Discussion

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.

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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

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

Singl

e

90K

Yes

10

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5.27

10.23

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