Data Mining: Data
Lecture Notes for Chapter 2
Introduction to Data Mining , 2nd Edition
by
Tan, Steinbach, Karpatne, Kumar
01/22/2018
Introduction to Data Mining, 2nd Edition
1
Outline
Attributes and Objects
Types of Data
Data Quality
Similarity and Distance
Data Preprocessing
01/22/2018
Introduction to Data Mining, 2nd Edition
2
What is Data?
Attributes
Collection of data objects
and their attributes
– Examples: eye color of a
person, temperature, etc.
– Attribute is also known as
variable, field, characteristic,
dimension, or feature
Objects
An attribute is a property
or characteristic of an
object
A collection of attributes
describe an object
– Object is also known as
record, point, case, sample,
entity, or instance
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
Single
90K
Yes
10 No
10
60K
A More Complete View of Data
Data may have parts
The different parts of the data may have
relationships
More generally, data may have structure
Data can be incomplete
We will discuss this in more detail later
01/22/2018
Introduction to Data Mining, 2nd Edition
4
Attribute Values
Attribute values are numbers or symbols
assigned to an attribute for a particular object
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
◆
01/22/2018
Introduction to Data Mining, 2nd Edition
5
Measurement of Length
The way you measure an attribute may not match the
attributes properties.
5
A
1
B
7
This scale
preserves
only the
ordering
property of
length.
2
C
8
3
D
10
4
E
15
5
This scale
preserves
the ordering
and additvity
properties of
length.
Types of Attributes
There are different types of attributes
– Nominal
◆
Examples: ID numbers, eye color, zip codes
– Ordinal
◆
Examples: rankings (e.g., taste of potato chips on a
scale from 1-10), grades, height {tall, medium, short}
– Interval
◆
Examples: calendar dates, temperatures in Celsius or
Fahrenheit.
– Ratio
◆
01/22/2018
Examples: temperature in Kelvin, length, time, counts
Introduction to Data Mining, 2nd Edition
7
Properties of Attribute Values
The type of an attribute depends on which of the
following properties/operations it possesses:
– Distinctness:
=
– Order:
< >
– Differences are
+ meaningful :
– Ratios are
meaningful
* /
– Nominal attribute: distinctness
– Ordinal attribute: distinctness & order
– Interval attribute: distinctness, order & meaningful
differences
– Ratio attribute: all 4 properties/operations
01/22/2018
Introduction to Data Mining, 2nd Edition
8
Difference Between Ratio and Interval
Is it physically meaningful to say that a
temperature of 10 ° is twice that of 5° on
– the Celsius scale?
– the Fahrenheit scale?
– the Kelvin scale?
Consider measuring the height above average
– If Bill’s height is three inches above average and
Bob’s height is six inches above average, then would
we say that Bob is twice as tall as Bill?
– Is this situation analogous to that of temperature?
01/22/2018
Introduction to Data Mining, 2nd Edition
9
Categorical
Qualitative
Attribute Description
Type
Nominal
Nominal attribute
values only
distinguish. (=, )
zip codes, employee
ID numbers, eye
color, sex: {male,
female}
Ordinal
Ordinal attribute
values also order
objects.
()
For interval
attributes,
differences between
values are
meaningful. (+, - )
For ratio variables,
both differences and
ratios are
meaningful. (*, /)
hardness of minerals,
{good, better, best},
grades, street
numbers
calendar dates,
temperature in
Celsius or Fahrenheit
Interval
Numeric
Quantitative
Examples
Ratio
Operations
mode, entropy,
contingency
correlation, 2
test
median,
percentiles, rank
correlation, run
tests, sign tests
mean, standard
deviation,
Pearson's
correlation, t and
F tests
temperature in Kelvin, geometric mean,
monetary quantities,
harmonic mean,
counts, age, mass,
percent variation
length, current
This categorization of attributes is due to S. S. Stevens
Numeric
Quantitative
Categorical
Qualitative
Attribute Transformation
Type
Comments
Nominal
Any permutation of values
If all employee ID numbers
were reassigned, would it
make any difference?
Ordinal
An order preserving change of
values, i.e.,
new_value = f(old_value)
where f is a monotonic function
An attribute encompassing
the notion of good, better best
can be represented equally
well by the values {1, 2, 3} or
by { 0.5, 1, 10}.
Interval
new_value = a * old_value + b
where a and b are constants
Ratio
new_value = a * old_value
Thus, the Fahrenheit and
Celsius temperature scales
differ in terms of where their
zero value is and the size of a
unit (degree).
Length can be measured in
meters or feet.
This categorization of attributes is due to S. S. Stevens
Discrete and Continuous Attributes
Discrete Attribute
– Has only a finite or countably infinite set of values
– Examples: zip codes, counts, or the set of words in a
collection of documents
– Often 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 floatingpoint variables.
01/22/2018
Introduction to Data Mining, 2nd Edition
12
Asymmetric Attributes
Only presence (a non-zero attribute value) is regarded as
important
◆
◆
Words present in documents
Items present in customer transactions
If we met a friend in the grocery store would we ever say the
following?
“I see our purchases are very similar since we didn’t buy most of the
same things.”
We need two asymmetric binary attributes to represent one
ordinary binary attribute
– Association analysis uses asymmetric attributes
Asymmetric attributes typically arise from objects that are
sets
01/22/2018
Introduction to Data Mining, 2nd Edition
13
Some Extensions and Critiques
Velleman, Paul F., and Leland Wilkinson. "Nominal,
ordinal, interval, and ratio typologies are misleading." The
American Statistician 47, no. 1 (1993): 65-72.
Mosteller, Frederick, and John W. Tukey. "Data analysis
and regression. A second course in statistics." AddisonWesley Series in Behavioral Science: Quantitative
Methods, Reading, Mass.: Addison-Wesley, 1977.
Chrisman, Nicholas R. "Rethinking levels of measurement
for cartography."Cartography and Geographic Information
Systems 25, no. 4 (1998): 231-242.
01/22/2018
Introduction to Data Mining, 2nd Edition
14
Critiques
Incomplete
– Asymmetric binary
– Cyclical
– Multivariate
– Partially ordered
– Partial membership
– Relationships between the data
Real data is approximate and noisy
– This can complicate recognition of the proper attribute type
– Treating one attribute type as another may be approximately
correct
01/22/2018
Introduction to Data Mining, 2nd Edition
15
Critiques …
Not a good guide for statistical analysis
– May unnecessarily restrict operations and results
◆
Statistical analysis is often approximate
◆
Thus, for example, using interval analysis for ordinal values
may be justified
– Transformations are common but don’t preserve
scales
◆
Can transform data to a new scale with better statistical
properties
◆
Many statistical analyses depend only on the distribution
01/22/2018
Introduction to Data Mining, 2nd Edition
16
More Complicated Examples
ID numbers
– Nominal, ordinal, or interval?
Number of cylinders in an automobile engine
– Nominal, ordinal, or ratio?
Biased Scale
– Interval or Ratio
01/22/2018
Introduction to Data Mining, 2nd Edition
17
Key Messages for Attribute Types
The types of operations you choose should be
“meaningful” for the type of data you have
– Distinctness, order, meaningful intervals, and meaningful ratios
are only four properties of data
– The data type you see – often numbers or strings – may not
capture all the properties or may suggest properties that are not
there
– Analysis may depend on these other properties of the data
◆
Many statistical analyses depend only on the distribution
– Many times what is meaningful is measured by statistical
significance
– But in the end, what is meaningful is measured by the domain
01/22/2018
Introduction to Data Mining, 2nd Edition
18
Types of data sets
Record
– Data Matrix
– Document Data
– Transaction Data
Graph
– World Wide Web
– Molecular Structures
Ordered
–
–
–
–
Spatial Data
Temporal Data
Sequential Data
Genetic Sequence Data
01/22/2018
Introduction to Data Mining, 2nd Edition
19
Important Characteristics of Data
– Dimensionality (number of attributes)
◆
High dimensional data brings a number of challenges
– Sparsity
◆
Only presence counts
– Resolution
◆
Patterns depend on the scale
– Size
◆
Type of analysis may depend on size of data
01/22/2018
Introduction to Data Mining, 2nd Edition
20
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
01/22/2018
Introduction to Data Mining, 2nd Edition
21
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
01/22/2018
Introduction to Data Mining, 2nd Edition
22
Document Data
Each document becomes a ‘term’ vector
– 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
play
ball
score
game
win
lost
timeout
season
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
01/22/2018
Introduction to Data Mining, 2nd Edition
23
Transaction Data
A special type of record data, where
– 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.
01/22/2018
TID
Items
1
Bread, Coke, Milk
2
3
4
5
Beer, Bread
Beer, Coke, Diaper, Milk
Beer, Bread, Diaper, Milk
Coke, Diaper, Milk
Introduction to Data Mining, 2nd Edition
24
Graph Data
Examples: Generic graph, a molecule, and webpages
2
1
5
2
5
Benzene Molecule: C6H6
01/22/2018
Introduction to Data Mining, 2nd Edition
25
Ordered Data
Sequences of transactions
Items/Events
An element of
the sequence
01/22/2018
Introduction to Data Mining, 2nd Edition
26
Ordered Data
Genomic sequence data
GGTTCCGCCTTCAGCCCCGCGCC
CGCAGGGCCCGCCCCGCGCCGTC
GAGAAGGGCCCGCCTGGCGGGCG
GGGGGAGGCGGGGCCGCCCGAGC
CCAACCGAGTCCGACCAGGTGCC
CCCTCTGCTCGGCCTAGACCTGA
GCTCATTAGGCGGCAGCGGACAG
GCCAAGTAGAACACGCGAAGCGC
TGGGCTGCCTGCTGCGACCAGGG
01/22/2018
Introduction to Data Mining, 2nd Edition
27
Ordered Data
Spatio-Temporal Data
Average Monthly
Temperature of
land and ocean
01/22/2018
Introduction to Data Mining, 2nd Edition
28
Data Quality
Poor data quality negatively affects many data processing
efforts
“The most important point is that poor data quality is an unfolding
disaster.
– Poor data quality costs the typical company at least ten
percent (10%) of revenue; twenty percent (20%) is
probably a better estimate.”
Thomas C. Redman, DM Review, August 2004
Data mining example: a classification model for detecting
people who are loan risks is built using poor data
– Some credit-worthy candidates are denied loans
– More loans are given to individuals that default
01/22/2018
Introduction to Data Mining, 2nd Edition
29
Data Quality …
What kinds of data quality problems?
How can we detect problems with the data?
What can we do about these problems?
Examples of data quality problems:
–
–
–
–
Noise and outliers
Missing values
Duplicate data
Wrong data
01/22/2018
Introduction to Data Mining, 2nd Edition
30
Noise
For objects, noise is an extraneous object
For attributes, noise refers to modification of original values
– Examples: distortion of a person’s voice when talking on a poor
phone and “snow” on television screen
Two Sine Waves
01/22/2018
Two Sine Waves + Noise
Introduction to Data Mining, 2nd Edition
31
Outliers
Outliers are data objects with characteristics that
are considerably different than most of the other
data objects in the data set
– Case 1: Outliers are
noise that interferes
with data analysis
– Case 2: Outliers are
the goal of our analysis
◆
Credit card fraud
◆
Intrusion detection
Causes?
01/22/2018
Introduction to Data Mining, 2nd Edition
32
Missing Values
Reasons for missing values
– Information is not collected
(e.g., people decline to give their age and weight)
– Attributes may not be applicable to all cases
(e.g., annual income is not applicable to children)
Handling missing values
– Eliminate data objects or variables
– Estimate missing values
Example: time series of temperature
◆ Example: census results
◆
– Ignore the missing value during analysis
01/22/2018
Introduction to Data Mining, 2nd Edition
33
Missing Values …
Missing completely at random (MCAR)
– Missingness of a value is independent of attributes
– Fill in values based on the attribute
– Analysis may be unbiased overall
Missing at Random (MAR)
– Missingness is related to other variables
– Fill in values based other values
– Almost always produces a bias in the analysis
Missing Not at Random (MNAR)
– Missingness is related to unobserved measurements
– Informative or non-ignorable missingness
Not possible to know the situation from the data
01/22/2018
Introduction to Data Mining, 2nd Edition
34
Duplicate Data
Data set may include data objects that are
duplicates, or almost duplicates of one another
– Major issue when merging data from heterogeneous
sources
Examples:
– Same person with multiple email addresses
Data cleaning
– Process of dealing with duplicate data issues
When should duplicate data not be removed?
01/22/2018
Introduction to Data Mining, 2nd Edition
35
Similarity and Dissimilarity Measures
Similarity measure
– Numerical measure of how alike two data objects are.
– Is higher when objects are more alike.
– Often falls in the range [0,1]
Dissimilarity measure
– 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
01/22/2018
Introduction to Data Mining, 2nd Edition
36
Similarity/Dissimilarity for Simple Attributes
The following table shows the similarity and dissimilarity
between two objects, x and y, with respect to a single, simple
attribute.
01/22/2018
Introduction to Data Mining, 2nd Edition
37
Euclidean Distance
Euclidean Distance
where n is the number of dimensions (attributes) and
xk and yk are, respectively, the kth attributes
(components) or data objects x and y.
Standardization is necessary, if scales differ.
01/22/2018
Introduction to Data Mining, 2nd Edition
38
Euclidean Distance
3
point
p1
p2
p3
p4
p1
2
p3
p4
1
p2
0
0
1
2
3
4
5
p1
p1
p2
p3
p4
0
2.828
3.162
5.099
x
0
2
3
5
y
2
0
1
1
6
p2
2.828
0
1.414
3.162
p3
3.162
1.414
0
2
p4
5.099
3.162
2
0
Distance Matrix
01/22/2018
Introduction to Data Mining, 2nd Edition
39
Minkowski Distance
Minkowski Distance is a generalization of Euclidean
Distance
Where r is a parameter, n is the number of dimensions
(attributes) and xk and yk are, respectively, the kth
attributes (components) or data objects x and y.
01/22/2018
Introduction to Data Mining, 2nd Edition
40
Minkowski Distance: Examples
r = 1. City block (Manhattan, taxicab, L1 norm) distance.
– A common example of this is the Hamming distance, which
is just the number of bits that are different between two
binary vectors
r = 2. Euclidean distance
r → . “supremum” (Lmax norm, L norm) distance.
– This is the maximum difference between any component of
the vectors
Do not confuse r with n, i.e., all these distances are
defined for all numbers of dimensions.
01/22/2018
Introduction to Data Mining, 2nd Edition
41
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
01/22/2018
Introduction to Data Mining, 2nd Edition
42
Mahalanobis Distance
𝐦𝐚𝐡𝐚𝐥𝐚𝐧𝐨𝐛𝐢𝐬 𝐱, 𝐲 = (𝐱 − 𝐲)𝑇 Ʃ−1 (𝐱 − 𝐲)
is the covariance matrix
For red points, the Euclidean distance is 14.7, Mahalanobis distance is 6.
01/22/2018
Introduction to Data Mining, 2nd Edition
43
Mahalanobis Distance
Covariance
Matrix:
C
0.3 0.2
=
0
.
2
0
.
3
A: (0.5, 0.5)
B
B: (0, 1)
A
C: (1.5, 1.5)
Mahal(A,B) = 5
Mahal(A,C) = 4
01/22/2018
Introduction to Data Mining, 2nd Edition
44
Common Properties of a Distance
Distances, such as the Euclidean distance,
have some well known properties.
1. d(x, y) 0 for all x and y and d(x, y) = 0 only if
x = y. (Positive definiteness)
2. d(x, y) = d(y, x) for all x and y. (Symmetry)
3. d(x, z) d(x, y) + d(y, z) for all points x, y, and z.
(Triangle Inequality)
where d(x, y) is the distance (dissimilarity) between
points (data objects), x and y.
A distance that satisfies these properties is a
metric
01/22/2018
Introduction to Data Mining, 2nd Edition
45
Common Properties of a Similarity
Similarities, also have some well known
properties.
1.
s(x, y) = 1 (or maximum similarity) only if x = y.
2.
s(x, y) = s(y, x) for all x and y. (Symmetry)
where s(x, y) is the similarity between points (data
objects), x and y.
01/22/2018
Introduction to Data Mining, 2nd Edition
46
Similarity Between Binary Vectors
Common situation is that objects, p and q, have only
binary attributes
Compute similarities using the following quantities
f01 = the number of attributes where p was 0 and q was 1
f10 = the number of attributes where p was 1 and q was 0
f00 = the number of attributes where p was 0 and q was 0
f11 = the number of attributes where p was 1 and q was 1
Simple Matching and Jaccard Coefficients
SMC = number of matches / number of attributes
= (f11 + f00) / (f01 + f10 + f11 + f00)
J = number of 11 matches / number of non-zero attributes
= (f11) / (f01 + f10 + f11)
01/22/2018
Introduction to Data Mining, 2nd Edition
47
SMC versus Jaccard: Example
x= 1000000000
y= 0000001001
f01 = 2 (the number of attributes where p was 0 and q was 1)
f10 = 1 (the number of attributes where p was 1 and q was 0)
f00 = 7 (the number of attributes where p was 0 and q was 0)
f11 = 0 (the number of attributes where p was 1 and q was 1)
SMC
= (f11 + f00) / (f01 + f10 + f11 + f00)
= (0+7) / (2+1+0+7) = 0.7
J = (f11) / (f01 + f10 + f11) = 0 / (2 + 1 + 0) = 0
01/22/2018
Introduction to Data Mining, 2nd Edition
48
Cosine Similarity
If d1 and d2 are two document vectors, then
cos( d1, d2 ) = / ||d1|| ||d2|| ,
where indicates inner product or vector dot
product of vectors, d1 and d2, and || d || is 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
= 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 = 2.449
cos(d1, d2 ) = 0.3150
01/22/2018
Introduction to Data Mining, 2nd Edition
49
Extended Jaccard Coefficient (Tanimoto)
Variation of Jaccard for continuous or count
attributes
– Reduces to Jaccard for binary attributes
01/22/2018
Introduction to Data Mining, 2nd Edition
50
Correlation measures the linear relationship
between objects
01/22/2018
Introduction to Data Mining, 2nd Edition
51
Visually Evaluating Correlation
Scatter plots
showing the
similarity from
–1 to 1.
01/22/2018
Introduction to Data Mining, 2nd Edition
52
Drawback of Correlation
x = (-3, -2, -1, 0, 1, 2, 3)
y = (9, 4, 1, 0, 1, 4, 9)
yi = xi2
mean(x) = 0, mean(y) = 4
std(x) = 2.16, std(y) = 3.74
corr = (-3)(5)+(-2)(0)+(-1)(-3)+(0)(-4)+(1)(-3)+(2)(0)+3(5) / ( 6 * 2.16 * 3.74 )
=0
01/22/2018
Introduction to Data Mining, 2nd Edition
53
Comparison of Proximity Measures
Domain of application
– Similarity measures tend to be specific to the type of
attribute and data
– Record data, images, graphs, sequences, 3D-protein
structure, etc. tend to have different measures
However, one can talk about various properties that
you would like a proximity measure to have
–
–
–
–
Symmetry is a common one
Tolerance to noise and outliers is another
Ability to find more types of patterns?
Many others possible
The measure must be applicable to the data and
produce results that agree with domain knowledge
01/22/2018
Introduction to Data Mining, 2nd Edition
54
Information Based Measures
Information theory is a well-developed and
fundamental disciple with broad applications
Some similarity measures are based on
information theory
– Mutual information in various versions
– Maximal Information Coefficient (MIC) and related
measures
– General and can handle non-linear relationships
– Can be complicated and time intensive to compute
01/22/2018
Introduction to Data Mining, 2nd Edition
55
Information and Probability
Information relates to possible outcomes of an event
– transmission of a message, flip of a coin, or measurement
of a piece of data
The more certain an outcome, the less information
that it contains and vice-versa
– For example, if a coin has two heads, then an outcome of
heads provides no information
– More quantitatively, the information is related the
probability of an outcome
◆
The smaller the probability of an outcome, the more information it
provides and vice-versa
– Entropy is the commonly used measure
01/22/2018
Introduction to Data Mining, 2nd Edition
56
Entropy
For
–
–
–
–
a variable (event), X,
with n possible values (outcomes), x1, x2 …, xn
each outcome having probability, p1, p2 …, pn
the entropy of X , H(X), is given by
𝑛
𝐻 𝑋 = − 𝑝𝑖 log 2 𝑝𝑖
𝑖=1
Entropy is between 0 and log2n and is measured in
bits
– Thus, entropy is a measure of how many bits it takes to
represent an observation of X on average
01/22/2018
Introduction to Data Mining, 2nd Edition
57
Entropy Examples
For a coin with probability p of heads and
probability q = 1 – p of tails
𝐻 = −𝑝 log 2 𝑝 − 𝑞 log 2 𝑞
– For p= 0.5, q = 0.5 (fair coin) H = 1
– For p = 1 or q = 1, H = 0
What is the entropy of a fair four-sided die?
01/22/2018
Introduction to Data Mining, 2nd Edition
58
Entropy for Sample Data: Example
Hair Color
Count
p
-plog2p
Black
75
0.75
0.3113
Brown
15
0.15
0.4105
Blond
5
0.05
0.2161
Red
0
0.00
0
Other
5
0.05
0.2161
Total
100
1.0
1.1540
Maximum entropy is log25 = 2.3219
01/22/2018
Introduction to Data Mining, 2nd Edition
59
Entropy for Sample Data
Suppose we have
– a number of observations (m) of some attribute, X,
e.g., the hair color of students in the class,
– where there are n different possible values
– And the number of observation in the ith category is mi
– Then, for this sample
𝑛
𝑚𝑖
𝑚𝑖
𝐻 𝑋 = − log 2
𝑚
𝑚
𝑖=1
For continuous data, the calculation is harder
01/22/2018
Introduction to Data Mining, 2nd Edition
60
Mutual Information
Information one variable provides about another
Formally, 𝐼 𝑋, 𝑌 = 𝐻 𝑋 + 𝐻 𝑌 − 𝐻(𝑋, 𝑌), where
H(X,Y) is the joint entropy of X and Y,
𝐻 𝑋, 𝑌 = − 𝑝𝑖𝑗log 2 𝑝𝑖𝑗
𝑖
𝑗
Where pij is the probability that the ith value of X and the jth value of Y
occur together
For discrete variables, this is easy to compute
Maximum mutual information for discrete variables is
log2(min( nX, nY ), where nX (nY) is the number of values of X (Y)
01/22/2018
Introduction to Data Mining, 2nd Edition
61
Mutual Information Example
Student Count
Status
p
-plog2p
Undergrad
45
0.45
0.5184
Grad
55
0.55
0.4744
Student Grade
Status
Count
p
-plog2p
Undergrad
A
5
0.05
0.2161
Undergrad
B
30
0.30
0.5211
Undergrad
C
10
0.10
0.3322
Total
100
1.00
0.9928
Grade
Count
p
-plog2p
Grad
A
30
0.30
0.5211
A
35
0.35
0.5301
Grad
B
20
0.20
0.4644
B
50
0.50
0.5000
Grad
C
5
0.05
0.2161
C
15
0.15
0.4105
Total
100
1.00
2.2710
Total
100
1.00
1.4406
Mutual information of Student Status and Grade = 0.9928 + 1.4406 - 2.2710 = 0.1624
01/22/2018
Introduction to Data Mining, 2nd Edition
62
Maximal Information Coefficient
Reshef, David N., Yakir A. Reshef, Hilary K. Finucane, Sharon R. Grossman, Gilean McVean, Peter
J. Turnbaugh, Eric S. Lander, Michael Mitzenmacher, and Pardis C. Sabeti. "Detecting novel
associations in large data sets." science 334, no. 6062 (2011): 1518-1524.
Applies mutual information to two continuous
variables
Consider the possible binnings of the variables into
discrete categories
– nX × nY ≤ N0.6 where
◆
◆
◆
nX is the number of values of X
nY is the number of values of Y
N is the number of samples (observations, data objects)
Compute the mutual information
– Normalized by log2(min( nX, nY )
Take the highest value
01/22/2018
Introduction to Data Mining, 2nd Edition
63
General Approach for Combining Similarities
Sometimes attributes are of many different types, but an
overall similarity is needed.
1: For the kth attribute, compute a similarity, sk(x, y), in the
range [0, 1].
2: Define an indicator variable, k, for the kth attribute as
follows:
k = 0 if the kth attribute is an asymmetric attribute and
both objects have a value of 0, or if one of the objects
has a missing value for the kth attribute
k = 1 otherwise
3. Compute
01/22/2018
Introduction to Data Mining, 2nd Edition
64
Using Weights to Combine Similarities
May not want to treat all attributes the same.
– Use non-negative weights 𝜔𝑘
– 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝐱, 𝐲 =
σ𝑛
𝑘=1 𝜔𝑘 𝛿𝑘 𝑠𝑘 (𝐱,𝐲)
σ𝑛
𝑘=1 𝜔𝑘 𝛿𝑘
Can also define a weighted form of distance
01/22/2018
Introduction to Data Mining, 2nd Edition
65
Density
Measures the degree to which data objects are close to
each other in a specified area
The notion of density is closely related to that of proximity
Concept of density is typically used for clustering and
anomaly detection
Examples:
– Euclidean density
◆
Euclidean density = number of points per unit volume
– Probability density
◆
Estimate what the distribution of the data looks like
– Graph-based density
◆ Connectivity
01/22/2018
Introduction to Data Mining, 2nd Edition
66
Euclidean Density: Grid-based Approach
Simplest approach is to divide region into a
number of rectangular cells of equal volume and
define density as # of points the cell contains
Grid-based density.
01/22/2018
Counts for each cell.
Introduction to Data Mining, 2nd Edition
67
Euclidean Density: Center-Based
Euclidean density is the number of points within a
specified radius of the point
Illustration of center-based density.
01/22/2018
Introduction to Data Mining, 2nd Edition
68
Data Preprocessing
Aggregation
Sampling
Dimensionality Reduction
Feature subset selection
Feature creation
Discretization and Binarization
Attribute Transformation
01/22/2018
Introduction to Data Mining, 2nd Edition
69
Aggregation
Combining two or more attributes (or objects) into
a single attribute (or object)
Purpose
– Data reduction
◆
Reduce the number of attributes or objects
– Change of scale
◆
Cities aggregated into regions, states, countries, etc.
◆
Days aggregated into weeks, months, or years
– More “stable” data
◆
01/22/2018
Aggregated data tends to have less variability
Introduction to Data Mining, 2nd Edition
70
Example: Precipitation in Australia
This example is based on precipitation in
Australia from the period 1982 to 1993.
The next slide shows
– A histogram for the standard deviation of average
monthly precipitation for 3,030 0.5◦ by 0.5◦ grid cells in
Australia, and
– A histogram for the standard deviation of the average
yearly precipitation for the same locations.
The average yearly precipitation has less
variability than the average monthly precipitation.
All precipitation measurements (and their
standard deviations) are in centimeters.
01/22/2018
Introduction to Data Mining, 2nd Edition
71
Example: Precipitation in Australia …
Variation of Precipitation in Australia
Standard Deviation of Average
Monthly Precipitation
01/22/2018
Standard Deviation of
Average Yearly Precipitation
Introduction to Data Mining, 2nd Edition
72
Sampling
Sampling is the main technique employed for data
reduction.
– It is often used for both the preliminary investigation of
the data and the final data analysis.
Statisticians often sample because obtaining the
entire set of data of interest is too expensive or
time consuming.
Sampling is typically used in data mining because
processing the entire set of data of interest is too
expensive or time consuming.
01/22/2018
Introduction to Data Mining, 2nd Edition
73
Sampling …
The key principle for effective sampling is the
following:
– Using a sample will work almost as well as using the
entire data set, if the sample is representative
– A sample is representative if it has approximately the
same properties (of interest) as the original set of data
01/22/2018
Introduction to Data Mining, 2nd Edition
74
Sample Size
8000 points
01/22/2018
2000 Points
Introduction to Data Mining, 2nd Edition
500 Points
75
Types of Sampling
Simple Random Sampling
– There is an equal probability of selecting any particular
item
– Sampling without replacement
◆ As each item is selected, it is removed from the
population
– Sampling with replacement
◆ Objects are not removed from the population as they
are selected for the sample.
◆ In sampling with replacement, the same object can
be picked up more than once
Stratified sampling
– Split the data into several partitions; then draw random
samples from each partition
01/22/2018
Introduction to Data Mining, 2nd Edition
76
Sample Size
What sample size is necessary to get at least one
object from each of 10 equal-sized groups.
01/22/2018
Introduction to Data Mining, 2nd Edition
77
Curse of Dimensionality
When dimensionality
increases, data becomes
increasingly sparse in the
space that it occupies
Definitions of density and
distance between points,
which are critical for
clustering and outlier
detection, become less
meaningful
•Randomly generate 500 points
•Compute difference between max and
min distance between any pair of points
01/22/2018
Introduction to Data Mining, 2nd Edition
78
Dimensionality Reduction
Purpose:
– Avoid curse of dimensionality
– Reduce amount of time and memory required by data
mining algorithms
– Allow data to be more easily visualized
– May help to eliminate irrelevant features or reduce
noise
Techniques
– Principal Components Analysis (PCA)
– Singular Value Decomposition
– Others: supervised and non-linear techniques
01/22/2018
Introduction to Data Mining, 2nd Edition
79
Dimensionality Reduction: PCA
Goal is to find a projection that captures the
largest amount of variation in data
x2
e
x1
01/22/2018
Introduction to Data Mining, 2nd Edition
80
Dimensionality Reduction: PCA
01/22/2018
Introduction to Data Mining, 2nd Edition
81
Feature Subset Selection
Another way to reduce dimensionality of data
Redundant features
– Duplicate much or all of the information contained in
one or more other attributes
– Example: purchase price of a product and the amount
of sales tax paid
Irrelevant features
– Contain no information that is useful for the data
mining task at hand
– Example: students' ID is often irrelevant to the task of
predicting students' GPA
Many techniques developed, especially for
classification
01/22/2018
Introduction to Data Mining, 2nd Edition
82
Feature Creation
Create new attributes that can capture the
important information in a data set much more
efficiently than the original attributes
Three general methodologies:
– Feature extraction
◆
Example: extracting edges from images
– Feature construction
◆
Example: dividing mass by volume to get density
– Mapping data to new space
◆
01/22/2018
Example: Fourier and wavelet analysis
Introduction to Data Mining, 2nd Edition
83
Mapping Data to a New Space
Fourier and wavelet transform
Frequency
Two Sine Waves + Noise
01/22/2018
Frequency
Introduction to Data Mining, 2nd Edition
84
Discretization
Discretization is the process of converting a
continuous attribute into an ordinal attribute
– A potentially infinite number of values are mapped
into a small number of categories
– Discretization is commonly used in classification
– Many classification algorithms work best if both
the independent and dependent variables have
only a few values
– We give an illustration of the usefulness of
discretization using the Iris data set
01/22/2018
Introduction to Data Mining, 2nd Edition
85
Iris Sample Data Set
Iris Plant data set.
– Can be obtained from the UCI Machine Learning Repository
http://www.ics.uci.edu/~mlearn/MLRepository.html
– From the statistician Douglas Fisher
– Three flower types (classes):
◆ Setosa
◆ Versicolour
◆ Virginica
– Four (non-class) attributes
◆ Sepal width and length
Virginica. Robert H. Mohlenbrock. USDA
◆ Petal width and length
01/22/2018
NRCS. 1995. Northeast wetland flora: Field
office guide to plant species. Northeast National
Technical Center, Chester, PA. Courtesy of
USDA NRCS Wetland Science Institute.
Introduction to Data Mining, 2nd Edition
86
Discretization: Iris Example
Petal width low or petal length low implies Setosa.
Petal width medium or petal length medium implies Versicolour.
Petal width high or petal length high implies Virginica.
Discretization: Iris Example …
How can we tell what the best discretization is?
– Unsupervised discretization: find breaks in the data
values
50
◆ Example:
40
Counts
Petal Length
30
20
10
0
0
2
4
6
Petal Length
8
– Supervised discretization: Use class labels to find
breaks
01/22/2018
Introduction to Data Mining, 2nd Edition
88
Discretization Without Using Class Labels
Data consists of four groups of points and two outliers. Data is onedimensional, but a random y component is added to reduce overlap.
01/22/2018
Introduction to Data Mining, 2nd Edition
89
Discretization Without Using Class Labels
Equal interval width approach used to obtain 4 values.
01/22/2018
Introduction to Data Mining, 2nd Edition
90
Discretization Without Using Class Labels
Equal frequency approach used to obtain 4 values.
01/22/2018
Introduction to Data Mining, 2nd Edition
91
Discretization Without Using Class Labels
K-means approach to obtain 4 values.
01/22/2018
Introduction to Data Mining, 2nd Edition
92
Binarization
Binarization maps a continuous or categorical
attribute into one or more binary variables
Typically used for association analysis
Often convert a continuous attribute to a
categorical attribute and then convert a
categorical attribute to a set of binary attributes
– Association analysis needs asymmetric binary
attributes
– Examples: eye color and height measured as
{low, medium, high}
01/22/2018
Introduction to Data Mining, 2nd Edition
93
Attribute Transformation
An attribute transform is a function that maps the
entire set of values of a given attribute to a new
set of replacement values such that each old
value can be identified with one of the new values
– Simple functions: xk, log(x), ex, |x|
– Normalization
Refers to various techniques to adjust to
differences among attributes in terms of frequency
of occurrence, mean, variance, range
◆ Take out unwanted, common signal, e.g.,
seasonality
– In statistics, standardization refers to subtracting off
the means and dividing by the standard deviation
◆
01/22/2018
Introduction to Data Mining, 2nd Edition
94
Example: Sample Time Series of Plant Growth
Minneapolis
Net Primary
Production (NPP)
is a measure of
plant growth used
by ecosystem
scientists.
Correlations between time series
Correlations between time series
Minneapolis
Minneapolis
1.0000
Atlanta
0.7591
Sao Paolo
-0.7581
01/22/2018
Atlanta
0.7591
1.0000
-0.5739
Sao Paolo
-0.7581
-0.5739
1.0000
Introduction to Data Mining, 2nd Edition
95
Seasonality Accounts for Much Correlation
Minneapolis
Normalized using
monthly Z Score:
Subtract off monthly
mean and divide by
monthly standard
deviation
Correlations between time series
Correlations between time series
Minneapolis
Minneapolis
1.0000
Atlanta
0.0492
Sao Paolo
0.0906
01/22/2018
Atlanta
0.0492
1.0000
-0.0154
Sao Paolo
0.0906
-0.0154
1.0000
Introduction to Data Mining, 2nd Edition
96
Purchase answer to see full
attachment