Machine Learning and Data Mining: 15 Data Exploration and Preparation

Data Exploration and Preparation
Machine Learning and Data Mining (Unit 15)

Prof. Pier Luca Lanzi

References 2

Jiawei Han and Micheline Kamber, "Data Mining, : Concepts
and Techniques", The Morgan Kaufmann Series in Data
Management Systems (Second Edition)
Tom M. Mitchell. “Machine Learning” McGraw Hill 1997
Pang-Ning Tan, Michael Steinbach, Vipin Kumar,
“Introduction to Data Mining”, Addison Wesley
Ian H. Witten, Eibe Frank. “Data Mining: Practical Machine
Learning Tools and Techniques with Java Implementations”
2nd Edition


Outline 3

Data Exploration
Types of data
Descriptive statistics
Visualization

Data Preprocessing
Aggregation
Sampling
Dimensionality Reduction
Feature creation
Discretization
Attribute Transformation
Compression
Concept hierarchies


Types of data sets 4

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 5
Structured Data

Dimensionality
Curse of Dimensionality

Sparsity
Only presence counts

Resolution
Patterns depend on the scale


Record Data 6

Data that consists of a collection of records, each of which
consists of a fixed set of attributes

Tid Refund Marital Taxable
Income Cheat
Status

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 7

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

Projection Projection Distance Load Thickness
of x Load of y load

10.23 5.27 15.22 2.7 1.2
12.65 6.25 16.22 2.2 1.1


Document Data 8

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.

timeout

season
coach

game
score
team

ball

lost
play

win
Document 1 3 0 5 0 2 6 0 2 0 2

Document 2 0 7 0 2 1 0 0 3 0 0

Document 3 0 1 0 0 1 2 2 0 3 0


Transaction Data 9

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 10
Generic graph and HTML Links

<a href="papers/papers.html#bbbb">
Data Mining </a>
<li>
2 <a href="papers/papers.html#aaaa">
Graph Partitioning </a>
1
5 <li>
<a href="papers/papers.html#aaaa">
2 Parallel Solution of Sparse Linear
System of Equations </a>
5 <li>
<a href="papers/papers.html#ffff">
N-Body Computation and Dense Linear
System Solvers


Chemical Data 11

Benzene Molecule: C6H6


Ordered Data 12

Sequences of transactions

Items/Events

An element of
the sequence


Ordered Data 13

Genomic sequence data

GGTTCCGCCTTCAGCCCCGCGCC
CGCAGGGCCCGCCCCGCGCCGTC
GAGAAGGGCCCGCCTGGCGGGCG
GGGGGAGGCGGGGCCGCCCGAGC
CCAACCGAGTCCGACCAGGTGCC
CCCTCTGCTCGGCCTAGACCTGA
GCTCATTAGGCGGCAGCGGACAG
GCCAAGTAGAACACGCGAAGCGC
TGGGCTGCCTGCTGCGACCAGGG


Ordered Data 14

Spatio-Temporal Data

Average Monthly
Temperature of
land and ocean


What is data exploration? 15

A preliminary exploration of the data to better understand its
characteristics.

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


Techniques Used In Data Exploration 16

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 17

Many of the exploratory data techniques are illustrated with the Iris
Plant data set 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.


Mining Data Descriptive Characteristics 18

Motivation
To better understand the data:
central tendency, variation and spread

Data dispersion characteristics
median, max, min, quantiles, outliers, variance, etc.

Numerical dimensions correspond to sorted intervals
Data dispersion: analyzed with multiple granularities of precision
Boxplot or quantile analysis on sorted intervals

Dispersion analysis on computed measures
Folding measures into numerical dimensions
Boxplot or quantile analysis on the transformed cube


Summary Statistics 19

They 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 20

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 21

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 xp of x such
that p% of the observed values of x are less than xp

For instance, the 50th percentile is the value x50% such that
x p

50% of all values of x are less than x50%


Measures of Location: Mean and Median 22

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 23

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 24

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 25

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


Representation 26

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 27

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


Selection 28

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 29

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 30

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 31

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

75th percentile

50th percentile
25th percentile

10th percentile


Example of Box Plots 32

Box plots can be used to compare attributes


Visualization Techniques: Scatter Plots 33

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


Scatter Plot Array of Iris Attributes 34


Visualization Techniques: Contour Plots 35

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.


Contour Plot Example: SST Dec, 1998 36

Celsius


Visualization Techniques: Matrix Plots 37

Can plot the data matrix

This can be useful when objects are sorted according to class

Typically, the attributes are normalized to prevent one
attribute from dominating the plot

Plots of similarity or distance matrices can also be useful for
visualizing the relationships between objects

Examples of matrix plots are presented on the next two slides


Visualization of the Iris Data Matrix 38

standard
deviation


Visualization of the Iris Correlation Matrix 39


Visualization Techniques: 40
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


Parallel Coordinates Plots for Iris Data 41


Other Visualization Techniques 42

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


Star Plots for Iris Data 43

Setosa

Versicolour

Virginica


Chernoff Faces for Iris Data 44

Setosa

Versicolour

Virginica


Why Data Preprocessing? 45

Data in the real world is dirty

Incomplete: lacking attribute values, lacking certain
attributes of interest, or containing only aggregate data
e.g., occupation=“ ”
Noisy: containing errors or outliers
e.g., Salary=“-10”

Inconsistent: containing discrepancies in codes or names
e.g., Age=“42” Birthday=“03/07/1997”
e.g., Was rating “1,2,3”, now rating “A, B, C”
e.g., discrepancy between duplicate records


Why Is Data Dirty? 46

Incomplete data may come from
“Not applicable” data value when collected
Different considerations between the time when the data
was collected and when it is analyzed.
Human/hardware/software problems
Noisy data (incorrect values) may come from
Faulty data collection instruments
Human or computer error at data entry
Errors in data transmission
Inconsistent data may come from
Different data sources
Functional dependency violation (e.g., modify some linked
data)
Duplicate records also need data cleaning


Why Is Data Preprocessing Important? 47

No quality data, no quality mining results!

Quality decisions must be based on quality data
E.g., duplicate or missing data may cause incorrect or
even misleading statistics.

Data warehouse needs consistent integration of quality data

Data extraction, cleaning, and transformation comprises the
majority of the work of building a data warehouse


Data Quality 48

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


Multi-Dimensional Measure of 49
Data Quality

A well-accepted multidimensional view:
Accuracy
Completeness
Consistency
Timeliness
Believability
Value added
Interpretability
Accessibility
Broad categories:
Intrinsic, contextual, representational, and accessibility


Noise 50

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 51

Outliers are data objects with characteristics that are
considerably different than most of the other data objects in
the data set


Missing Values 52

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 53

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 Cleaning as a Process 54

Data discrepancy detection
Use metadata (e.g., domain, range, dependency, distribution)
Check field overloading
Check uniqueness rule, consecutive rule and null rule
Use commercial tools
• Data scrubbing: use simple domain knowledge (e.g., postal code,
spell-check) to detect errors and make corrections
• Data auditing: by analyzing data to discover rules and relationship
to detect violators (e.g., correlation and clustering to find outliers)

Data migration and integration
Data migration tools: allow transformations to be specified
ETL (Extraction/Transformation/Loading) tools: allow users to
specify transformations through a graphical user interface

Integration of the two processes
Iterative and interactive (e.g., Potter’s Wheels)


Data Preprocessing 55

Aggregation
Sampling
Dimensionality Reduction
Feature subset selection
Feature creation
Discretization and Binarization
Attribute Transformation


Aggregation 56

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


Aggregation 57

Variation of Precipitation in Australia

Standard Deviation of Average Standard Deviation of Average
Monthly Precipitation Yearly Precipitation


Sampling 58

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


Key principles for Effective Sampling 59

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 60

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


Sample Size 61

8000 points 2000 Points 500 Points


Sample Size 62

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


Curse of Dimensionality 63

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


Dimensionality Reduction 64

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: 65
Principal Component Analysis (PCA)

Given N data vectors from n-dimensions, find k ≤ n orthogonal
vectors (principal components) that can be best used to represent
data
Steps
Normalize input data: Each attribute falls within the same range
Compute k orthonormal (unit) vectors, i.e., principal
components
Each input data (vector) is a linear combination of the k
principal component vectors
The principal components are sorted in order of decreasing
“significance” or strength
Since the components are sorted, the size of the data can be
reduced by eliminating the weak components, i.e., those with
low variance. (i.e., using the strongest principal components, it
is possible to reconstruct a good approximation of the original
data
Works for numeric data only
Used when the number of dimensions is large


Principal Component Analysis 66

X2

Y1
Y2

X1


Dimensionality Reduction: PCA 67

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

x2

e

x1


Dimensionality Reduction: PCA 68

Find the eigenvectors of the covariance matrix
The eigenvectors define the new space

x2

e

x1


Feature Subset Selection 69

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 70

Brute-force approach
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 using a procedure that is
independent from a specific data mining algorithm
E.g., attributes are selected based on correlation measures

Wrapper approaches:
Use a data mining algorithm as a black box
to find best subset of attributes
E.g., apply a genetic algorithm and an algorithm for decision tree
to find the best set of features for a decision tree


Feature Creation 71

Create new attributes that can capture the important
information in a data set much more efficiently than the
original attributes
E.g., given the birthday, create the attribute age

Three general methodologies:
Feature Extraction: domain-specific
Mapping Data to New Space
Feature Construction: combining features


Mapping Data to a New Space 72

Fourier transform
Wavelet transform

Two Sine Waves Two Sine Waves + Noise Frequency


Discretization 73

Three types of attributes:
Nominal: values from an unordered set, e.g., color
Ordinal: values from an ordered set,
e.g., military or academic rank
Continuous: real numbers,
e.g., integer or real numbers

Discretization:
Divide the range of a continuous attribute into intervals
Some classification algorithms only accept categorical
attributes
Reduce data size by discretization
Prepare for further analysis


Discretization Approaches 74

Supervised
Attributes are discretized using the class information
Generates intervals that tries to minimize the loss of
information about the class

Unsupervised
Attributes are discretized solely based on their values


Discretization Using Class Labels 75

Entropy based approach

3 categories for both x and y 5 categories for both x and y


Discretization Without Using Class Labels 76

Data Equal interval width

Equal frequency K-means


Discretization and Concept Hierarchy 77

Discretization
Reduce the number of values for a given continuous attribute
by dividing the range of the attribute into intervals
Interval labels can then be used to replace actual data values
Supervised vs. unsupervised
Split (top-down) vs. merge (bottom-up)
Discretization can be performed recursively on an attribute

Concept hierarchy formation
Recursively reduce the data by collecting and replacing low level
concepts (such as numeric values for age) by higher level
concepts (such as young, middle-aged, or senior)


Typical Discretization Methods 78

Entropy-based discretization:
supervised, top-down split

Interval merging by χ2 Analysis:
supervised, bottom-up merge

Segmentation by natural partitioning:
top-down split, unsupervised


Entropy-Based Discretization 79

Given a set of samples S, if S is partitioned into two intervals S1
and S2 using boundary T, the information gain after partitioning is

|S |
| S1 |
I (S , T ) = Entropy ( S 1) + 2 Entropy ( S 2 )
|S| |S|
Entropy is calculated based on class distribution of the samples in
the set. Given m classes, the entropy of S1 is
m
Entropy ( S 1 ) = − ∑ p i log 2 ( p i )
i =1

where pi is the probability of class i in S1
The boundary that minimizes the entropy function over all possible
boundaries is selected as a binary discretization
The process is recursively applied to partitions obtained until some
stopping criterion is met
Such a boundary may reduce data size and improve classification
accuracy


Interval Merge by χ2 Analysis 80

Merging-based (bottom-up) vs. splitting-based methods
Merge: Find the best neighboring intervals and merge them
to form larger intervals recursively
ChiMerge [Kerber AAAI 1992, See also Liu et al. DMKD 2002]
Initially, each distinct value of a numerical attr. A is
considered to be one interval
χ2 tests are performed for every pair of adjacent intervals
Adjacent intervals with the least χ2 values are merged
together, since low χ2 values for a pair indicate similar
class distributions
This merge process proceeds recursively until a
predefined stopping criterion is met (such as significance
level, max-interval, max inconsistency, etc.)


Segmentation by Natural Partitioning 81

A simply 3-4-5 rule can be used to segment numeric data
into relatively uniform, “natural” intervals.

If an interval covers 3, 6, 7 or 9 distinct values at the most
significant digit, partition the range into 3 equi-width
intervals

If it covers 2, 4, or 8 distinct values at the most significant
digit, partition the range into 4 intervals

If it covers 1, 5, or 10 distinct values at the most significant
digit, partition the range into 5 intervals


Example of 3-4-5 Rule 82

count

-$351 -$159 profit $1,838 $4,700
Step 1:
Min Low (i.e, 5%-tile) High(i.e, 95%-0 tile) Max
Step 2: msd=1,000 Low=-$1,000 High=$2,000

(-$1,000 - $2,000)
Step 3:

(-$1,000 - 0) ($1,000 - $2,000)
(0 -$ 1,000)

(-$400 -$5,000)
Step 4:

($2,000 - $5, 000)
(-$400 - 0) ($1,000 - $2, 000)
(0 - $1,000)
(0 -
($1,000 -
(-$400 - $200)
($2,000 -
$1,200)
-$300)
$3,000)
($200 -
($1,200 -
$400)
(-$300 - $1,400)
($3,000 -
-$200)
$4,000)
($1,400 -
($400 -
(-$200 - $1,600)
$600) ($4,000 -
-$100) $5,000)
($600 - ($1,600 -
($1,800 -
$800) ($800 - $1,800)
$2,000)
(-$100 - $1,000)
0)

Attribute Transformation 83

A function that maps the entire set of values of a given
attribute to a new set of replacement values such that each
old value can be identified with one of the new values
Simple functions: xk, log(x), ex, |x|
Standardization and Normalization


On-Line Analytical Processing (OLAP) 84

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 85

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.


Example: Iris data 86

We show how the attributes, petal length, petal width, and
species type can be converted to a multidimensional array
First, we discretized the petal width and length to have
categorical values: low, medium, and high
We get the following table - note the count attribute


Example: Iris data (continued) 87

Each unique tuple of petal width, petal length, and species
type identifies one element of the array.
This element is assigned the corresponding count value.
The figure illustrates
the result.
All non-specified
tuples are 0.


Example: Iris data (continued) 88

Slices of the multidimensional array are shown by the
following cross-tabulations
What do these tables tell us?


OLAP Operations: Data Cube 89

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.


Data Cube Example 90

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 (continued) 91

The following figure table shows one of the two dimensional
aggregates, along with two of the one-dimensional
aggregates, and the overall total


OLAP Operations: Slicing and Dicing 92

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 93

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 94

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.


Data Compression 95

String compression
There are extensive theories and well-tuned algorithms
Typically lossless
But only limited manipulation is possible without
expansion
Audio/video compression
Typically lossy compression, with progressive refinement
Sometimes small fragments of signal can be
reconstructed without reconstructing the whole
Time sequence is not audio
Typically short and vary slowly with time


96
Dimensionality Reduction:
Wavelet Transformation

Haar2 Daubechie4
Discrete wavelet transform (DWT): linear signal processing, multi-
resolutional analysis
Compressed approximation: store only a small fraction of the
strongest of the wavelet coefficients
Similar to discrete Fourier transform (DFT), but better lossy
compression, localized in space
Method:
Length, L, must be an integer power of 2 (padding with 0’s, when
necessary)
Each transform has 2 functions: smoothing, difference
Applies to pairs of data, resulting in two set of data of length L/2
Applies two functions recursively, until reaches the desired length


Concept Hierarchy Generation 97
for Categorical Data

Specification of a partial/total ordering of attributes explicitly
at the schema level by users or experts
street < city < state < country

Specification of a hierarchy for a set of values by explicit
data grouping
{Urbana, Champaign, Chicago} < Illinois

Specification of only a partial set of attributes
E.g., only street < city, not others

Automatic generation of hierarchies (or attribute levels) by
the analysis of the number of distinct values
E.g., for a set of attributes: {street, city, state, country}


Automatic Concept Hierarchy Generation 98

Some hierarchies can be automatically generated based on
the analysis of the number of distinct values per attribute in
the data set
The attribute with the most distinct values is placed at the
lowest level of the hierarchy
Exceptions, e.g., weekday, month, quarter, year

Country 15 distinct values

Province 365 distinct values

City 3567 distinct values

Street 674,339 distinct values


Summary 99

Data exploration and preparation, or preprocessing, is a big
issue for both data warehousing and data mining
Descriptive data summarization is need for quality data
preprocessing
Data preparation includes
Data cleaning and data integration
Data reduction and feature selection
Discretization
A lot a methods have been developed but data preprocessing
still an active area of research


Machine Learning and Data Mining: 15 Data Exploration and Preparation

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