Data Mining: Exploring Data
Lecture Notes for Chapter 3
Introduction to Data Mining
by
Tan, Steinbach, Kumar

What is data exploration?
Key motivations of data exploration include
Helping to select the right tool for preprocessing or analysis
Making use of humans’ abilities to recognize patterns
People can recognize patterns not captured by data analysis tools
Related to the area of Exploratory Data Analysis (EDA)
Created by statistician John Tukey
Seminal book is Exploratory Data Analysis by Tukey
A nice online introduction can be found in Chapter 1 of the NIST Engineering Statistics Handbook

http://www.itl.nist.gov/div898/handbook/index.htm
A preliminary exploration of the data to better understand its characteristics.

Techniques Used In Data Exploration
In EDA, as originally defined by Tukey
The focus was on visualization
Clustering and anomaly detection were viewed as exploratory techniques
In data mining, clustering and anomaly detection are major areas of interest, and not thought of as just exploratory
In our discussion of data exploration, we focus on
Summary statistics
Visualization
Online Analytical Processing (OLAP)

Iris Sample Data Set
Many of the exploratory data techniques are illustrated with the 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
Virginica
Versicolour
Four (non-class) attributes
Sepal width and length
Petal width and length

Virginica. Robert H. Mohlenbrock. USDA NRCS. 1995. Northeast wetland flora: Field office guide to plant species. Northeast National Technical Center, Chester, PA. Courtesy of USDA NRCS Wetland Science Institute.

Summary Statistics
Summary statistics are numbers that summarize properties of the data
Summarized properties include frequency, location and spread
Examples: location – mean
spread – standard deviation
Most summary statistics can be calculated in a single pass through the data

Frequency and Mode
The frequency of an attribute value is the percentage of time the value occurs in the
data set
For example, given the attribute ‘gender’ and a representative population of people, the gender ‘female’ occurs about 50% of the time.
The mode of a an attribute is the most frequent attribute value
The notions of frequency and mode are typically used with categorical data

Percentiles
For continuous data, the notion of a percentile is more useful.

Given an ordinal or continuous attribute x and a number p between 0 and 100, the pth percentile is a value of x such that p% of the observed values of x are less than .
For instance, the 50th percentile is the value such that 50% of all values of x are less than .

Measures of Location: Mean and Median
The mean is the most common measure of the location of a set of points.
However, the mean is very sensitive to outliers.
Thus, the median or a trimmed mean is also commonly used.

Measures of Spread: Range and Variance
Range is the difference between the max and min
The variance or standard deviation is the most common measure of the spread of a set of points.

However, this is also sensitive to outliers, so that other measures are often used.

Visualization
Visualization is the conversion of data into a visual or tabular format so that the characteristics of the data and the relationships among data items or attributes can be analyzed or reported.

Visualization of data is one of the most powerful and appealing techniques for data exploration.
Humans have a well developed ability to analyze large amounts of information that is presented visually
Can detect general patterns and trends
Can detect outliers and unusual patterns

Example: Sea Surface Temperature
The following shows the Sea Surface Temperature (SST) for July 1982
Tens of thousands of data points are summarized in a single figure

Representation
Is the mapping of information to a visual format
Data objects, their attributes, and the relationships among data objects are translated into graphical elements such as points, lines, shapes, and colors.
Example:
Objects are often represented as points
Their attribute values can be represented as the position of the points or the characteristics of the points, e.g., color, size, and shape
If position is used, then the relationships of points, i.e., whether they form groups or a point is an outlier, is easily perceived.

Arrangement
Is the placement of visual elements within a display
Can make a large difference in how easy it is to understand the data

Selection
Is the elimination or the de-emphasis of certain objects and attributes
Selection may involve the chossing a subset of attributes
Dimensionality reduction is often used to reduce the number of dimensions to two or three
Alternatively, pairs of attributes can be considered
Selection may also involve choosing a subset of objects
A region of the screen can only show so many points
Can sample, but want to preserve points in sparse areas

Visualization Techniques: Histograms
Histogram
Usually shows the distribution of values of a single variable
Divide the values into bins and show a bar plot of the number of objects in each bin.
The height of each bar indicates the number of objects
Shape of histogram depends on the number of bins
Example: Petal Width (10 and 20 bins, respectively)

Two-Dimensional Histograms
Show the joint distribution of the values of two attributes
Example: petal width and petal length
What does this tell us?

Visualization Techniques: Box Plots
Box Plots
Invented by J. Tukey
Another way of displaying the distribution of data
Following figure shows the basic part of a box plot

outlier
10th percentile
25th percentile
75th percentile
50th percentile
10th percentile

Example of Box Plots
Box plots can be used to compare attributes

Visualization Techniques: Scatter Plots
Scatter plots
Attributes values determine the position
Two-dimensional scatter plots most common, but can have three-dimensional scatter plots
Often additional attributes can be displayed by using the size, shape, and color of the markers that represent the objects
It is useful to have arrays of scatter plots can compactly summarize the relationships of several pairs of attributes
See example on the next slide

Visualization Techniques: Contour Plots
Contour plots
Useful when a continuous attribute is measured on a spatial grid
They partition the plane into regions of similar values
The contour lines that form the boundaries of these regions connect points with equal values
The most common example is contour maps of elevation
Can also display temperature, rainfall, air pressure, etc.
An example for Sea Surface Temperature (SST) is provided on the next slide

Visualization Techniques: Parallel Coordinates
Parallel Coordinates
Used to plot the attribute values of high-dimensional data
Instead of using perpendicular axes, use a set of parallel axes
The attribute values of each object are plotted as a point on each corresponding coordinate axis and the points are connected by a line
Thus, each object is represented as a line
Often, the lines representing a distinct class of objects group together, at least for some attributes
Ordering of attributes is important in seeing such groupings

Other Visualization Techniques
Star Plots
Similar approach to parallel coordinates, but axes radiate from a central point
The line connecting the values of an object is a polygon
Chernoff Faces
Approach created by Herman Chernoff
This approach associates each attribute with a characteristic of a face
The values of each attribute determine the appearance of the corresponding facial characteristic
Each object becomes a separate face
Relies on human’s ability to distinguish faces

OLAP
On-Line Analytical Processing (OLAP) was proposed by E. F. Codd, the father of the relational database.
Relational databases put data into tables, while OLAP uses a multidimensional array representation.
Such representations of data previously existed in statistics and other fields
There are a number of data analysis and data exploration operations that are easier with such a data representation.

Creating a Multidimensional Array
Two key steps in converting tabular data into a multidimensional array.
First, identify which attributes are to be the dimensions and which attribute is to be the target attribute whose values appear as entries in the multidimensional array.
The attributes used as dimensions must have discrete values
The target value is typically a count or continuous value, e.g., the cost of an item
Can have no target variable at all except the count of objects that have the same set of attribute values
Second, find the value of each entry in the multidimensional array by summing the values (of the target attribute) or count of all objects that have the attribute values corresponding to that entry.

OLAP Operations: Data Cube
The key operation of a OLAP is the formation of a data cube
A data cube is a multidimensional representation of data, together with all possible aggregates.
By all possible aggregates, we mean the aggregates that result by selecting a proper subset of the dimensions and summing over all remaining dimensions.
For example, if we choose the species type dimension of the Iris data and sum over all other dimensions, the result will be a one-dimensional entry with three entries, each of which gives the number of flowers of each type.

Consider a data set that records the sales of products at a number of company stores at various dates.
This data can be represented
as a 3 dimensional array
There are 3 two-dimensional
aggregates (3 choose 2 ),
3 one-dimensional aggregates,
and 1 zero-dimensional
aggregate (the overall total)
Data Cube Example

The following figure table shows one of the two dimensional aggregates, along with two of the one-dimensional aggregates, and the overall total
Data Cube Example (continued)

OLAP Operations: Slicing and Dicing
Slicing is selecting a group of cells from the entire multidimensional array by specifying a specific value for one or more dimensions.
Dicing involves selecting a subset of cells by specifying a range of attribute values.
This is equivalent to defining a subarray from the complete array.
In practice, both operations can also be accompanied by aggregation over some dimensions.

OLAP Operations: Roll-up and Drill-down
Attribute values often have a hierarchical structure.
Each date is associated with a year, month, and week.
A location is associated with a continent, country, state (province, etc.), and city.
Products can be divided into various categories, such as clothing, electronics, and furniture.
Note that these categories often nest and form a tree or lattice
A year contains months which contains day
A country contains a state which contains a city

OLAP Operations: Roll-up and Drill-down
This hierarchical structure gives rise to the roll-up and drill-down operations.
For sales data, we can aggregate (roll up) the sales across all the dates in a month.
Conversely, given a view of the data where the time dimension is broken into months, we could split the monthly sales totals (drill down) into daily sales totals.
Likewise, we can drill down or roll up on the location or product ID attributes.

 
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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�
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�

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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Data Mining