Data Mining

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ITS-632Intro to Data Mining

Dr. Oner Celepcikay
Dept. of Information Technology &

School of Computer and Information Sciences
University of the Cumberlands

Chapter 2 Assignment

1. What’s noise? How can noise be reduced in a dataset?

2. Define outlier. Describe 2 different approaches to detect outliers in a dataset.

3. Give 2 examples in which aggregation is useful.

4. What’s stratified sampling? Why is it preferred?

5. Provide a brief description of what Principal Components Analysis (PCA) does. [Hint: See
Appendix A and your lecture notes.] State what’s the input and what the output of PCA is.

6. What’s the difference between dimensionality reduction and feature selection?

7. What’s the difference between feature selection and feature extraction?

8. Give two examples of data in which feature extraction would be useful.

9. What’s data discretization and when is it needed?

10. How are the Correlation and Covariance, used in data pre-processing (see pp. 76-78).

©Tan,St

inbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

Dr. Oner Celepcikay

ITS 63

Data Mining

Summer 2019Week 3: Data and Data Exploration

Chapter 2: Data

What is Data?

● 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 feature

● 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 60K No

7 Yes Divorced 220K No

8 No Single 85K Yes

9 No Married 75K No

10 No Single 90K Yes
10

Attributes

Objects

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
u Example: height can be measured in feet or meters

– Different attributes can be mapped to the same set of
values
u Example: Attribute values for ID and age are integers
u But properties of attribute values can be different

– ID has no limit but age has a maximum and minimum value
– Some operations are meaningful on age but meaningless on ID

Types of Attributes

● There are different types of attributes

– Nominal

u Examples: ID numbers, eye color, zip codes

– Ordinal

u Examples: rankings (e.g., taste of potato chips on a scale

from 1-10), grades, height in {tall, medium, short}

– Interval

u Examples: calendar dates, temperatures in Celsius or

Fahrenheit.

– Ratio

u Examples: temperature in Kelvin, length, time, counts

Attribute
Type

Description Examples Operations

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. (=, ¹)

zip codes, employee
ID numbers, eye color,
sex: {male, female}

mode, entropy,
contingency
correlation, c2 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

Ratio 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

Attribute
Level

Transformation 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

Thus, the Fahrenheit and
Celsius temperature scales
differ in terms of where
their zero value is and the
size of a unit (degree).

Ratio new_value = a * old_value Length can be measured in
meters or feet.

Properties of Attribute Values

● The type of an attribute depends on which of the

following properties it possesses:

– Distinctness: = ¹
– Order: < >

– Addition: + –

– Multiplication: * /

– Nominal attribute: distinctness

– Ordinal attribute: distinctness & order

– Interval attribute: distinctness, order & addition

– Ratio attribute: all 4 properties

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 floating-point

variables.

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

Important Characteristics of Structured Data

– Dimensionality
u

Curse of Dimensionality

– Sparsity
u Only presence counts

– Resolution
u Patterns depend on the scale

– Examples: Texas data, Aleks, Simpson’s Paradox

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 60K No
7 Yes Divorced 220K No
8 No Single 85K Yes
9 No Married 75K No
10 No Single 90K Yes
10

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

1.12.216.226.2512.6

1.22.715.225.2710.2

Thickness LoadDistanceProjection
of y load

Projection
of x Load

1.12.216.226.2512.65
1.22.715.225.2710.23
Thickness LoadDistanceProjection
of y load
Projection
of x Load

● 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.
– In practice only non-0 is stored

Document

season

tim
eout

lost

w
in

gam
e

score

ball

play

coach

team

Document 2

Document 3

3 0 5 0 2 6 0 2 0 2

7 0 2 1 0 0 3 0 0

1 0 0 1 2 2 0 3 0

● 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.

TID Items

1 Bread, Coke, Milk
2 Beer, Bread
3 Beer, Coke, Diaper, Milk
4 Beer, Bread, Diaper, Milk
5 Coke, Diaper, Milk

Graph Data

● Examples: Generic graph and HTML Links

● Data objects are nodes, links are properties

5
2

1
2

Data Mining

Graph Partitioning

Parallel Solution of Sparse Linear System of Equations

N-Body Computation and Dense Linear System Solvers

Chemical Data

● Benzene Molecule: C6H6

● Nodes are atoms, links are chemical bonds
● helps to identify substructures.

Ordered Data

● Sequences of transactions

An element of
the sequence

Items/Events

● Genomic sequence data

● Similar to sequential data but no time stamps

GGTTCCGCCTTCAGCCCCGCGCC
CGCAGGGCCCGCCCCGCGCCGTC
GAGAAGGGCCCGCCTGGCGGGCG
GGGGGAGGCGGGGCCGCCCGAGC
CCAACCGAGTCCGACCAGGTGCC
CCCTCTGCTCGGCCTAGACCTGA
GCTCATTAGGCGGCAGCGGACAG
GCCAAGTAGAACACGCGAAGCGC
TGGGCTGCCTGCTGCGACCAGGG

Ordered Data

● Spatio-Temporal Data

Average Monthly
Temperature of
land and ocean

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

● Precision: The closeness of repeated measurements (of
the same quantity) to other measurements.

● Bias: A systematic variation of measurements from the
quantity being measured.

● Accuracy: The closeness of measurements to the true
value of the quantity being measurement.

Noise

● 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 Two Sine Waves + Noise

Outliers

● Outliers are data objects with characteristics that
are considerably different than most of the other
data objects in the data set (diff. than noise)

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 (unless many missing)

– Estimate Missing Values (avg., most common val.)

– Ignore the Missing Value During Analysis

– Replace with all possible values (weighted by their

probabilities)

Duplicate Data

● Data set may include data objects that are
duplicates, or almost duplicates of one another
– Major issue when merging data from heterogeous

sources
– Also attention needed to avoid combining 2 very

similar objects into 1.

● Examples:
– Same person with multiple email addresses

● Data cleaning
– Process of dealing with duplicate data issues

Data Preprocessing

●

Aggregation

● Sampling

●

Dimensionality Reduction

● Feature subset selection

● Feature creation

● Discretization and Binarization

● Attribute Transformation

● Combining two or more attributes (or objects) into
a single attribute (or object)

● Purpose
– Data reduction

u Reduce the number of attributes or objects

– Change of scale
u Cities aggregated into regions, states, countries, etc

– More �stable� data
u Aggregated data tends to have less variability

Aggregation-Why?

● Less memory & less processing times
– Aggregation allows to use very expensive Algorithms

● High level view of the data
– Store example

● Behavior of groups of objects often more stable
than individual objects.
– A disadvantage of this is losing information or

patterns,
– e.g. if you aggregate days into months, you might

miss the sales peak in Valentine’s Day.

Standard Deviation of Average
Monthly Precipitation

Standard Deviation of Average
Yearly Precipitation

Variation of Precipitation in Australia

Sampling

● Sampling is the main technique employed for data selection.
– It is often used for both the preliminary investigation of the data
and the final data analysis.

● Statisticians sample because obtaining the entire set of data
of interest is too expensive or time consuming.

● Sampling is used in data mining because processing the
entire set of data of interest is too expensive or time
consuming.

Sampling …

● The key principle for effective sampling is the

following:

– using a sample will work almost as well as using the

entire data sets, if the sample is representative

– A sample is representative if it has approximately the

same property (of interest) as the original set of data

– If mean is of interest then the mean of the sample,

should be similar to mean of the full data.

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.
u In sampling with replacement, the same object can be picked up
more than once (easier to analyze, probability is constant)

● Stratified sampling
– Split the data into several partitions; then draw random samples

from each partition (handles representation of less freq. objects)

Sample Size

8000 points 2000 Points 500 Points

● What sample size is necessary to get at least one
object from each of 10 groups.

● When dimensionality
increases, data becomes
increasingly sparse in the
space that it occupies

● Definitions of density and
distance between points,
which is 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

● 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
– Principle Component Analysis
– Singular Value Decomposition
– Others: supervised and non-linear techniques

Dimensionality Reduction: PCA

● Goal is to find a projection that captures the
largest amount of variation in data

● Find the eigenvectors of the covariance matrix

● The eigenvectors define the new space

● Tends to identify strongest patterns in data.

x2
x1
e

Dimensions = 10Dimensions = 40Dimensions = 80Dimensions = 120Dimensions = 160Dimensions = 206

Dimensionality Reduction: PCA

Face detection and recognition

Detection Recognition “Sally”

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

● Techniques:
– Brute-force approch:

uTry all possible feature subsets as input to data mining algorithm

– Embedded approaches:
u Feature selection occurs naturally as part of the data mining
algorithm

– Filter approaches:
u Features are selected before data mining algorithm is run

– Wrapper approaches:
u Use the data mining algorithm as a black box to find best subset
of attributes

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

u domain-specific

– Mapping Data to New Space
– Feature Construction

u combining features (pixels à edges for face recognition)
u e.g. using density instead of mass, volume in identifying
artifacts such as gold, bronze, clay, etc…

Similarity and Dissimilarity

● Similarity
– Numerical measure of how alike two data objects are.
– Is higher when objects are more alike.
– Often falls in the range [0,1]

● Dissimilarity
– 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

Similarity/Dissimilarity for Simple Attributes

p and q are the attribute values for two data objects.

● An example: quality of a product (e.g. candy)
{poor, fair, OK, good, wonderful}

● P1->Wonderful, P->2 good, P3->OK
● P1 is closer to P2 than it is to P3
● Map ordinal attributes into integers:

{poor=0, fair=1, OK=2, good=3, wonderful=4}
● Estimate the distance values for each pair.
● Normalize if you want [1,1] interval

Euclidean

Distance

● Euclidean Distance

Where n is the number of dimensions (attributes) and pk and q

are, respectively, the kth attributes (components) or data
objects p and q.

● Standardization is necessary, if scales differ.

å
=

-=
n

k
kk qpdist

2)(

0 1 2 3 4 5 6

p3 p4

point x y
p1 0 2
p2 2 0
p3 3 1
p4 5 1

Distance Matrix

p1 p2 p3 p4
p1 0 2.828 3.162 5.099
p2 2.828 0 1.414 3.162
p3 3.162 1.414 0 2
p4 5.099 3.162 2 0

Minkowski Distance

● Minkowski Distance is a generalization of Euclidean

Distance

Where r is a parameter, n is the number of dimensions
(attributes) and pk and qk are, respectively, the kth attributes
(components) or data objects p and q.

r
n

r
kk qpdist

1
)||( å

=
-=

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.

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
Minkowski Distance
Distance Matrix
point x y
p1 0 2
p2 2 0
p3 3 1
p4 5 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 0 2.828 3.162 5.099
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 0 2 3 5
p2 2 0 1 3
p3 3 1 0 2
p4 5 3 2 0

Common Properties of a Distance

● Distances, such as the Euclidean distance,

have some well known properties.

1. d(p, q) ³ 0 for all p and q and d(p, q) = 0 only if
p = q. (Positive definiteness)

2. d(p, q) = d(q, p) for all p and q. (Symmetry)
3. d(p, r) £ d(p, q) + d(q, r) for all points p, q, and r.

(Triangle Inequality)

where d(p, q) is the distance (dissimilarity) between
points (data objects), p and q.

● A distance that satisfies these properties is a

metric

● Examples 2.14 and 2.15

Common Properties of a Similarity

● Similarities, also have some well known
properties.

1. s(p, q) = 1 (or maximum similarity) only if p = q.

2. s(p, q) = s(q, p) for all p and q. (Symmetry)

where s(p, q) is the similarity between points (data
objects), p and q.

SMC versus Jaccard: Example

p = 1 0 0 0 0 0 0 0 0 0
q = 0 0 0 0 0 0 1 0 0 1

M01 = 2 (the number of attributes where p was 0 and q was 1)
M10 = 1 (the number of attributes where p was 1 and q was 0)
M00 = 7 (the number of attributes where p was 0 and q was 0)
M11 = 0 (the number of attributes where p was 1 and q was 1)

SMC = (M11 + M00)/(M01 + M10 + M11 + M00) = (0+7) / (2+1+0+7) = 0.7

J = (M11) / (M01 + M10 + M11) = 0 / (2 + 1 + 0) = 0

Cosine Similarity

● If d1 and d2 are two document vectors, then
cos( d1, d2 ) = (d1 • d2) / ||d1|| ||d2|| ,

where • indicates vector dot product 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

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

Correlation

● Correlation measures the linear relationship
between objects

● To compute correlation, we standardize data
objects, p and q, and then take their dot product

)(/))(( pstdpmeanpp kk -=¢

)(/))(( qstdqmeanqq kk -=¢

qpqpncorrelatio ¢•¢=),(

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
Correlation
● Correlation measures the linear relationship
between objects
● To compute correlation, we standardize data
objects, p and q, and then take their dot product

Visually Evaluating Correlation

Scatter plots
showing the
similarity from
–1 to 1.

Density

● Density-based clustering require a notion of
density

● Examples:
– Euclidean density

u Euclidean density = number of points per unit volume

– Probability density

– Graph-based density

Euclidean Density – Cell-based

● Simplest approach is to divide region into a
number of rectangular cells of equal volume and
define density as # of points the cell contains

Euclidean Density – Center-based

● Euclidean density is the number of points within a
specified radius of the point

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