程序代写代做代考 SQL AI Bayesian scheme chain Functional Dependencies data mining algorithm database decision tree 3Data Preprocessing

3Data Preprocessing
Today’s real-world databases are highly susceptible to noisy, missing, and inconsistent data

due to their typically huge size (often several gigabytes or more) and their likely origin
from multiple, heterogenous sources. Low-quality data will lead to low-quality mining
results. “How can the data be preprocessed in order to help improve the quality of the data
and, consequently, of the mining results? How can the data be preprocessed so as to improve
the efficiency and ease of the mining process?”

There are several data preprocessing techniques. Data cleaning can be applied to
remove noise and correct inconsistencies in data. Data integration merges data from
multiple sources into a coherent data store such as a data warehouse. Data reduction
can reduce data size by, for instance, aggregating, eliminating redundant features, or
clustering. Data transformations (e.g., normalization) may be applied, where data are
scaled to fall within a smaller range like 0.0 to 1.0. This can improve the accuracy and
efficiency of mining algorithms involving distance measurements. These techniques are
not mutually exclusive; they may work together. For example, data cleaning can involve
transformations to correct wrong data, such as by transforming all entries for a date field
to a common format.

In Chapter 2, we learned about the different attribute types and how to use basic
statistical descriptions to study data characteristics. These can help identify erroneous
values and outliers, which will be useful in the data cleaning and integration steps.
Data processing techniques, when applied before mining, can substantially improve the
overall quality of the patterns mined and/or the time required for the actual mining.

In this chapter, we introduce the basic concepts of data preprocessing in Section 3.1.
The methods for data preprocessing are organized into the following categories: data
cleaning (Section 3.2), data integration (Section 3.3), data reduction (Section 3.4), and
data transformation (Section 3.5).

c© 2012 Elsevier Inc. All rights reserved.

Data Mining: Concepts and Techniques 83

84 Chapter 3 Data Preprocessing

3.1 Data Preprocessing: An Overview
This section presents an overview of data preprocessing. Section 3.1.1 illustrates the
many elements defining data quality. This provides the incentive behind data prepro-
cessing. Section 3.1.2 outlines the major tasks in data preprocessing.

3.1.1 Data Quality: Why Preprocess the Data?
Data have quality if they satisfy the requirements of the intended use. There are many
factors comprising data quality, including accuracy, completeness, consistency, timeliness,
believability, and interpretability.

Imagine that you are a manager at AllElectronics and have been charged with ana-
lyzing the company’s data with respect to your branch’s sales. You immediately set out
to perform this task. You carefully inspect the company’s database and data warehouse,
identifying and selecting the attributes or dimensions (e.g., item, price, and units sold)
to be included in your analysis. Alas! You notice that several of the attributes for various
tuples have no recorded value. For your analysis, you would like to include informa-
tion as to whether each item purchased was advertised as on sale, yet you discover that
this information has not been recorded. Furthermore, users of your database system
have reported errors, unusual values, and inconsistencies in the data recorded for some
transactions. In other words, the data you wish to analyze by data mining techniques are
incomplete (lacking attribute values or certain attributes of interest, or containing only
aggregate data); inaccurate or noisy (containing errors, or values that deviate from the
expected); and inconsistent (e.g., containing discrepancies in the department codes used
to categorize items). Welcome to the real world!

This scenario illustrates three of the elements defining data quality: accuracy, com-
pleteness, and consistency. Inaccurate, incomplete, and inconsistent data are common-
place properties of large real-world databases and data warehouses. There are many
possible reasons for inaccurate data (i.e., having incorrect attribute values). The data col-
lection instruments used may be faulty. There may have been human or computer errors
occurring at data entry. Users may purposely submit incorrect data values for manda-
tory fields when they do not wish to submit personal information (e.g., by choosing
the default value “January 1” displayed for birthday). This is known as disguised missing
data. Errors in data transmission can also occur. There may be technology limitations
such as limited buffer size for coordinating synchronized data transfer and consump-
tion. Incorrect data may also result from inconsistencies in naming conventions or data
codes, or inconsistent formats for input fields (e.g., date). Duplicate tuples also require
data cleaning.

Incomplete data can occur for a number of reasons. Attributes of interest may not
always be available, such as customer information for sales transaction data. Other data
may not be included simply because they were not considered important at the time
of entry. Relevant data may not be recorded due to a misunderstanding or because of
equipment malfunctions. Data that were inconsistent with other recorded data may

3.1 Data Preprocessing: An Overview 85

have been deleted. Furthermore, the recording of the data history or modifications may
have been overlooked. Missing data, particularly for tuples with missing values for some
attributes, may need to be inferred.

Recall that data quality depends on the intended use of the data. Two different users
may have very different assessments of the quality of a given database. For example, a
marketing analyst may need to access the database mentioned before for a list of cus-
tomer addresses. Some of the addresses are outdated or incorrect, yet overall, 80% of
the addresses are accurate. The marketing analyst considers this to be a large customer
database for target marketing purposes and is pleased with the database’s accuracy,
although, as sales manager, you found the data inaccurate.

Timeliness also affects data quality. Suppose that you are overseeing the distribu-
tion of monthly sales bonuses to the top sales representatives at AllElectronics. Several
sales representatives, however, fail to submit their sales records on time at the end of
the month. There are also a number of corrections and adjustments that flow in after
the month’s end. For a period of time following each month, the data stored in the
database are incomplete. However, once all of the data are received, it is correct. The fact
that the month-end data are not updated in a timely fashion has a negative impact on
the data quality.

Two other factors affecting data quality are believability and interpretability. Believ-
ability reflects how much the data are trusted by users, while interpretability reflects
how easy the data are understood. Suppose that a database, at one point, had several
errors, all of which have since been corrected. The past errors, however, had caused
many problems for sales department users, and so they no longer trust the data. The
data also use many accounting codes, which the sales department does not know how to
interpret. Even though the database is now accurate, complete, consistent, and timely,
sales department users may regard it as of low quality due to poor believability and
interpretability.

3.1.2 Major Tasks in Data Preprocessing
In this section, we look at the major steps involved in data preprocessing, namely, data
cleaning, data integration, data reduction, and data transformation.

Data cleaning routines work to “clean” the data by filling in missing values, smooth-
ing noisy data, identifying or removing outliers, and resolving inconsistencies. If users
believe the data are dirty, they are unlikely to trust the results of any data mining that has
been applied. Furthermore, dirty data can cause confusion for the mining procedure,
resulting in unreliable output. Although most mining routines have some procedures
for dealing with incomplete or noisy data, they are not always robust. Instead, they may
concentrate on avoiding overfitting the data to the function being modeled. Therefore,
a useful preprocessing step is to run your data through some data cleaning routines.
Section 3.2 discusses methods for data cleaning.

Getting back to your task at AllElectronics, suppose that you would like to include
data from multiple sources in your analysis. This would involve integrating multiple
databases, data cubes, or files (i.e., data integration). Yet some attributes representing a

86 Chapter 3 Data Preprocessing

given concept may have different names in different databases, causing inconsistencies
and redundancies. For example, the attribute for customer identification may be referred
to as customer id in one data store and cust id in another. Naming inconsistencies may
also occur for attribute values. For example, the same first name could be registered as
“Bill” in one database, “William” in another, and “B.” in a third. Furthermore, you sus-
pect that some attributes may be inferred from others (e.g., annual revenue). Having
a large amount of redundant data may slow down or confuse the knowledge discov-
ery process. Clearly, in addition to data cleaning, steps must be taken to help avoid
redundancies during data integration. Typically, data cleaning and data integration are
performed as a preprocessing step when preparing data for a data warehouse. Addi-
tional data cleaning can be performed to detect and remove redundancies that may have
resulted from data integration.

“Hmmm,” you wonder, as you consider your data even further. “The data set I have
selected for analysis is HUGE, which is sure to slow down the mining process. Is there a
way I can reduce the size of my data set without jeopardizing the data mining results?”
Data reduction obtains a reduced representation of the data set that is much smaller in
volume, yet produces the same (or almost the same) analytical results. Data reduction
strategies include dimensionality reduction and numerosity reduction.

In dimensionality reduction, data encoding schemes are applied so as to obtain a
reduced or “compressed” representation of the original data. Examples include data
compression techniques (e.g., wavelet transforms and principal components analysis),
attribute subset selection (e.g., removing irrelevant attributes), and attribute construction
(e.g., where a small set of more useful attributes is derived from the original set).

In numerosity reduction, the data are replaced by alternative, smaller representa-
tions using parametric models (e.g., regression or log-linear models) or nonparametric
models (e.g., histograms, clusters, sampling, or data aggregation). Data reduction is the
topic of Section 3.4.

Getting back to your data, you have decided, say, that you would like to use a distance-
based mining algorithm for your analysis, such as neural networks, nearest-neighbor
classifiers, or clustering.1 Such methods provide better results if the data to be ana-
lyzed have been normalized, that is, scaled to a smaller range such as [0.0, 1.0]. Your
customer data, for example, contain the attributes age and annual salary. The annual
salary attribute usually takes much larger values than age. Therefore, if the attributes
are left unnormalized, the distance measurements taken on annual salary will generally
outweigh distance measurements taken on age. Discretization and concept hierarchy gen-
eration can also be useful, where raw data values for attributes are replaced by ranges or
higher conceptual levels. For example, raw values for age may be replaced by higher-level
concepts, such as youth, adult, or senior.

Discretization and concept hierarchy generation are powerful tools for data min-
ing in that they allow data mining at multiple abstraction levels. Normalization, data

1Neural networks and nearest-neighbor classifiers are described in Chapter 9, and clustering is discussed
in Chapters 10 and 11.

3.2 Data Preprocessing: An Overview 87

discretization, and concept hierarchy generation are forms of data transformation.
You soon realize such data transformation operations are additional data preprocessing
procedures that would contribute toward the success of the mining process. Data
integration and data discretization are discussed in Sections 3.5.

Figure 3.1 summarizes the data preprocessing steps described here. Note that the pre-
vious categorization is not mutually exclusive. For example, the removal of redundant
data may be seen as a form of data cleaning, as well as data reduction.

In summary, real-world data tend to be dirty, incomplete, and inconsistent. Data pre-
processing techniques can improve data quality, thereby helping to improve the accuracy
and efficiency of the subsequent mining process. Data preprocessing is an important step
in the knowledge discovery process, because quality decisions must be based on qual-
ity data. Detecting data anomalies, rectifying them early, and reducing the data to be
analyzed can lead to huge payoffs for decision making.

Data cleaning

Data integration

Data reduction
Attributes Attributes

A1 A2 A3 … A126

T1
T2
T3
T4
…
T2000

T
ra

ns
ac

ti
on

s

T
ra

ns
ac

ti
on

s

T1
T4
…
T1456

A1 A3 … A115

Data transformation �2, 32, 100, 59, 48 �0.02, 0.32, 1.00, 0.59, 0.48

Figure 3.1 Forms of data preprocessing.

88 Chapter 3 Data Preprocessing

3.2 Data Cleaning
Real-world data tend to be incomplete, noisy, and inconsistent. Data cleaning (or data
cleansing) routines attempt to fill in missing values, smooth out noise while identi-
fying outliers, and correct inconsistencies in the data. In this section, you will study
basic methods for data cleaning. Section 3.2.1 looks at ways of handling missing values.
Section 3.2.2 explains data smoothing techniques. Section 3.2.3 discusses approaches to
data cleaning as a process.

3.2.1 Missing Values
Imagine that you need to analyze AllElectronics sales and customer data. You note that
many tuples have no recorded value for several attributes such as customer income. How
can you go about filling in the missing values for this attribute? Let’s look at the following
methods.

1. Ignore the tuple: This is usually done when the class label is missing (assuming the
mining task involves classification). This method is not very effective, unless the tuple
contains several attributes with missing values. It is especially poor when the percent-
age of missing values per attribute varies considerably. By ignoring the tuple, we do
not make use of the remaining attributes’ values in the tuple. Such data could have
been useful to the task at hand.

2. Fill in the missing value manually: In general, this approach is time consuming and
may not be feasible given a large data set with many missing values.

3. Use a global constant to fill in the missing value: Replace all missing attribute values
by the same constant such as a label like “Unknown” or −∞. If missing values are
replaced by, say, “Unknown,” then the mining program may mistakenly think that
they form an interesting concept, since they all have a value in common—that of
“Unknown.” Hence, although this method is simple, it is not foolproof.

4. Use a measure of central tendency for the attribute (e.g., the mean or median) to
fill in the missing value: Chapter 2 discussed measures of central tendency, which
indicate the “middle” value of a data distribution. For normal (symmetric) data dis-
tributions, the mean can be used, while skewed data distribution should employ
the median (Section 2.2). For example, suppose that the data distribution regard-
ing the income of AllElectronics customers is symmetric and that the mean income is
$56,000. Use this value to replace the missing value for income.

5. Use the attribute mean or median for all samples belonging to the same class as
the given tuple: For example, if classifying customers according to credit risk, we
may replace the missing value with the mean income value for customers in the same
credit risk category as that of the given tuple. If the data distribution for a given class
is skewed, the median value is a better choice.

6. Use the most probable value to fill in the missing value: This may be determined
with regression, inference-based tools using a Bayesian formalism, or decision tree

3.2 Data Cleaning 89

induction. For example, using the other customer attributes in your data set, you
may construct a decision tree to predict the missing values for income. Decision trees
and Bayesian inference are described in detail in Chapters 8 and 9, respectively, while
regression is introduced in Section 3.4.5.

Methods 3 through 6 bias the data—the filled-in value may not be correct. Method 6,
however, is a popular strategy. In comparison to the other methods, it uses the most
information from the present data to predict missing values. By considering the other
attributes’ values in its estimation of the missing value for income, there is a greater
chance that the relationships between income and the other attributes are preserved.

It is important to note that, in some cases, a missing value may not imply an error
in the data! For example, when applying for a credit card, candidates may be asked to
supply their driver’s license number. Candidates who do not have a driver’s license may
naturally leave this field blank. Forms should allow respondents to specify values such
as “not applicable.” Software routines may also be used to uncover other null values
(e.g., “don’t know,” “?” or “none”). Ideally, each attribute should have one or more rules
regarding the null condition. The rules may specify whether or not nulls are allowed
and/or how such values should be handled or transformed. Fields may also be inten-
tionally left blank if they are to be provided in a later step of the business process. Hence,
although we can try our best to clean the data after it is seized, good database and data
entry procedure design should help minimize the number of missing values or errors in
the first place.

3.2.2 Noisy Data
“What is noise?” Noise is a random error or variance in a measured variable. In
Chapter 2, we saw how some basic statistical description techniques (e.g., boxplots
and scatter plots), and methods of data visualization can be used to identify outliers,
which may represent noise. Given a numeric attribute such as, say, price, how can we
“smooth” out the data to remove the noise? Let’s look at the following data smoothing
techniques.

Binning: Binning methods smooth a sorted data value by consulting its “neighbor-
hood,” that is, the values around it. The sorted values are distributed into a number
of “buckets,” or bins. Because binning methods consult the neighborhood of values,
they perform local smoothing. Figure 3.2 illustrates some binning techniques. In this
example, the data for price are first sorted and then partitioned into equal-frequency
bins of size 3 (i.e., each bin contains three values). In smoothing by bin means, each
value in a bin is replaced by the mean value of the bin. For example, the mean of the
values 4, 8, and 15 in Bin 1 is 9. Therefore, each original value in this bin is replaced
by the value 9.

Similarly, smoothing by bin medians can be employed, in which each bin value
is replaced by the bin median. In smoothing by bin boundaries, the minimum and
maximum values in a given bin are identified as the bin boundaries. Each bin value
is then replaced by the closest boundary value. In general, the larger the width, the

90 Chapter 3 Data Preprocessing

Sorted data for price (in dollars): 4, 8, 15, 21, 21, 24, 25, 28, 34

Partition into (equal-frequency) bins:

Bin 1: 4, 8, 15
Bin 2: 21, 21, 24
Bin 3: 25, 28, 34

Smoothing by bin means:

Bin 1: 9, 9, 9
Bin 2: 22, 22, 22
Bin 3: 29, 29, 29

Smoothing by bin boundaries:

Bin 1: 4, 4, 15
Bin 2: 21, 21, 24
Bin 3: 25, 25, 34

Figure 3.2 Binning methods for data smoothing.

greater the effect of the smoothing. Alternatively, bins may be equal width, where the
interval range of values in each bin is constant. Binning is also used as a discretization
technique and is further discussed in Section 3.5.

Regression: Data smoothing can also be done by regression, a technique that con-
forms data values to a function. Linear regression involves finding the “best” line to
fit two attributes (or variables) so that one attribute can be used to predict the other.
Multiple linear regression is an extension of linear regression, where more than two
attributes are involved and the data are fit to a multidimensional surface. Regression
is further described in Section 3.4.5.

Outlier analysis: Outliers may be detected by clustering, for example, where similar
values are organized into groups, or “clusters.” Intuitively, values that fall outside of
the set of clusters may be considered outliers (Figure 3.3). Chapter 12 is dedicated to
the topic of outlier analysis.

Many data smoothing methods are also used for data discretization (a form of data
transformation) and data reduction. For example, the binning techniques described
before reduce the number of distinct values per attribute. This acts as a form of data
reduction for logic-based data mining methods, such as decision tree induction, which
repeatedly makes value comparisons on sorted data. Concept hierarchies are a form of
data discretization that can also be used for data smoothing. A concept hierarchy for
price, for example, may map real price values into inexpensive, moderately priced, and
expensive, thereby reducing the number of data values to be handled by the mining

3.2 Data Cleaning 91

Figure 3.3 A 2-D customer data plot with respect to customer locations in a city, showing three data
clusters. Outliers may be detected as values that fall outside of the cluster sets.

process. Data discretization is discussed in Section 3.5. Some methods of classification
(e.g., neural networks) have built-in data smoothing mechanisms. Classification is the
topic of Chapters 8 and 9.

3.2.3 Data Cleaning as a Process
Missing values, noise, and inconsistencies contribute to inaccurate data. So far, we have
looked at techniques for handling missing data and for smoothing data. “But data clean-
ing is a big job. What about data cleaning as a process? How exactly does one proceed in
tackling this task? Are there any tools out there to help?”

The first step in data cleaning as a process is discrepancy detection. Discrepancies can
be caused by several factors, including poorly designed data entry forms that have many
optional fields, human error in data entry, deliberate errors (e.g., respondents not want-
ing to divulge information about themselves), and data decay (e.g., outdated addresses).
Discrepancies may also arise from inconsistent data representations and inconsistent use
of codes. Other sources of discrepancies include errors in instrumentation devices that
record data and system errors. Errors can also occur when the data are (inadequately)
used for purposes other than originally intended. There may also be inconsistencies due
to data integration (e.g., where a given attribute can have different names in different
databases).2

2Data integration and the removal of redundant data that can result from such integration are further
described in Section 3.3.

92 Chapter 3 Data Preprocessing

“So, how can we proceed with discrepancy detection?” As a starting point, use any
knowledge you may already have regarding properties of the data. Such knowledge or
“data about data” is referred to as metadata. This is where we can make use of the know-
ledge we gained about our data in Chapter 2. For example, what are the data type and
domain of each attribute? What are the acceptable values for each attribute? The basic
statistical data descriptions discussed in Section 2.2 are useful here to grasp data trends
and identify anomalies. For example, find the mean, median, and mode values. Are the
data symmetric or skewed? What is the range of values? Do all values fall within the
expected range? What is the standard deviation of each attribute? Values that are more
than two standard deviations away from the mean for a given attribute may be flagged
as potential outliers. Are there any known dependencies between attributes? In this step,
you may write your own scripts and/or use some of the tools that we discuss further later.
From this, you may find noise, outliers, and unusual values that need investigation.

As a data analyst, you should be on the lookout for the inconsistent use of codes and
any inconsistent data representations (e.g., “2010/12/25” and “25/12/2010” for date).
Field overloading is another error source that typically results when developers squeeze
new attribute definitions into unused (bit) portions of already defined attributes (e.g.,
an unused bit of an attribute that has a value range that uses only, say, 31 out of
32 bits).

The data should also be examined regarding unique rules, consecutive rules, and null
rules. A unique rule says that each value of the given attribute must be different from
all other values for that attribute. A consecutive rule says that there can be no miss-
ing values between the lowest and highest values for the attribute, and that all values
must also be unique (e.g., as in check numbers). A null rule specifies the use of blanks,
question marks, special characters, or other strings that may indicate the null condition
(e.g., where a value for a given attribute is not available), and how such values should
be handled. As mentioned in Section 3.2.1, reasons for missing values may include
(1) the person originally asked to provide a value for the attribute refuses and/or finds
that the information requested is not applicable (e.g., a license number attribute left
blank by nondrivers); (2) the data entry person does not know the correct value; or (3)
the value is to be provided by a later step of the process. The null rule should specify how
to record the null condition, for example, such as to store zero for numeric attributes, a
blank for character attributes, or any other conventions that may be in use (e.g., entries
like “don’t know” or “?” should be transformed to blank).

There are a number of different commercial tools that can aid in the discrepancy
detection step. Data scrubbing tools use simple domain knowledge (e.g., knowledge
of postal addresses and spell-checking) to detect errors and make corrections in the
data. These tools rely on parsing and fuzzy matching techniques when cleaning data
from multiple sources. Data auditing tools find discrepancies by analyzing the data to
discover rules and relationships, and detecting data that violate such conditions. They
are variants of data mining tools. For example, they may employ statistical analysis to
find correlations, or clustering to identify outliers. They may also use the basic statistical
data descriptions presented in Section 2.2.

Some data inconsistencies may be corrected manually using external references.
For example, errors made at data entry may be corrected by performing a paper

3.3 Data Integration 93

trace. Most errors, however, will require data transformations. That is, once we find
discrepancies, we typically need to define and apply (a series of) transformations to
correct them.

Commercial tools can assist in the data transformation step. Data migration tools
allow simple transformations to be specified such as to replace the string “gender” by
“sex.” ETL (extraction/transformation/loading) tools allow users to specify transforms
through a graphical user interface (GUI). These tools typically support only a restricted
set of transforms so that, often, we may also choose to write custom scripts for this step
of the data cleaning process.

The two-step process of discrepancy detection and data transformation (to correct
discrepancies) iterates. This process, however, is error-prone and time consuming. Some
transformations may introduce more discrepancies. Some nested discrepancies may only
be detected after others have been fixed. For example, a typo such as “20010” in a year
field may only surface once all date values have been converted to a uniform format.
Transformations are often done as a batch process while the user waits without feedback.
Only after the transformation is complete can the user go back and check that no new
anomalies have been mistakenly created. Typically, numerous iterations are required
before the user is satisfied. Any tuples that cannot be automatically handled by a given
transformation are typically written to a file without any explanation regarding the rea-
soning behind their failure. As a result, the entire data cleaning process also suffers from
a lack of interactivity.

New approaches to data cleaning emphasize increased interactivity. Potter’s Wheel,
for example, is a publicly available data cleaning tool that integrates discrepancy detec-
tion and transformation. Users gradually build a series of transformations by composing
and debugging individual transformations, one step at a time, on a spreadsheet-like
interface. The transformations can be specified graphically or by providing examples.
Results are shown immediately on the records that are visible on the screen. The user
can choose to undo the transformations, so that transformations that introduced addi-
tional errors can be “erased.” The tool automatically performs discrepancy checking in
the background on the latest transformed view of the data. Users can gradually develop
and refine transformations as discrepancies are found, leading to more effective and
efficient data cleaning.

Another approach to increased interactivity in data cleaning is the development of
declarative languages for the specification of data transformation operators. Such work
focuses on defining powerful extensions to SQL and algorithms that enable users to
express data cleaning specifications efficiently.

As we discover more about the data, it is important to keep updating the metadata
to reflect this knowledge. This will help speed up data cleaning on future versions of the
same data store.

3.3 Data Integration
Data mining often requires data integration—the merging of data from multiple data
stores. Careful integration can help reduce and avoid redundancies and inconsistencies

94 Chapter 3 Data Preprocessing

in the resulting data set. This can help improve the accuracy and speed of the subsequent
data mining process.

The semantic heterogeneity and structure of data pose great challenges in data inte-
gration. How can we match schema and objects from different sources? This is the
essence of the entity identification problem, described in Section 3.3.1. Are any attributes
correlated? Section 3.3.2 presents correlation tests for numeric and nominal data. Tuple
duplication is described in Section 3.3.3. Finally, Section 3.3.4 touches on the detection
and resolution of data value conflicts.

3.3.1 Entity Identification Problem
It is likely that your data analysis task will involve data integration, which combines data
from multiple sources into a coherent data store, as in data warehousing. These sources
may include multiple databases, data cubes, or flat files.

There are a number of issues to consider during data integration. Schema integration
and object matching can be tricky. How can equivalent real-world entities from multiple
data sources be matched up? This is referred to as the entity identification problem.
For example, how can the data analyst or the computer be sure that customer id in one
database and cust number in another refer to the same attribute? Examples of metadata
for each attribute include the name, meaning, data type, and range of values permitted
for the attribute, and null rules for handling blank, zero, or null values (Section 3.2).
Such metadata can be used to help avoid errors in schema integration. The metadata
may also be used to help transform the data (e.g., where data codes for pay type in one
database may be “H” and “S” but 1 and 2 in another). Hence, this step also relates to
data cleaning, as described earlier.

When matching attributes from one database to another during integration, special
attention must be paid to the structure of the data. This is to ensure that any attribute
functional dependencies and referential constraints in the source system match those in
the target system. For example, in one system, a discount may be applied to the order,
whereas in another system it is applied to each individual line item within the order.
If this is not caught before integration, items in the target system may be improperly
discounted.

3.3.2 Redundancy and Correlation Analysis
Redundancy is another important issue in data integration. An attribute (such as annual
revenue, for instance) may be redundant if it can be “derived” from another attribute
or set of attributes. Inconsistencies in attribute or dimension naming can also cause
redundancies in the resulting data set.

Some redundancies can be detected by correlation analysis. Given two attributes,
such analysis can measure how strongly one attribute implies the other, based on the
available data. For nominal data, we use the χ2 (chi-square) test. For numeric attributes,
we can use the correlation coefficient and covariance, both of which access how one
attribute’s values vary from those of another.

3.3 Data Integration 95

χ2 Correlation Test for Nominal Data
For nominal data, a correlation relationship between two attributes, A and B, can be
discovered by aχ2 (chi-square) test. Suppose A has c distinct values, namely a1,a2, . . .ac .
B has r distinct values, namely b1,b2, . . .br . The data tuples described by A and B can be
shown as a contingency table, with the c values of A making up the columns and the r
values of B making up the rows. Let (Ai ,Bj) denote the joint event that attribute A takes
on value ai and attribute B takes on value bj , that is, where (A= ai ,B = bj). Each and
every possible (Ai ,Bj) joint event has its own cell (or slot) in the table. The χ

2 value
(also known as the Pearson χ2 statistic) is computed as

χ
2
=

c∑
i=1

r∑
j=1

(oij − eij)
2

eij
, (3.1)

where oij is the observed frequency (i.e., actual count) of the joint event (Ai ,Bj) and eij is
the expected frequency of (Ai ,Bj), which can be computed as

eij =
count(A= ai)× count(B = bj)

n
, (3.2)

where n is the number of data tuples, count(A= ai) is the number of tuples having value
ai for A, and count(B = bj) is the number of tuples having value bj for B. The sum in
Eq. (3.1) is computed over all of the r× c cells. Note that the cells that contribute the
most to the χ2 value are those for which the actual count is very different from that
expected.

The χ2 statistic tests the hypothesis that A and B are independent, that is, there is no
correlation between them. The test is based on a significance level, with (r− 1)× (c− 1)
degrees of freedom. We illustrate the use of this statistic in Example 3.1. If the hypothesis
can be rejected, then we say that A and B are statistically correlated.

Example 3.1 Correlation analysis of nominal attributes using χ2. Suppose that a group of 1500
people was surveyed. The gender of each person was noted. Each person was polled as
to whether his or her preferred type of reading material was fiction or nonfiction. Thus,
we have two attributes, gender and preferred reading. The observed frequency (or count)
of each possible joint event is summarized in the contingency table shown in Table 3.1,
where the numbers in parentheses are the expected frequencies. The expected frequen-
cies are calculated based on the data distribution for both attributes using Eq. (3.2).

Using Eq. (3.2), we can verify the expected frequencies for each cell. For example,
the expected frequency for the cell (male, fiction) is

e11 =
count(male)× count(fiction)

n
=

300× 450

1500
= 90,

and so on. Notice that in any row, the sum of the expected frequencies must equal the
total observed frequency for that row, and the sum of the expected frequencies in any
column must also equal the total observed frequency for that column.

96 Chapter 3 Data Preprocessing

Table 3.1 Example 2.1’s 2× 2 Contingency Table Data

male female Total

fiction 250 (90) 200 (360) 450

non fiction 50 (210) 1000 (840) 1050

Total 300 1200 1500

Note: Are gender and preferred reading correlated?

Using Eq. (3.1) for χ2 computation, we get

χ
2
=
(250− 90)2

90
+
(50− 210)2

210
+
(200− 360)2

360
+
(1000− 840)2

840

= 284.44+ 121.90+ 71.11+ 30.48= 507.93.

For this 2× 2 table, the degrees of freedom are (2− 1)(2− 1)= 1. For 1 degree of free-
dom, the χ2 value needed to reject the hypothesis at the 0.001 significance level is 10.828
(taken from the table of upper percentage points of the χ2 distribution, typically avail-
able from any textbook on statistics). Since our computed value is above this, we can
reject the hypothesis that gender and preferred reading are independent and conclude
that the two attributes are (strongly) correlated for the given group of people.

Correlation Coefficient for Numeric Data
For numeric attributes, we can evaluate the correlation between two attributes, A and B,
by computing the correlation coefficient (also known as Pearson’s product moment
coefficient, named after its inventer, Karl Pearson). This is

rA,B =

n∑
i=1

(ai − Ā)(bi − B̄)

nσAσB
=

n∑
i=1

(aibi)− nĀB̄

nσAσB
, (3.3)

where n is the number of tuples, ai and bi are the respective values of A and B in tuple i,
Ā and B̄ are the respective mean values of A and B, σA and σB are the respective standard
deviations of A and B (as defined in Section 2.2.2), and 6(aibi) is the sum of the AB
cross-product (i.e., for each tuple, the value for A is multiplied by the value for B in that
tuple). Note that −1≤ rA,B ≤+1. If rA,B is greater than 0, then A and B are positively
correlated, meaning that the values of A increase as the values of B increase. The higher
the value, the stronger the correlation (i.e., the more each attribute implies the other).
Hence, a higher value may indicate that A (or B) may be removed as a redundancy.

If the resulting value is equal to 0, then A and B are independent and there is no
correlation between them. If the resulting value is less than 0, then A and B are negatively
correlated, where the values of one attribute increase as the values of the other attribute
decrease. This means that each attribute discourages the other. Scatter plots can also be
used to view correlations between attributes (Section 2.2.3). For example, Figure 2.8’s

3.3 Data Integration 97

scatter plots respectively show positively correlated data and negatively correlated data,
while Figure 2.9 displays uncorrelated data.

Note that correlation does not imply causality. That is, if A and B are correlated, this
does not necessarily imply that A causes B or that B causes A. For example, in analyzing a
demographic database, we may find that attributes representing the number of hospitals
and the number of car thefts in a region are correlated. This does not mean that one
causes the other. Both are actually causally linked to a third attribute, namely, population.

Covariance of Numeric Data
In probability theory and statistics, correlation and covariance are two similar measures
for assessing how much two attributes change together. Consider two numeric attributes
A and B, and a set of n observations {(a1,b1), . . . ,(an,bn)}. The mean values of A and B,
respectively, are also known as the expected values on A and B, that is,

E(A)= Ā=

∑n
i=1 ai
n

and

E(B)= B̄ =

∑n
i=1 bi
n

.

The covariance between A and B is defined as

Cov(A,B)= E((A− Ā)(B− B̄))=

∑n
i=1(ai − Ā)(bi − B̄)

n
. (3.4)

If we compare Eq. (3.3) for rA,B (correlation coefficient) with Eq. (3.4) for covariance,
we see that

rA,B =
Cov(A,B)

σAσB
, (3.5)

where σA and σB are the standard deviations of A and B, respectively. It can also be
shown that

Cov(A,B)= E(A ·B)− ĀB̄. (3.6)

This equation may simplify calculations.
For two attributes A and B that tend to change together, if A is larger than Ā (the

expected value of A), then B is likely to be larger than B̄ (the expected value of B).
Therefore, the covariance between A and B is positive. On the other hand, if one of
the attributes tends to be above its expected value when the other attribute is below its
expected value, then the covariance of A and B is negative.

If A and B are independent (i.e., they do not have correlation), then E(A ·B)= E(A) ·
E(B). Therefore, the covariance is Cov(A,B)= E(A ·B)− ĀB̄ = E(A) · E(B)− ĀB̄ = 0.
However, the converse is not true. Some pairs of random variables (attributes) may have
a covariance of 0 but are not independent. Only under some additional assumptions

98 Chapter 3 Data Preprocessing

Table 3.2 Stock Prices for AllElectronics and HighTech

Time point AllElectronics HighTech

t1 6 20

t2 5 10

t3 4 14

t4 3 5

t5 2 5

(e.g., the data follow multivariate normal distributions) does a covariance of 0 imply
independence.

Example 3.2 Covariance analysis of numeric attributes. Consider Table 3.2, which presents a sim-
plified example of stock prices observed at five time points for AllElectronics and
HighTech, a high-tech company. If the stocks are affected by the same industry trends,
will their prices rise or fall together?

E(AllElectronics)=
6+ 5+ 4+ 3+ 2

5
=

20

5
= $4

and

E(HighTech)=
20+ 10+ 14+ 5+ 5

5
=

54

5
= $10.80.

Thus, using Eq. (3.4), we compute

Cov(AllElectroncis,HighTech)=
6× 20+ 5× 10+ 4× 14+ 3× 5+ 2× 5

5
− 4× 10.80

= 50.2− 43.2= 7.

Therefore, given the positive covariance we can say that stock prices for both companies
rise together.

Variance is a special case of covariance, where the two attributes are identical (i.e., the
covariance of an attribute with itself). Variance was discussed in Chapter 2.

3.3.3 Tuple Duplication
In addition to detecting redundancies between attributes, duplication should also be
detected at the tuple level (e.g., where there are two or more identical tuples for a given
unique data entry case). The use of denormalized tables (often done to improve per-
formance by avoiding joins) is another source of data redundancy. Inconsistencies often
arise between various duplicates, due to inaccurate data entry or updating some but not
all data occurrences. For example, if a purchase order database contains attributes for

3.4 Data Reduction 99

the purchaser’s name and address instead of a key to this information in a purchaser
database, discrepancies can occur, such as the same purchaser’s name appearing with
different addresses within the purchase order database.

3.3.4 Data Value Conflict Detection and Resolution
Data integration also involves the detection and resolution of data value conflicts. For
example, for the same real-world entity, attribute values from different sources may dif-
fer. This may be due to differences in representation, scaling, or encoding. For instance,
a weight attribute may be stored in metric units in one system and British imperial
units in another. For a hotel chain, the price of rooms in different cities may involve
not only different currencies but also different services (e.g., free breakfast) and taxes.
When exchanging information between schools, for example, each school may have its
own curriculum and grading scheme. One university may adopt a quarter system, offer
three courses on database systems, and assign grades from A+ to F, whereas another
may adopt a semester system, offer two courses on databases, and assign grades from 1
to 10. It is difficult to work out precise course-to-grade transformation rules between
the two universities, making information exchange difficult.

Attributes may also differ on the abstraction level, where an attribute in one sys-
tem is recorded at, say, a lower abstraction level than the “same” attribute in another.
For example, the total sales in one database may refer to one branch of All Electronics,
while an attribute of the same name in another database may refer to the total sales
for All Electronics stores in a given region. The topic of discrepancy detection is further
described in Section 3.2.3 on data cleaning as a process.

3.4 Data Reduction
Imagine that you have selected data from the AllElectronics data warehouse for analysis.
The data set will likely be huge! Complex data analysis and mining on huge amounts of
data can take a long time, making such analysis impractical or infeasible.

Data reduction techniques can be applied to obtain a reduced representation of the
data set that is much smaller in volume, yet closely maintains the integrity of the original
data. That is, mining on the reduced data set should be more efficient yet produce the
same (or almost the same) analytical results. In this section, we first present an overview
of data reduction strategies, followed by a closer look at individual techniques.

3.4.1 Overview of Data Reduction Strategies
Data reduction strategies include dimensionality reduction, numerosity reduction, and
data compression.

Dimensionality reduction is the process of reducing the number of random variables
or attributes under consideration. Dimensionality reduction methods include wavelet

100 Chapter 3 Data Preprocessing

transforms (Section 3.4.2) and principal components analysis (Section 3.4.3), which
transform or project the original data onto a smaller space. Attribute subset selection is a
method of dimensionality reduction in which irrelevant, weakly relevant, or redundant
attributes or dimensions are detected and removed (Section 3.4.4).

Numerosity reduction techniques replace the original data volume by alternative,
smaller forms of data representation. These techniques may be parametric or non-
parametric. For parametric methods, a model is used to estimate the data, so that
typically only the data parameters need to be stored, instead of the actual data. (Out-
liers may also be stored.) Regression and log-linear models (Section 3.4.5) are examples.
Nonparametric methods for storing reduced representations of the data include his-
tograms (Section 3.4.6), clustering (Section 3.4.7), sampling (Section 3.4.8), and data
cube aggregation (Section 3.4.9).

In data compression, transformations are applied so as to obtain a reduced or “com-
pressed” representation of the original data. If the original data can be reconstructed
from the compressed data without any information loss, the data reduction is called
lossless. If, instead, we can reconstruct only an approximation of the original data, then
the data reduction is called lossy. There are several lossless algorithms for string com-
pression; however, they typically allow only limited data manipulation. Dimensionality
reduction and numerosity reduction techniques can also be considered forms of data
compression.

There are many other ways of organizing methods of data reduction. The computa-
tional time spent on data reduction should not outweigh or “erase” the time saved by
mining on a reduced data set size.

3.4.2 Wavelet Transforms
The discrete wavelet transform (DWT) is a linear signal processing technique that,
when applied to a data vector X, transforms it to a numerically different vector, X′, of
wavelet coefficients. The two vectors are of the same length. When applying this tech-
nique to data reduction, we consider each tuple as an n-dimensional data vector, that
is, X = (x1,x2, . . . ,xn), depicting n measurements made on the tuple from n database
attributes.3

“How can this technique be useful for data reduction if the wavelet transformed data are
of the same length as the original data?” The usefulness lies in the fact that the wavelet
transformed data can be truncated. A compressed approximation of the data can be
retained by storing only a small fraction of the strongest of the wavelet coefficients.
For example, all wavelet coefficients larger than some user-specified threshold can be
retained. All other coefficients are set to 0. The resulting data representation is therefore
very sparse, so that operations that can take advantage of data sparsity are computa-
tionally very fast if performed in wavelet space. The technique also works to remove
noise without smoothing out the main features of the data, making it effective for data

3In our notation, any variable representing a vector is shown in bold italic font; measurements depicting
the vector are shown in italic font.

3.4 Data Reduction 101

cleaning as well. Given a set of coefficients, an approximation of the original data can be
constructed by applying the inverse of the DWT used.

The DWT is closely related to the discrete Fourier transform (DFT), a signal process-
ing technique involving sines and cosines. In general, however, the DWT achieves better
lossy compression. That is, if the same number of coefficients is retained for a DWT and
a DFT of a given data vector, the DWT version will provide a more accurate approxima-
tion of the original data. Hence, for an equivalent approximation, the DWT requires less
space than the DFT. Unlike the DFT, wavelets are quite localized in space, contributing
to the conservation of local detail.

There is only one DFT, yet there are several families of DWTs. Figure 3.4 shows
some wavelet families. Popular wavelet transforms include the Haar-2, Daubechies-4,
and Daubechies-6. The general procedure for applying a discrete wavelet transform uses
a hierarchical pyramid algorithm that halves the data at each iteration, resulting in fast
computational speed. The method is as follows:

1. The length, L, of the input data vector must be an integer power of 2. This condition
can be met by padding the data vector with zeros as necessary (L ≥ n).

2. Each transform involves applying two functions. The first applies some data smooth-
ing, such as a sum or weighted average. The second performs a weighted difference,
which acts to bring out the detailed features of the data.

3. The two functions are applied to pairs of data points in X, that is, to all pairs of
measurements (x2i,x2i+1). This results in two data sets of length L/2. In general,
these represent a smoothed or low-frequency version of the input data and the high-
frequency content of it, respectively.

4. The two functions are recursively applied to the data sets obtained in the previous
loop, until the resulting data sets obtained are of length 2.

5. Selected values from the data sets obtained in the previous iterations are designated
the wavelet coefficients of the transformed data.

0 2 4 6

0.8

0.6

0.4

0.2

0.0

�1.0 �0.5 0.0 0.5
(a) Haar-2 (b) Daubechies-4

1.0 1.5 2.0

0.6

0.4

0.2

0.0

Figure 3.4 Examples of wavelet families. The number next to a wavelet name is the number of vanishing
moments of the wavelet. This is a set of mathematical relationships that the coefficients must
satisfy and is related to the number of coefficients.

102 Chapter 3 Data Preprocessing

Equivalently, a matrix multiplication can be applied to the input data in order to
obtain the wavelet coefficients, where the matrix used depends on the given DWT. The
matrix must be orthonormal, meaning that the columns are unit vectors and are mutu-
ally orthogonal, so that the matrix inverse is just its transpose. Although we do not have
room to discuss it here, this property allows the reconstruction of the data from the
smooth and smooth-difference data sets. By factoring the matrix used into a product of
a few sparse matrices, the resulting “fast DWT” algorithm has a complexity of O(n) for
an input vector of length n.

Wavelet transforms can be applied to multidimensional data such as a data cube. This
is done by first applying the transform to the first dimension, then to the second, and so
on. The computational complexity involved is linear with respect to the number of cells
in the cube. Wavelet transforms give good results on sparse or skewed data and on data
with ordered attributes. Lossy compression by wavelets is reportedly better than JPEG
compression, the current commercial standard. Wavelet transforms have many real-
world applications, including the compression of fingerprint images, computer vision,
analysis of time-series data, and data cleaning.

3.4.3 Principal Components Analysis
In this subsection we provide an intuitive introduction to principal components analy-
sis as a method of dimesionality reduction. A detailed theoretical explanation is beyond
the scope of this book. For additional references, please see the bibliographic notes
(Section 3.8) at the end of this chapter.

Suppose that the data to be reduced consist of tuples or data vectors described
by n attributes or dimensions. Principal components analysis (PCA; also called the
Karhunen-Loeve, or K-L, method) searches for k n-dimensional orthogonal vectors that
can best be used to represent the data, where k ≤ n. The original data are thus projected
onto a much smaller space, resulting in dimensionality reduction. Unlike attribute sub-
set selection (Section 3.4.4), which reduces the attribute set size by retaining a subset of
the initial set of attributes, PCA “combines” the essence of attributes by creating an alter-
native, smaller set of variables. The initial data can then be projected onto this smaller
set. PCA often reveals relationships that were not previously suspected and thereby
allows interpretations that would not ordinarily result.

The basic procedure is as follows:

1. The input data are normalized, so that each attribute falls within the same range. This
step helps ensure that attributes with large domains will not dominate attributes with
smaller domains.

2. PCA computes k orthonormal vectors that provide a basis for the normalized input
data. These are unit vectors that each point in a direction perpendicular to the others.
These vectors are referred to as the principal components. The input data are a linear
combination of the principal components.

3. The principal components are sorted in order of decreasing “significance” or
strength. The principal components essentially serve as a new set of axes for the data,

3.4 Data Reduction 103

X2

X1

Y1Y2

Figure 3.5 Principal components analysis. Y1 and Y2 are the first two principal components for the
given data.

providing important information about variance. That is, the sorted axes are such
that the first axis shows the most variance among the data, the second axis shows the
next highest variance, and so on. For example, Figure 3.5 shows the first two princi-
pal components, Y1 and Y2, for the given set of data originally mapped to the axes X1
and X2. This information helps identify groups or patterns within the data.

4. Because the components are sorted in decreasing order of “significance,” the data size
can be reduced by eliminating the weaker components, that is, those with low vari-
ance. Using the strongest principal components, it should be possible to reconstruct
a good approximation of the original data.

PCA can be applied to ordered and unordered attributes, and can handle sparse data
and skewed data. Multidimensional data of more than two dimensions can be han-
dled by reducing the problem to two dimensions. Principal components may be used
as inputs to multiple regression and cluster analysis. In comparison with wavelet trans-
forms, PCA tends to be better at handling sparse data, whereas wavelet transforms are
more suitable for data of high dimensionality.

3.4.4 Attribute Subset Selection
Data sets for analysis may contain hundreds of attributes, many of which may be irrel-
evant to the mining task or redundant. For example, if the task is to classify customers
based on whether or not they are likely to purchase a popular new CD at AllElectronics
when notified of a sale, attributes such as the customer’s telephone number are likely to
be irrelevant, unlike attributes such as age or music taste. Although it may be possible for
a domain expert to pick out some of the useful attributes, this can be a difficult and time-
consuming task, especially when the data’s behavior is not well known. (Hence, a reason
behind its analysis!) Leaving out relevant attributes or keeping irrelevant attributes may
be detrimental, causing confusion for the mining algorithm employed. This can result
in discovered patterns of poor quality. In addition, the added volume of irrelevant or
redundant attributes can slow down the mining process.

104 Chapter 3 Data Preprocessing

Attribute subset selection4 reduces the data set size by removing irrelevant or
redundant attributes (or dimensions). The goal of attribute subset selection is to find
a minimum set of attributes such that the resulting probability distribution of the data
classes is as close as possible to the original distribution obtained using all attributes.
Mining on a reduced set of attributes has an additional benefit: It reduces the number
of attributes appearing in the discovered patterns, helping to make the patterns easier to
understand.

“How can we find a ‘good’ subset of the original attributes?” For n attributes, there are
2n possible subsets. An exhaustive search for the optimal subset of attributes can be pro-
hibitively expensive, especially as n and the number of data classes increase. Therefore,
heuristic methods that explore a reduced search space are commonly used for attribute
subset selection. These methods are typically greedy in that, while searching through
attribute space, they always make what looks to be the best choice at the time. Their
strategy is to make a locally optimal choice in the hope that this will lead to a globally
optimal solution. Such greedy methods are effective in practice and may come close to
estimating an optimal solution.

The “best” (and “worst”) attributes are typically determined using tests of statistical
significance, which assume that the attributes are independent of one another. Many
other attribute evaluation measures can be used such as the information gain measure
used in building decision trees for classification.5

Basic heuristic methods of attribute subset selection include the techniques that
follow, some of which are illustrated in Figure 3.6.

Forward selection

Initial attribute set:
{A1, A2, A3, A4, A5, A6}

Initial reduced set:
{}
=> {A1}
=> {A1, A4}
=> Reduced attribute set:
{A1, A4, A6}

Initial attribute set:
{A1, A2, A3, A4, A5, A6}

=> {A1, A3, A4, A5, A6}
=> {A1, A4, A5, A6}
=> Reduced attribute set:
{A1, A4, A6}

Initial attribute set:
{A1, A2, A3, A4, A5, A6}

=> Reduced attribute set:
{A1, A4, A6}

Backward elimination Decision tree induction

A4?

A1? A6?

Class 1 Class 2 Class 1 Class 2

Y N

Y N Y N

Figure 3.6 Greedy (heuristic) methods for attribute subset selection.

4In machine learning, attribute subset selection is known as feature subset selection.
5The information gain measure is described in detail in Chapter 8.

3.4 Data Reduction 105

1. Stepwise forward selection: The procedure starts with an empty set of attributes as
the reduced set. The best of the original attributes is determined and added to the
reduced set. At each subsequent iteration or step, the best of the remaining original
attributes is added to the set.

2. Stepwise backward elimination: The procedure starts with the full set of attributes.
At each step, it removes the worst attribute remaining in the set.

3. Combination of forward selection and backward elimination: The stepwise for-
ward selection and backward elimination methods can be combined so that, at each
step, the procedure selects the best attribute and removes the worst from among the
remaining attributes.

4. Decision tree induction: Decision tree algorithms (e.g., ID3, C4.5, and CART) were
originally intended for classification. Decision tree induction constructs a flowchart-
like structure where each internal (nonleaf) node denotes a test on an attribute, each
branch corresponds to an outcome of the test, and each external (leaf) node denotes a
class prediction. At each node, the algorithm chooses the “best” attribute to partition
the data into individual classes.

When decision tree induction is used for attribute subset selection, a tree is con-
structed from the given data. All attributes that do not appear in the tree are assumed
to be irrelevant. The set of attributes appearing in the tree form the reduced subset
of attributes.

The stopping criteria for the methods may vary. The procedure may employ a threshold
on the measure used to determine when to stop the attribute selection process.

In some cases, we may want to create new attributes based on others. Such attribute
construction6 can help improve accuracy and understanding of structure in high-
dimensional data. For example, we may wish to add the attribute area based on the
attributes height and width. By combining attributes, attribute construction can dis-
cover missing information about the relationships between data attributes that can be
useful for knowledge discovery.

3.4.5 Regression and Log-Linear Models: Parametric
Data Reduction

Regression and log-linear models can be used to approximate the given data. In (simple)
linear regression, the data are modeled to fit a straight line. For example, a random
variable, y (called a response variable), can be modeled as a linear function of another
random variable, x (called a predictor variable), with the equation

y = wx+ b, (3.7)

where the variance of y is assumed to be constant. In the context of data mining, x and y
are numeric database attributes. The coefficients, w and b (called regression coefficients),

6In the machine learning literature, attribute construction is known as feature construction.

106 Chapter 3 Data Preprocessing

specify the slope of the line and the y-intercept, respectively. These coefficients can
be solved for by the method of least squares, which minimizes the error between the
actual line separating the data and the estimate of the line. Multiple linear regression
is an extension of (simple) linear regression, which allows a response variable, y, to be
modeled as a linear function of two or more predictor variables.

Log-linear models approximate discrete multidimensional probability distributions.
Given a set of tuples in n dimensions (e.g., described by n attributes), we can con-
sider each tuple as a point in an n-dimensional space. Log-linear models can be used
to estimate the probability of each point in a multidimensional space for a set of dis-
cretized attributes, based on a smaller subset of dimensional combinations. This allows
a higher-dimensional data space to be constructed from lower-dimensional spaces.
Log-linear models are therefore also useful for dimensionality reduction (since the
lower-dimensional points together typically occupy less space than the original data
points) and data smoothing (since aggregate estimates in the lower-dimensional space
are less subject to sampling variations than the estimates in the higher-dimensional
space).

Regression and log-linear models can both be used on sparse data, although their
application may be limited. While both methods can handle skewed data, regression
does exceptionally well. Regression can be computationally intensive when applied to
high-dimensional data, whereas log-linear models show good scalability for up to 10 or
so dimensions.

Several software packages exist to solve regression problems. Examples include SAS
(www.sas.com), SPSS (www.spss.com), and S-Plus (www.insightful.com). Another useful
resource is the book Numerical Recipes in C, by Press, Teukolsky, Vetterling, and Flannery
[PTVF07], and its associated source code.

3.4.6 Histograms
Histograms use binning to approximate data distributions and are a popular form
of data reduction. Histograms were introduced in Section 2.2.3. A histogram for an
attribute, A, partitions the data distribution of A into disjoint subsets, referred to as
buckets or bins. If each bucket represents only a single attribute–value/frequency pair, the
buckets are called singleton buckets. Often, buckets instead represent continuous ranges
for the given attribute.

Example 3.3 Histograms. The following data are a list of AllElectronics prices for commonly sold
items (rounded to the nearest dollar). The numbers have been sorted: 1, 1, 5, 5, 5,
5, 5, 8, 8, 10, 10, 10, 10, 12, 14, 14, 14, 15, 15, 15, 15, 15, 15, 18, 18, 18, 18, 18,
18, 18, 18, 20, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 25, 25, 25, 25, 25, 28, 28, 30,
30, 30.

Figure 3.7 shows a histogram for the data using singleton buckets. To further reduce
the data, it is common to have each bucket denote a continuous value range for
the given attribute. In Figure 3.8, each bucket represents a different $10 range for
price.

3.4 Data Reduction 107

5 10

10

9

8

7

6

5

4

3

2

1

0
15 20 25 30

price ($)

co
u
n
t

Figure 3.7 A histogram for price using singleton buckets—each bucket represents one price–value/
frequency pair.

1–10 11–20 21–30

price ($)

co
u
n
t

25

20

15

10

5

0

Figure 3.8 An equal-width histogram for price, where values are aggregated so that each bucket has a
uniform width of $10.

“How are the buckets determined and the attribute values partitioned?” There are
several partitioning rules, including the following:

Equal-width: In an equal-width histogram, the width of each bucket range is
uniform (e.g., the width of $10 for the buckets in Figure 3.8).

Equal-frequency (or equal-depth): In an equal-frequency histogram, the buckets are
created so that, roughly, the frequency of each bucket is constant (i.e., each bucket
contains roughly the same number of contiguous data samples).

108 Chapter 3 Data Preprocessing

Histograms are highly effective at approximating both sparse and dense data, as
well as highly skewed and uniform data. The histograms described before for single
attributes can be extended for multiple attributes. Multidimensional histograms can cap-
ture dependencies between attributes. These histograms have been found effective in
approximating data with up to five attributes. More studies are needed regarding the
effectiveness of multidimensional histograms for high dimensionalities.

Singleton buckets are useful for storing high-frequency outliers.

3.4.7 Clustering
Clustering techniques consider data tuples as objects. They partition the objects into
groups, or clusters, so that objects within a cluster are “similar” to one another and “dis-
similar” to objects in other clusters. Similarity is commonly defined in terms of how
“close” the objects are in space, based on a distance function. The “quality” of a cluster
may be represented by its diameter, the maximum distance between any two objects in
the cluster. Centroid distance is an alternative measure of cluster quality and is defined
as the average distance of each cluster object from the cluster centroid (denoting the
“average object,” or average point in space for the cluster). Figure 3.3 showed a 2-D plot
of customer data with respect to customer locations in a city. Three data clusters are
visible.

In data reduction, the cluster representations of the data are used to replace the actual
data. The effectiveness of this technique depends on the data’s nature. It is much more
effective for data that can be organized into distinct clusters than for smeared data.

There are many measures for defining clusters and cluster quality. Clustering meth-
ods are further described in Chapters 10 and 11.

3.4.8 Sampling
Sampling can be used as a data reduction technique because it allows a large data set to
be represented by a much smaller random data sample (or subset). Suppose that a large
data set, D, contains N tuples. Let’s look at the most common ways that we could sample
D for data reduction, as illustrated in Figure 3.9.

Simple random sample without replacement (SRSWOR) of size s: This is created
by drawing s of the N tuples from D (s < N), where the probability of drawing any tuple in D is 1/N , that is, all tuples are equally likely to be sampled. Simple random sample with replacement (SRSWR) of size s: This is similar to SRSWOR, except that each time a tuple is drawn from D, it is recorded and then replaced. That is, after a tuple is drawn, it is placed back in D so that it may be drawn again. Cluster sample: If the tuples in D are grouped into M mutually disjoint “clusters,” then an SRS of s clusters can be obtained, where s