical method
Consists of a set of rectangles that reflect the counts or
frequencies of the classes present in the given data
Data Mining: Concepts and Techniques
27
Histograms Often Tells More than Boxplots

The two histograms
shown in the left may
have the same boxplot
representation
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
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The same values
for: min, Q1,
median, Q3, max
But they have rather
different data
distributions
Data Mining: Concepts and Techniques
28
Quantile Plot
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Displays all of the data (allowing the user to assess both
the overall behavior and unusual occurrences)
Plots quantile information
 For a data xi data sorted in increasing order, fi
indicates that approximately 100 fi% of the data are
below or equal to the value xi
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Data Mining: Concepts and Techniques
29
Quantile-Quantile (Q-Q) Plot
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Graphs the quantiles of one univariate distribution against
the corresponding quantiles of another
Allows the user to view whether there is a shift in going
from one distribution to another
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Data Mining: Concepts and Techniques
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Scatter plot

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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
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Loess Curve


Adds a smooth curve to a scatter plot in order to
provide better perception of the pattern of dependence
Loess curve is fitted by setting two parameters: a
smoothing parameter, and the degree of the
polynomials that are fitted by the regression
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Data Mining: Concepts and Techniques
32
Positively and Negatively Correlated Data

The left half fragment is positively
correlated

March 21, 2017
The right half is negative correlated
Data Mining: Concepts and Techniques
33
Not Correlated Data
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Data Mining: Concepts and Techniques
34
Data Visualization and Its Methods

Why data visualization?

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Gain insight into an information space by mapping data onto
graphical primitives
Provide qualitative overview of large data sets
Search for patterns, trends, structure, irregularities, relationships
among data
Help find interesting regions and suitable parameters for further
quantitative analysis
Provide a visual proof of computer representations derived
Typical visualization methods:

Geometric techniques

Icon-based techniques

Hierarchical techniques
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35
Geometric Techniques
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Visualization of geometric transformations and projections
of the data
Methods

Landscapes
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Projection pursuit technique
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Finding meaningful projections of multidimensional
data
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Scatterplot matrices
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Prosection views
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Hyperslice
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Parallel coordinates
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Data Mining: Concepts and Techniques
36
Used by ermission of M. Ward, Worcester Polytechnic Institute
Scatterplot Matrices
Matrix of scatterplots (x-y-diagrams) of the k-dim. data
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Data Mining: Concepts and Techniques
37
Used by permission of B. Wright, Visible Decisions Inc.
Landscapes
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
news articles
visualized as
a landscape
Visualization of the data as perspective landscape
The data needs to be transformed into a (possibly artificial) 2D
spatial representation which preserves the characteristics of the data
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Data Mining: Concepts and Techniques
38
Parallel Coordinates



n equidistant axes which are parallel to one of the screen axes and
correspond to the attributes
The axes are scaled to the [minimum, maximum]: range of the
corresponding attribute
Every data item corresponds to a polygonal line which intersects each
of the axes at the point which corresponds to the value for the
attribute
• • •
Attr. 1
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Attr. 2
Attr. 3
Data Mining: Concepts and Techniques
Attr. k
39
Parallel Coordinates of a Data Set
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Data Mining: Concepts and Techniques
40
Icon-based Techniques

Visualization of the data values as features of icons

Methods:

Chernoff Faces
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Stick Figures

Shape Coding:

Color Icons:

TileBars: The use of small icons representing the
relevance feature vectors in document retrieval
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Data Mining: Concepts and Techniques
41
Chernoff Faces




A way to display variables on a two-dimensional surface, e.g., let x be
eyebrow slant, y be eye size, z be nose length, etc.
The figure shows faces produced using 10 characteristics--head
eccentricity, eye size, eye spacing, eye eccentricity, pupil size,
eyebrow slant, nose size, mouth shape, mouth size, and mouth
opening): Each assigned one of 10 possible values, generated using
Mathematica (S. Dickson)
REFERENCE: Gonick, L. and Smith, W. The
Cartoon Guide to Statistics. New York:
Harper Perennial, p. 212, 1993
Weisstein, Eric W. "Chernoff Face." From
MathWorld--A Wolfram Web Resource.
mathworld.wolfram.com/ChernoffFace.html
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Data Mining: Concepts and Techniques
42
Hierarchical Techniques


Visualization of the data using a hierarchical
partitioning into subspaces.
Methods

Dimensional Stacking

Worlds-within-Worlds

Treemap

Cone Trees

InfoCube
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Data Mining: Concepts and Techniques
43
Tree-Map


Screen-filling method which uses a hierarchical partitioning
of the screen into regions depending on the attribute values
The x- and y-dimension of the screen are partitioned
alternately according to the attribute values (classes)
MSR Netscan Image
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Data Mining: Concepts and Techniques
44
Tree-Map of a File System (Schneiderman)
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Data Mining: Concepts and Techniques
45

General data characteristics

Basic data description and exploration

Measuring data similarity (Sec. 7.2)
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Data Mining: Concepts and Techniques
46
Similarity and Dissimilarity

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
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 (i.e., distance)
 Numerical measure of how different are two data
objects
 Lower when objects are more alike
 Minimum dissimilarity is often 0
 Upper limit varies
Proximity refers to a similarity or dissimilarity
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Data Mining: Concepts and Techniques
47
Data Matrix and Dissimilarity Matrix


Data matrix
 n data points with p
dimensions
 Two modes
Dissimilarity matrix
 n data points, but
registers only the
distance
 A triangular matrix
 Single mode
March 21, 2017
 x11

 ...
x
 i1
 ...
x
 n1
...
x1f
...
...
...
...
xif
...
...
...
...
... xnf
...
...
 0
 d(2,1)
0

 d(3,1) d ( 3,2) 0

:
:
 :
d ( n,1) d ( n,2) ...
Data Mining: Concepts and Techniques
x1p 

... 
xip 

... 
xnp 







... 0
48
Example: Data Matrix and Distance Matrix
3
point
p1
p2
p3
p4
p1
2
p3
p4
1
p2
0
0
1
2
3
4
5
p1
p2
p3
p4
0
2.828
3.162
5.099
y
2
0
1
1
Data Matrix
6
p1
x
0
2
3
5
p2
2.828
0
1.414
3.162
p3
3.162
1.414
0
2
p4
5.099
3.162
2
0
Distance Matrix (i.e., Dissimilarity Matrix) for Euclidean Distance
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49
Minkowski Distance

Minkowski distance: A popular distance measure
d (i, j)  q (| x  x |q  | x  x |q ... | x  x |q )
i1 j1
i2
j2
ip
jp
where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two
p-dimensional data objects, and q is the order


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
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50
Special Cases of Minkowski Distance

q = 1: Manhattan (city block, L1 norm) distance

E.g., the Hamming distance: the number of bits that are
different between two binary vectors
d (i, j) | x  x |  | x  x | ... | x  x |
i1 j1
i2 j 2
ip
jp

q= 2: (L2 norm) Euclidean distance
d (i, j)  (| x  x |2  | x  x |2 ... | x  x |2 )
i1 j1
i2
j2
ip
jp

q  . “supremum” (Lmax norm, L norm) distance.
This is the maximum difference between any component of the
vectors
Do not confuse q with n, i.e., all these distances are defined for all
numbers of dimensions.
Also, one can use weighted distance, parametric Pearson product
moment correlation, or other dissimilarity measures



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Data Mining: Concepts and Techniques
51
Example: Minkowski Distance
point
p1
p2
p3
p4
x
0
2
3
5
y
2
0
1
1
L1
p1
p2
p3
p4
p1
0
4
4
6
p2
4
0
2
4
p3
4
2
0
2
p4
6
4
2
0
L2
p1
p2
p3
p4
p1
p2
2.828
0
1.414
3.162
p3
3.162
1.414
0
2
p4
5.099
3.162
2
0
L
p1
p2
p3
p4
p1
p2
p3
p4
0
2.828
3.162
5.099
0
2
3
5
2
0
1
3
3
1
0
2
5
3
2
0
Distance Matrix
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52
Binary Variables


1
0
a
b
A contingency table for binary data Object i 1
0
c
d
sum a  c b  d
Distance measure for symmetric
d (i, j) 
binary variables:

Distance measure for asymmetric
binary variables:

Jaccard coefficient (similarity
measure for asymmetric binary
variables):

Object j
d (i, j) 
sum
a b
cd
p
bc
a bc  d
bc
a bc
simJaccard (i, j) 
a
a b c
A binary variable is symmetric if
both of its states are equally
valuable and carry the same weight.
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Dissimilarity between Binary Variables

Example
Name
Jack
Mary
Jim



Gender
M
F
M
Fever
Y
Y
Y
Cough
N
N
P
Test-1
P
P
N
Test-2
N
N
N
Test-3
N
P
N
Test-4
N
N
N
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
01
 0.33
2 01
11
d ( jack , jim ) 
 0.67
111
1 2
d ( jim , mary ) 
 0.75
11 2
d ( jack , mary ) 
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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
m
d (i, j)  p 
p

Method 2: Use a large number of binary variables

creating a new binary variable for each of the M
nominal states
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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
zif

rif {1,...,M f }
rif 1

M f 1
compute the dissimilarity using methods for intervalscaled variables
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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?—the scale can be distorted)
apply logarithmic transformation
yif = log(xif)

March 21, 2017
treat them as continuous ordinal data treat their rank
as interval-scaled
Data Mining: Concepts and Techniques
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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
 pf  1 ij( f ) dij( f )
d (i, j) 
 pf  1 ij( f )



f is binary or nominal:
dij(f) = 0 if xif = xjf , or dij(f) = 1 otherwise
f is interval-based: use the normalized distance
f is ordinal or ratio-scaled
 Compute ranks rif and
r
1
zif 
M 1
 Treat zif as interval-scaled
if
f
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Vector Objects: Cosine Similarity




Vector objects: keywords in documents, gene features in micro-arrays, …
Applications: information retrieval, biologic taxonomy, ...
Cosine measure: If d1 and d2 are two vectors, then
cos(d1, d2) = (d1  d2) /||d1|| ||d2|| ,
where  indicates vector dot product, ||d||: the length of vector d
Example:
d1 = 3 2 0 5 0 0 0 2 0 0
d2 = 1 0 0 0 0 0 0 1 0 2
d1d2 = 3*1+2*0+0*0+5*0+0*0+0*0+0*0+2*1+0*0+0*2 = 5
||d1||= (3*3+2*2+0*0+5*5+0*0+0*0+0*0+2*2+0*0+0*0)0.5=(42)0.5
= 6.481
||d2|| = (1*1+0*0+0*0+0*0+0*0+0*0+0*0+1*1+0*0+2*2)0.5=(6) 0.5
= 2.245
cos( d1, d2 ) = .3150
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Correlation Analysis (Numerical Data)

Correlation coefficient (also called Pearson’s product
moment coefficient)
rp ,q
( p  p)( q  q)  ( pq)  n p q



(n  1) p q
(n  1) p q
where n is the number of tuples, p and q are the respective
means of p and q, σp and σq are the respective standard deviation
of p and q, and Σ(pq) is the sum of the pq cross-product.


If rp,q > 0, p and q are positively correlated (p’s values
increase as q’s). The higher, the stronger correlation.
rp,q = 0: independent; rpq < 0: negatively correlated
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Correlation (viewed as linear relationship)


Correlation measures the linear relationship
between objects
To compute correlation, we standardize data
objects, p and q, and then take their dot product
pk  ( pk  mean( p)) / std ( p)
qk  (qk  mean(q)) / std (q)
correlation( p, q)  p  q
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Visually Evaluating Correlation
Scatter plots
showing the
similarity from
–1 to 1.
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