Data Mining: Data

Lecture Notes for Chapter 2

Introduction to Data Mining

by

Tan, Steinbach, Kumar

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

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

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

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 in {tall, medium, short}

Interval

Examples: calendar dates, temperatures in Celsius or Fahrenheit.

Ratio

Examples: temperature in Kelvin, length, time, counts

Properties of Attribute Values

The type of an attribute depends on which of the following properties it possesses:

Distinctness: =

Order: < >

Addition: + –

Multiplication: * /

Nominal attribute: distinctness

Ordinal attribute: distinctness & order

Interval attribute: distinctness, order & addition

Ratio attribute: all 4 properties

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. (=, )

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

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.

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

Curse of Dimensionality

Sparsity

Only presence counts

Resolution

Patterns depend on the scale

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

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.

Document 1�

season�

timeout�

lost�

win�

game�

score�

ball�

play�

coach�

team�

Document 2�

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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.

Graph Data

Examples: Generic graph and HTML Links

Ordered Data

Sequences of transactions

An element of the sequence

Items/Events

Ordered Data

Genomic sequence data

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

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

Estimate Missing Values

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

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

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

More “stable” data

Aggregated data tends to have less variability

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

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

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

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

Feature Subset Selection

Techniques:

Brute-force approch:

Try all possible feature subsets as input to data mining algorithm

Embedded approaches:

Feature selection occurs naturally as part of the data mining algorithm

Filter approaches:

Features are selected before data mining algorithm is run

Wrapper approaches:

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

domain-specific

Mapping Data to New Space

Feature Construction

combining features

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.

Euclidean Distance

Euclidean Distance

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

Standardization is necessary, if scales differ.

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.

Common Properties of a Distance

Distances, such as the Euclidean distance, have some well known properties.

d(p, q) 0 for all p and q and d(p, q) = 0 only if

p = q. (Positive definiteness)

d(p, q) = d(q, p) for all p and q. (Symmetry)

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

Common Properties of a Similarity

Similarities, also have some well known properties.

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

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.

Similarity Between Binary Vectors

Common situation is that objects, p and q, have only binary attributes

Compute similarities using the following quantities

M01 = the number of attributes where p was 0 and q was 1

M10 = the number of attributes where p was 1 and q was 0

M00 = the number of attributes where p was 0 and q was 0

M11 = the number of attributes where p was 1 and q was 1

Simple Matching and Jaccard Coefficients

SMC = number of matches / number of attributes

= (M11 + M00) / (M01 + M10 + M11 + M00)

J = number of 11 matches / number of not-both-zero attributes values

= (M11) / (M01 + M10 + M11)

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

General Approach for Combining Similarities

Sometimes attributes are of many different types, but an overall similarity is needed.

Density

Density-based clustering require a notion of density

Examples:

Euclidean density

Euclidean density = number of points per unit volume

Probability density

Graph-based density

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Refund

Marital

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Taxable

Income

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Yes

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Single

70K

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4

Yes

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5

No

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95K

Yes

6

No

Married

60K

No

7

Yes

Divorced

220K

No

8

No

Single

85K

Yes

9

No

Married

75K

No

10

No

Singl

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90K

Yes

10

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