Data Mining

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ITS-632Intro to Data Mining

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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

e

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

Dr. Oner Celepcikay

ITS 63

2

Data Mining

Summer 2019Week 3: Data and Data Exploration

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

Chapter 2: Data

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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.

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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.

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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

5

1.22.715.225.2710.2

3

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

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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.
– In practice only non-0 is stored

Document

1

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

0

0

7 0 2 1 0 0 3 0 0

1 0 0 1 2 2 0 3 0

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
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.

TID Items

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

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

Graph Data

● Examples: Generic graph and HTML Links

● Data objects are nodes, links are properties

5
2

1
2

5


Data Mining


  • Graph Partitioning


  • Parallel Solution of Sparse Linear System of Equations


  • N-Body Computation and Dense Linear System Solvers

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Chemical Data

    ● Benzene Molecule: C6H6

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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Ordered Data

    ● Sequences of transactions

    An element of
    the sequence

    Items/Events

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Ordered Data

    ● Genomic sequence data

    ● Similar to sequential data but no time stamps

    GGTTCCGCCTTCAGCCCCGCGCC
    CGCAGGGCCCGCCCCGCGCCGTC
    GAGAAGGGCCCGCCTGGCGGGCG
    GGGGGAGGCGGGGCCGCCCGAGC
    CCAACCGAGTCCGACCAGGTGCC
    CCCTCTGCTCGGCCTAGACCTGA
    GCTCATTAGGCGGCAGCGGACAG
    GCCAAGTAGAACACGCGAAGCGC
    TGGGCTGCCTGCTGCGACCAGGG

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Ordered Data

    ● Spatio-Temporal Data

    Average Monthly
    Temperature of
    land and ocean

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Data Quality

    ● 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.

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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)

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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)

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Data Preprocessing

    Aggregation

    ● Sampling

    Dimensionality Reduction

    ● Feature subset selection

    ● Feature creation

    ● Discretization and Binarization

    ● Attribute Transformation

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Aggregation

    ● 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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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.

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Aggregation

    Standard Deviation of Average
    Monthly Precipitation

    Standard Deviation of Average
    Yearly Precipitation

    Variation of Precipitation in Australia

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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.

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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.

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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)

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Sample Size

    8000 points 2000 Points 500 Points

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Sample Size

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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Curse of Dimensionality

    ● 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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    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
    – Principle Component Analysis
    – Singular Value Decomposition
    – Others: supervised and non-linear techniques

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Dimensionality Reduction: PCA

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

    x2

    x1

    e

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Dimensionality Reduction: PCA

    ● Find the eigenvectors of the covariance matrix

    ● The eigenvectors define the new space

    ● Tends to identify strongest patterns in data.

    x2
    x1
    e

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

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

    Dimensionality Reduction: PCA

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Face detection and recognition

    Detection Recognition “Sally”

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Feature Subset Selection

    ● 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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Feature Subset Selection

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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…

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Similarity/Dissimilarity for Simple Attributes

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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Similarity/Dissimilarity for Simple Attributes

    ● 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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Euclidean

    Distance

    ● Euclidean Distance

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

    k

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

    ● Standardization is necessary, if scales differ.

    å
    =

    -=
    n

    k
    kk qpdist

    1

    2)(

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›
    Euclidean Distance
    0
    1
    2
    3

    0 1 2 3 4 5 6

    p1

    p2

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    k

    r
    kk qpdist

    1

    1
    )||( å

    =
    -=

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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.

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © 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

    )(/))(( 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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Visually Evaluating Correlation

    Scatter plots
    showing the
    similarity from
    –1 to 1.

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    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

    © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹#›

    Euclidean Density – Center-based

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

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