Content uploaded by Yoav Benjamini

Author content

All content in this area was uploaded by Yoav Benjamini

Content may be subject to copyright.

Chapter 1

STATISTICAL METHODS FOR DATA MINING

Yoav Benjamini

Department of Statistics, School of Mathematical Sciences, Sackler Faculty for Exact

Sciences

Tel Aviv University

ybenja@post.tau.ac.il

Moshe Leshno

Faculty of Management and Sackler Faculty of Medicine

Tel Aviv University

leshnom@post.tau.ac.il

Abstract The aim of this chapter is to present the main statistical issues in Data

mining (DM) and Knowledge Data Discovery (KDD) and to examine

whether traditional statistics approach and methods substantially diﬀer

from the new trend of KDD and DM. We address and emphasize some

central issues of statistics which are highly relevant to DM and have

much to oﬀer to DM.

Keywords: Statistics, Regression Models, False Discovery Rate (FDR), Model se-

lection and False Discovery Rate (FDR)

1. Introduction

In the words of anonymous saying there are two problems in mod-

ern science: too many people using diﬀerent terminology to solve the

same problems and even more people using the same terminology to

address completely diﬀerent issues. This is particularly relevant to the

relationship between traditional statistics and the new emerging ﬁeld of

knowledge data discovery (KDD) and data mining (DM). The explosive

growth of interest and research in the domain of KDD and DM of recent

1

2

years is not surprising given the proliferation of low-cost computers and

the requisite software, low-cost database technology (for collecting and

storing data) and the ample data that has been and continues to be

collected and organized in databases and on the web. Indeed, the imple-

mentation of KDD and DM in business and industrial organizations has

increased dramatically, although their impact on these organizations is

not clear. The aim of this chapter is mainly to present the main sta-

tistical issues in DM and KDD and to examine the role of traditional

statistics approach and methods in the new trend of KDD and DM. We

argue that data miners should be familiar with statistical themes and

models and statisticians should be aware of the capabilities and limi-

tation of data mining and the ways in which data mining diﬀers from

traditional statistics.

Statistics is the traditional ﬁeld that deals with the quantiﬁcation,

collection, analysis, interpretation, and drawing conclusions from data.

Data mining is an interdisciplinary ﬁeld that draws on computer sci-

ences (data base, artiﬁcial intelligence, machine learning, graphical and

visualization models), statistics and engineering (pattern recognition,

neural networks). DM involves the analysis of large existing data bases

in order to discover patterns and relationships in the data, and other

ﬁndings (unexpected, surprising, and useful). Typically, it diﬀers from

traditional statistics on two issues: the size of the data set and the fact

that the data were initially collected for purpose other than the that

of the DM analysis. Thus, experimental design, a very important topic

in traditional statistics, is usually irrelevant to DM. On the other hand

asymptotic analysis, sometimes criticized in statistics as being irrelevant,

becomes very relevant in DM.

While in traditional statistics a data set of 100 to 10

4

entries is con-

sidered large, in DM even 10

4

may be considered a small set set ﬁt to

be used as an example, rather than a problem encountered in practice.

Problem sizes of 10

7

to 10

10

are more typical. It is important to empha-

size, though, that data set sizes are not all created equal. One needs to

distinguish between the number of cases (observations) in a large data

set (n), and the number of features (variables) available for each case

(m). In a large data set,n ,m or both can be large, and it do es matter

which, a point on which we will elaborate in the continuation. Moreover

these deﬁnitions may change when the same data set is being used for

two diﬀerent purposes. A nice demonstration of such an instance can

be found in the 2001 KDD competition, where in one task the number

of cases was the number of purchasing customers, the click information

being a subset of the features, and in the other task the clicks were the

cases.

Statistical Methods for Data Mining 3

Our aim in this chapter is to indicate certain focal areas where sta-

tistical thinking and practice have much to oﬀer to DM. Some of them

are well known, whereas others are not. We will cover some of them

in depth, and touch upon others only marginally. We will address the

following issues which are highly relevant to DM:

Size

Curse of Dimensionality

Assessing uncertainty

Automated analysis

Algorithms for data analysis in Statistics

Visualization

Scalability

Sampling

Modelling relationships

Model selection

We brieﬂy discuss these issues in the next section and then devote

special sections to three of them. In section 3 we explain and present

how the most basic of statistical metho dologies, namely regression anal-

ysis, has developed over the years to create a very ﬂexible tool to model

relationships, in the form of Generalized Linear Models (GLMs). In sec-

tion 4 we discuss the False Discovery Rate (FDR) as a scalable approach

to hypothesis testing. In section 5 we discuss how FDR ideas contribute

to ﬂexible model selection in GLM. We conclude the chapter by asking

whether the concepts and methods of KDD and DM diﬀer from those of

traditional statistical, and how statistics and DM should act together.

2. Statistical Issues in DM

2.1 Size of the Data and Statistical Theory

Traditional statistics emphasizes the mathematical formulation and

validation of a methodology, and views simulations and empirical or

practical evidence as a less form of validation. The emphasis on rigor

has required proof that a proposed method will work prior to its use.

In contrast, computer science and machine learning use exp erimental

validation methods. In many cases mathematical analysis of the per-

formance of a statistical algorithm is not feasible in a speciﬁc setting,

4

but becomes so when analyzed asymptotically. At the same time, when

size becomes extremely large, studying performance by simulations is

also not feasible. It is therefore in settings typical of DM problems that

asymptotic analysis becomes both feasible and appropriate. Interest-

ingly, in classical asymptotic analysis the number of cases n tends to

inﬁnity. In more contemporary literature there is a shift of emphasis to

asymptotic analysis where the number of variables m tends to inﬁnity.

It is a shift that has occurred because of the interest of statisticians and

applied mathematicians in wavelet analysis (see Chapter ...), where the

number of parameters (wavelet coeﬃcients) equals the number of cases,

and has proved highly successful in areas such as the analysis of gene

expression data from microarrays.

2.2 The curse of dimensionality and approaches

to address it

The curse of dimensionality is a well documented and often cited

fundamental problem. Not only do algorithms face more diﬃculties as

the the data increases in dimension, but the structure of the data it-

self changes. Take, for example, data uniformly distributed in a high-

dimensional ball. It turns out that (in some precise way, see Meilijson,

1991) most of the data points are very close to the surface of the ball.

This phenomenon becomes very evident when looking for the k-Nearest

Neighbors of a point in high-dimensional space. The points are so far

away from each other that the radius of the neighborhood b ecomes ex-

tremely large.

The main remedy oﬀered for the curse of dimensionality is to use only

part of the available variables per case, or to combine variables in the

data set in a way that will summarize the relevant information with

fewer variables. This dimension reduction is the essence of what goes

on in the data warehousing stage of the DM process, along with the

cleansing of the data. It is an important and time-consuming stage of

the DM operations, accounting for 80-90% of the time devoted to the

analysis.

The dimension reduction comprises two types of activities: the ﬁrst

is quantifying and summarizing information into a number of variables,

and the second is further reducing the variables thus constructed into a

workable number of combined variables. Consider, for instance, a phone

company that has at its disposal the entire history of calls made by a

customer. How should this history be reﬂected in just a few variables?

Should it be by monthly summaries of the number of calls per month

for each of the last 12 months, such as their means (or medians), their

Statistical Methods for Data Mining 5

maximal number, and a certain percentile? Maybe we should use the

mean, standard deviation and the number of calls below two standard

deviations from the mean? Or maybe we should use none of these but

rather variables capturing the monetary values of the activity? If we

take this last approach, should we work with the cost itself or will it be

more useful to transfer the cost data to the log scale? Statistical theory

and practice have much to oﬀer in this respect, both in measurement

theory, and in data analysis practices and tools. The variables thus

constructed now have to be further reduced into a workable number of

combined variables. This stage may still involve judgmental combination

of previously deﬁned variables, such as cost per number of customers

using a phone lines, but more often will require more automatic methods

such as principal components or independent components analysis (for

a further discussion of principle component analysis see Roberts and

Everson, 2001).

We cannot conclude the discussion on this topic without noting that

occasionally we also start getting the blessing of dimensionality, a term

coined by David Donoho (Donoho, 2000) to describe the phenomenon of

the high dimension helping rather than hurting that we often encounter

as we proceed up the scale in working with very high dimensional data.

For example, for large m if the data we study is pure noise, the i-th

largest observation is very close to its expectations under the model

for the noise! Another case in point is microarray analysis, where the

many non-relevant genes analyzed give ample information about the

distribution of the noise, making it easier to identify real discoveries.

We shall see a third case below.

2.3 Assessing uncertainty

Assessing the uncertainty surrounding knowledge derived from data

is recognized as a the central theme in statistics. The concern about

the uncertainty is down-weighted in KDD, often because of the myth

that all relevant data is available in DM. Thus, standard errors of av-

erages, for example, will be ridiculously low, as will prediction errors.

On the other hand experienced users of DM tools are aware of the vari-

ability and uncertainty involved. They simply tend to rely on seemingly

”non-statistical” technologies such as the use of a training sample and

a test sample. Interestingly the latter is a methodology widely used in

statistics, with origins going back to the 1950s. The use of such valida-

tion methods, in the form of cross-validation for smaller data sets, has

been a common practice in exploratory data analysis when dealing with

medium size data sets.

6

Some of the insights gained over the years in statistics regarding the

use of these tools have not yet found their way into DM. Take, for

example, data on food store baskets, available for the last four years,

where the goal is to develop a prediction model. A typical analysis will

involve taking a random training sample from the data, then testing the

model on the training sample, with the results guiding us as to the choice

of the most appropriate model. However, the model will be used next

year, not last year. The main uncertainty surrounding its conclusions

may not stem from the person to person variability captured by the

diﬀerences between the values in the training sample, but rather follow

from the year to year variability. If this is the case, we have all the

data, but only four observations. The choice of the data for validation

and training samples should reﬂect the higher sources of variability in

the data, by each time setting the data of one year aside to serve as the

source for the test sample (for an illustrated yet profound discussion of

these issues in exploratory data analysis see Mosteller and Tukey, 1977,

Ch. 7,8).

2.4 Automated analysis

The inherent dangers of the necessity to rely on automatic strategies

for analyzing the data, another main theme in DM, have been demon-

strated again and again. There are many examples where trivial non-

relevant variables, such as case number, turned out to be the best predic-

tors in automated analysis. Similarly, variables displaying a major role

in predicting a variable of interest in the past, may turn out to be useless

because they reﬂect some strong phenomenon not expected to occur in

the future (see for example the conclusions using the onion metaphor

from the 2002 KDD competition). In spite of these warnings, it is clear

that large parts of the analysis should be automated, especially at the

warehousing stage of the DM.

This may raise new dangers. It is well known in statistics that having

even a small proportion of outliers in the data can seriously distort its

numerical summary. Such unreasonable values, deviating from the main

structure of the data, can usually b e identiﬁed by a careful human data

analyst, and excluded from the analysis. But once we have to warehouse

information about millions of customers, summarizing the information

about each customer by a few numbers has to be automated and the

analysis should rather deal automatically with the possible impact of a

few outliers.

Statistical theory and methodology supply the framework and the

tools for this endeavor. A numerical summary of the data that is not

Statistical Methods for Data Mining 7

unboundedly inﬂuenced by a negligible proportion of the data is called

a resistant summary. According to this deﬁnition the average is not re-

sistant, for even one straying data value can have an unbounded eﬀect

on it. In contrast, the median is resistant. A resistant summary that

retains its good properties under less than ideal situations is called a

robust summary, the α-trimmed mean (rather than the median) being

an example of such. The concepts of robustness and resistance, and the

development of robust statistical tools for summarizing location, scale,

and relationships, were developed during the 1970’s and the 1980’s, and

resulting theory is quite mature (see, for instance, Ronchetti (Ronchetti

et al., 1986; Dell’Aquila and Ronchetti, 2004), even though robustness

remains an active area of contemporary research in statistics. Robust

summaries, rather than merely averages, standard deviations, and simple

regression coeﬃcients, are indispensable in DM. Here too, some adap-

tation of the computations to size may be needed, but eﬀorts in this

direction are being made in the statistical literature.

2.5 Algorithms for data analysis in statistics

Computing has always been a fundamental to statistic, and it re-

mained so even in times when mathematical rigorousity was most highly

valued quality of a data analytic tool. Some of the important computa-

tional tools for data analysis, rooted in classical statistics, can be found

in the following list: eﬃcient estimation by maximum likelihood, least

squares and least absolute deviation estimation, and the EM algorithm;

analysis of variance (ANOVA, MANOVA, ANCOVA), and the analy-

sis of repeated measurements; nonparametric statistics; log-linear anal-

ysis of categorial data; linear regression analysis, generalized additive

and linear models, logistic regression, survival analysis, and discrimi-

nant analysis; frequency domain (spectrum) and time domain (ARIMA)

methods for the analysis of time series; multivariate analysis tools such as

factor analysis, principal component and later independent component

analyses, and cluster analysis; density estimation, smoothing and de-

noising, and classiﬁcation and regression trees (decision trees); Bayesian

networks and the Monte Carlo Markov Chain (MCMC) algorithm for

Bayesian inference.

For an overview of most of these topics, with an eye to the DM commu-

nity see Hastie, Tibshirani and Friedman, 2001. Some of the algorithms

used in DM which were not included in classical statistic, are considered

by some statisticians to be part of statistics (Friedman, 1998). For exam-

ple, rule induction (AQ, CN2, Recon, etc.), associate rules, neural net-

8

works, genetic algorithms and self-organization maps may be attributed

to classical statistics.

2.6 Visualization

Visualization of the data and its structure, as well as visualization of

the conclusions drawn from the data, are another central theme in DM.

Visualization of quantitative data as a major activity ﬂourished in the

statistics of the 19th century, faded out of favor through most of the 20th

century, and began to regain importance in the early 1980s. This impor-

tance in reﬂected in the development of the Journal of Computational

and Graphical Statistics of the American Statistical Association. Both

the theory of visualizing quantitative data and the practice have dramat-

ically changed in recent years. Spinning data to gain a 3-dimensional

understanding of pointclouds, or the use of projection pursuit are just

two examples of visualization technologies that emerged from statistics.

It is therefore quite frustrating to see how much KDD software de-

viates from known principles of good visualization practices. Thus, for

instance the fundamental principle that the retinal variable in a graphi-

cal display (length of line, or the position of a point on a scale) should be

proportional to the quantitative variable it represents is often violated

by introducing a dramatic p erspective. Add colors to the display and

the result is even harder to understand.

Much can be gained in DM by mining the knowledge about visualiza-

tion available in statistics, though the visualization tools of statistics are

usually not calibrated for the size of the data sets commonly dealt within

DM. Take for example the extremely eﬀective Boxplots display, used for

the visual comparisons of batches of data. A well-known rule determines

two fences for each batch, and points outside the fences are individually

displayed. There is a traditional default value in most statistical soft-

ware, even though the rule was developed with batches of very small size

in mind (in DM terms). In order to adapt the visualization technique

for routine use in DM, some other rule which will probably be adaptive

to the size of the batch should be developed. As this small example

demonstrates, visualization is an area where joint work may prove to be

extremely fruitful.

2.7 Scalability

In machine learning and data mining scalability relates to the ability of

an algorithm to scale up with size, an essential condition being that the

storage requirement and running time should not become infeasible as

the size of the problem increases. Even simple problems like multivariate

Statistical Methods for Data Mining 9

histograms become a serious task, and may beneﬁt from complex algo-

rithms that scale up with size. Designing scalable algorithms for more

complex tasks, such as decision tree modeling, optimization algorithms,

and the mining of association rules, has been the most active research

area in DM. Altogether, scalability is clearly a fundamental problem in

DM mostly viewed with regard to its algorithmic aspects. We want to

highlight the duality of the problem by suggesting that concepts should

be scalable as well. In this respect, consider the general belief that hy-

pothesis testing is a statistical concept that has nothing to oﬀer in DM.

The usual argument is that data sets are so large that every hypothesis

tested will turn out to be statistically signiﬁcant - even if diﬀerences or

relationships are minuscule. Using association rules as an example, one

may wonder whether an observed lift for a given rule is ”really diﬀer-

ent from 1”, but then ﬁnd that at the traditional level of signiﬁcance

used (the mythological 0.05) an extremely large number of rules are in-

deed signiﬁcant. Such ﬁndings brought David Hand (Hand, 1998) to

ask ”what should replace hypothesis testing?” in DM. We shall discuss

two such important scalable concepts in the continuation: the testing

of multiple hypotheses using the False Discovery Rate and the penalty

concept in model selection.

2.8 Sampling

Sampling is the ultimate scalable statistical tool: if the number of

cases n is very large the conclusions drawn from the sample depend only

on the size of the sample and not on the size of the data set. It is often

used to get a ﬁrst impression of the data, visualize its main features, and

reach decisions as to the strategy of analysis. In spite of its scalability

and usefulness sampling has been attacked in the KDD community for its

inability to ﬁnd very rare yet extremely interesting pieces of knowledge.

Sampling is a very well developed area of statistics (see for example

Cochran, 1977), but is usually used in DM at the very basic level. Strat-

iﬁed sampling, where the probability of picking a case changes from one

stratum to another, is hardly ever used. But the questions are relevant

even in the simplest settings: should we sample from the few positive re-

sponses at the same rate that we sample from the negative ones? When

studying faulty loans, should we sample larger loans at a higher rate?

A thorough investigations of such questions, phrased in the realm of

particular DM applications may prove to be very beneﬁcial.

Even greater beneﬁts might be realized when more advanced sampling

models, especially those related to super populations, are utilized in

DM. The idea here is that the population of customers we view each

10

year, and from which we sample, can itself b e viewed as a sample of

the same super population. Hence next year’s customers will again be a

population sampled from the super population. We leave this issue wide

open.

3. Modeling Relationships using Regression

Models

Demonstrating that statistics, like data mining, is concerned with

turning data into information and knowledge, even though the terminol-

ogy may diﬀer, in this section we present a major statistical approach

being used in data mining, namely regression analysis. In the late 1990s,

statistical methodologies such as regression analysis were not included

in commercial data mining packages. Nowadays, most commercial data

mining software includes many statistical to ols and in particular re-

gression analysis. Although regression analysis may seem simple and

anachronistic, it is a very p owerful tool in DM with large data sets,

especially in the form of the generalized linear models (GLMs). We

emphasize the assumptions of the models being used and how the un-

derlying approach diﬀers from that of machine learning. The reader

is referred to McCullagh and Nelder, 1991 and Chapters ... for more

detailed information on the speciﬁc statistical methods.

3.1 Linear Regression Analysis

Regression analysis is the process of determining how a variable y

is related to one, or more, other variables x

1

, . . . , x

k

. The y is usually

called the dependent variable and the x

i

’s are called the independent or

explanatory variables. In a linear regression model we assume that

y

i

= β

0

+

k

X

j=1

β

j

x

ji

+ ε

i

i = 1, . . . , M (1.1)

and that the ε

i

’s are independent and are identically distributed as

N (0, σ

2

) and M is the number of data points. The expected value of y

i

is given by

E(y

i

) = β

0

+

k

X

j=1

β

j

x

ji

(1.2)

To estimate the coeﬃcients of the linear regression model we use the

least square estimation which gives results equivalent to the estimators

obtained by the maximum likelihood method. Note that for the linear

regression model there is an explicit formula of the β’s. We can write

Statistical Methods for Data Mining 11

(1.1) in matrix form by Y = X · β

t

+ ε

t

where β is the transpose of the

vector [β

0

, β

1

, . . . , β

k

], ε is the transpose of the vector ε= [ε

1

, . . . , ε

M

]

and the matrix X is given by

X =

1 x

11

· · · x

1k

1 x

21

· · · x

2k

1

.

.

.

.

.

.

.

.

.

1 x

k1

· · · x

Mk

(1.3)

The estimates of the β’s are given (in matrix form) by

ˆ

β=(X

t

X)

−1

X

t

Y .

Note that in linear regression analysis we assume that for a given x

1

, . . . , x

k

y

i

is distributed as N (β

0

+

P

k

j=1

β

j

x

ji

, σ

2

). There is a large class of gen-

eral regression models where the relationship between the y

i

s and the

vector x is not assumed to be linear, that can be converted to a linear

model.

Machine learning approach compared to regression analysis aims to

select a function f ∈ F from a given set of functions F, that best approx-

imates or ﬁts the given data. Machine learning assumes that the given

data (x

i

, y

i

), (i = 1, . . . , M) is obtained by a data generator, producing

the data according to an unknown distribution p(x, y) = p(x)p(y|x).

Given a loss function Ψ(y − f(x)), the quality of an approximation pro-

duced by the machine learning is measured by the expected loss, the

expectation being below the unknown distribution p(x, y). The subject

of statistical machine learning is the following optimization problem:

min

f∈F

Z

Ψ(y − f(x))dp(x, y) (1.4)

when the density function p(x, y) is unknown but a random indepen-

dent sample of (x

i

, y

i

) is given. The problem of minimizing (1.4) on

the basis of the data is the subject of statistical machine learning. If

F is the set of all linear function of x and Ψ(y − f(x)) = (y − f(x))

2

then if p(y|x) is normally distributed then the minimization of (1.4) is

equivalent to linear regression analysis.

3.2 Generalized Linear Models

Although in many cases the set of linear function is good enough to

model the relationship between the stochastic response y as a function

of x it may not always suﬃce to represent the relationship. The general-

ized linear model increases the family of functions F that may represent

the relationship between the response y and x. The tradeoﬀ is between

12

having a simple model and a more complex model representing the rela-

tionship between y and x. In the general linear model the distribution of

y given x does not have to be normal, but can be any of the distributions

in the exponential family (see McCullagh and Nelder, 1991). Instead of

the expected value of y|x being a linear function, we have

g(E(y

i

)) = β

0

+

k

X

j=1

β

j

x

ji

(1.5)

where g(·) is a monotone diﬀerentiable function.

In the generalized additive models, g(E(y

i

)) need not to be a linear

function of x but has the form:

g(E(y

i

)) = β

0

+

k

X

j=1

σ

j

(x

ji

) (1.6)

where σ(·)’s are smooth functions. Note that neural networks are a

special case of the generalized additive linear models. For example the

function that a multilayer feedforward neural network with one hidden

layer computes is (see Chapter ... for detailed information):

y

i

= f(x) =

m

X

l=1

β

j

· σ

k

X

j=1

w

jl

x

ji

− θ

j

(1.7)

where m is the number of processing-units in the hidden layer. The

family of functions that can be computed depends on the numb er of

neurons in the hidden layer and the activation function σ. Note that

a standard multilayer feedforward network with a smooth activation

function σ can approximate any continuous function on a compact set

to any degree of accuracy if and only if the network’s activation function

σ is not a polynomial (Leshno et al., 1993).

There are methods for ﬁtting generalized additive models. However,

unlike linear models for which there exits a framework of statistical in-

ference, for machine learning algorithms as well as generalized additive

methods, no such framework have yet been developed. For example,

using a statistical inference framework in linear regression one can test

the hypothesis that all or part of the coeﬃcients are zero.

The total sum of squares (SST ) is equal to the sum of squares due to

regression (SSR ) plus the residual sum of square (RSS

k

), i.e.

Statistical Methods for Data Mining 13

M

X

i=1

(y

i

− y)

2

| {z }

SST

=

M

X

i=1

( ˆy

i

− y)

2

| {z }

SSR

+

M

X

i=1

(y

i

− ˆy

i

)

2

| {z }

RSS

k

(1.8)

The percentage of variance explained by the regression is a very popu-

lar method to measure the goodness-of-ﬁt of the model. More speciﬁcally

R

2

and the adjusted R

2

deﬁned below are used to measure the goodness

of ﬁt.

R

2

=

P

M

i=1

(ˆy

i

− y)

2

P

M

i=1

(y

i

− y)

2

= 1 −

RSS

k

SST

(1.9)

Adjusted-R

2

= 1 − (1 − R

2

)

M − 1

M − k − 1

(1.10)

We next turn to a special case of the general additive model that is

very popular and powerful tool in cases where the responses are binary

values.

3.3 Logistic regression

In logistic regression the y

i

s are binary variables and thus not nor-

mally distributed. The distribution of y

i

given x is assumed to follow a

binomial distribution such that:

log

µ

p(y

i

= 1|x)

1 − p(y

i

= 1|x)

¶

= β

0

+

k

X

j=1

β

j

x

ji

(1.11)

If we denote π(x) = p(y = 1|x) and the real valued function g(t) =

t

1−t

then g(π(x)) is a linear function of x. Note that we can write y = π(x)+ε

such that if y = 1 then ε = 1 − π(x) with probability π(x), and if y = 0

then ε = −π(x) with probability 1 − π(x). Thus, π(x) = E(y|x) and

π(x) =

e

β

0

+

P

k

j=1

β

j

x

j

1 + e

β

0

+

P

k

j=1

β

j

x

j

(1.12)

Of the several methods to estimates the β’s, the method of maximum

likelihood is one most commonly used in the logistic regression routine

of the major software packages.

In linear regression, interest focuses on the size of R

2

or adjusted-R

2

.

The guiding principle in logistic regression is similar: the comparison of

observed to predicted values is based on the log likelihood function. To

14

compare two models - a full model and a reduced model, one uses the

following likelihood ratio:

D = −2 ln

µ

likelihod of the reduced model

likelihod of the full model

¶

(1.13)

The statistic D in equation (1.13), is called the deviance (McCullagh

and Nelder, 1991). Logistic regression is a very powerful tool for classiﬁ-

cation problems in discriminant analysis and is applied in many medical

and clinical research studies.

3.4 Survival analysis

Survival analysis addresses the question of how long it takes for a

particular event to happen. In many medical applications the most im-

portant response variable often involves time; the event is some hazard

or death and thus we analyze the patient’s survival time. In business

application the event may be a failure of a machine or market entry

of a competitor. There are two main characteristics of survival anal-

ysis that make it diﬀerent from regression analysis. The ﬁrst is that

the presence of censored observation, where the event (e.g. death) has

not necessarily occurred by the end of the study. Censored observa-

tion may also occur when patients are lost to follow-up for one reason

or another. If the output is censored, we do not have the value of the

output, but we do have some information about it. The second, is that

the distribution of survival times is often skewed or far from normal-

ity. These features require special methods of analysis of survival data,

two functions describing the distribution of survival times being of cen-

tral importance: the hazard function and the survival function. Using

T to represent survival time, the survival function denoted by S(t), is

deﬁned as the probability of survival time to be greater than t, i.e.

S(t) = Pr(T > t) = 1 − F (t), where F (t) is the cumulative distribu-

tion function of the output. The hazard function, h(t), is deﬁned as the

probability density of the output at time t conditional upon survival to

time t, that is h(t) = f (t)/S(t), where f(t) is the probability density

of the output. Is is also known as the instantaneous failure rate and

presents the probability that an event will happen in a small time in-

terval ∆t, given that the individual has survived up to the beginning of

this interval, i.e. h(t) = lim

∆t↓0

Pr(t≤T <t+∆t|t≤T )

∆t

= f (t)/S(t). The haz-

ard function may remain constant, increase, decrease or take some more

complex shape. Most modeling of survival data is done using a propor-

Statistical Methods for Data Mining 15

tional hazard model. A proportional-hazard model, which assumes that

the hazard function is of the form

h(t) = α(t) exp

Ã

β

0

+

n

X

i=1

β

i

x

i

!

(1.14)

α(t) is a hazard function on its own, called the baseline hazard func-

tion, corresponding to that for the average value of all the covariates

x

1

, . . . , x

n

. This is called a proportional-hazard model, because the haz-

ard function for two diﬀerent patients have a constant ratio. The inter-

pretation of the β’s in this model is that the eﬀect is multiplicative.

There are several approaches to survival data analysis. The simplest it

to assume that the baseline hazard function is constant which is equiv-

alent to assuming exponential distribution. Another simple approach

would be to assume that the baseline hazard function is of the two-

parameter family of function, like the Weibull distribution. In these

cases the standard methods such as maximum likelihood can be used.

In other cases one may restrict α(t) for example by assuming it to be

monotonic. In business application, the baseline hazard function can

be determined by experimentation, but in medical situations it is not

practical to carry out an experiment to determine the shape of the base-

line hazard function. The Cox proportional hazards model ( Cox, 1972),

introduced to overcome this problem, has become the most commonly

used procedure for modelling the relationship of covariates to a survival

outcome and it is used in almost all medical analyses of survival data.

Estimation of the β’s is based on the partial likelihood function intro-

duced by Cox ( Cox, 1972; Therneau and Grambsch, 2000).

There are many other important statistical themes that are highly

relevant to DM, among them: statistical classiﬁcation methods, spline

and wavelets, decision trees and others (see Chapters ... for more detailed

information on these issues). In the next section we elaborate on the

False Discovery Rate (FDR) metho d (Benjamini and Hochberg, 1995),

a most salient feature of DM.

4. False Discovery Rate (FDR) Control in

Hypotheses testing

As noted before there is a feeling that the testing of a hypothesis is

irrelevant in DM. However the problem of separating a real phenomenon

from its background noise is just as fundamental a concern in DM as in

statistics. Take for example an association rule, with an observed lift

which is bigger than 1, as desired. Is it also signiﬁcantly bigger than 1

in the statistical sense, that is beyond what is expected to happen as a

16

result of noise? The answer to this question is given by the testing of the

hypothesis that the lift is 1. However, in DM a hypothesis is rarely tested

alone, as the above point demonstrates. The tested hypothesis is always

a member of a larger family of similar hypotheses, all association rules

of at least a given support and conﬁdence being tested simultaneously.

Thus, the testing of hypotheses in DM always invokes the ”Multiple

Comparisons Problem” so often discussed in statistics which is interest-

ing in itself as the ﬁrst DM problem in the statistics of 50 years ago

did just that: when a feature of interest (a variable) is measured on

10 subgroups (treatments), and the mean values are compared to some

reference value (such as 0), the problem is a small one, but take these

same means and search among all pairwise comparisons between the

treatments to ﬁnd a signiﬁcant diﬀerence, and the number of compar-

isons increases to 10*(10-1)/2=45 - which is in general quadratic in the

number of treatments. It becomes clear that if we allow an .05 proba-

bility of deciding that a diﬀerence exists in a single comparison even if

it really does not, thereby making a false discovery (or a type I error in

statistical terms), we can expect to ﬁnd on the average 2.25 such errors

in our pool of discoveries. No wonder this DM activity is sometimes

described in statistics as ”post hoc analysis” - a nice deﬁnition for DM

with a traditional ﬂavor.

The attitude that has been taken during 45 years of statistical re-

search is that in such problems the probability of making even one false

discover should be controlled, that is controlling the Family Wise Error

rate (FWE) as it is called. The simplest way to address the multiple

comparisons problem, and oﬀer FWE control at some desired level α, is

to use the Bonferroni procedure: conduct each of the m tests at level

α/m. In problems where m becomes very large the penalty to the re-

searcher from the extra caution becomes heavy, in the sense that the

probability of making any discovery becomes very small, and so it is not

uncommon to avoid the need to adjust for multiplicity.

The False Discovery Rate (FDR), namely the expectation of the pro-

portion of false discoveries (rejected true null hypotheses) among the

discoveries (the rejected hypotheses), was developed by Benjamini and

Hochberg, 1995 to bridge these two extremes. When the null hypothesis

is true for all hypotheses - the FDR and FWE criteria are equivalent.

However, when there are some hypotheses for which the null hypotheses

are false, an FDR controlling procedure may yield many more discoveries

at the expense of having a small proportion of false discoveries.

Formally, let H

0i

, i = 1, . . . m be the tested null hypotheses. For

i = 1, . . . m

0

the null hypotheses are true, and for the remaining m

1

=

m−m

0

hypotheses they are not. Thus, any discovery about a hypothesis

Statistical Methods for Data Mining 17

from the ﬁrst set is a false discovery, while a discover about a hypothesis

from the second set is a true discovery. Let V denote the number of false

discoveries and R the total number of discoveries. Let the proportion of

false discoveries be

Q =

½

V/R if R > 0

0 if R = 0

,

and deﬁne F DR = E(Q).

Benjamini and Hochberg advocated that the FDR should be con-

trolled at some desirable level q, while maximizing the number of dis-

coveries made. They oﬀered the linear step-up procedure as a simple and

general procedure that controls the FDR. The linear step-up procedure

makes use of the m p-values, P = (P

1

, . . . P

m

) so in a sense it is very

general, as it compares the ordered values P

(1)

≤ . . . ≤ P

(m)

to the set

of constants linearly interpolated between q and q/m.:

Deﬁnition 4.1 The Linear step-up Procedure: Let k = max{i : P

(i)

≤

iq/m}, and reject the k hypotheses associated with P

(1)

, . . . P

(k)

. If no

such a k exists reject none.

The procedure was ﬁrst suggested by Eklund (Seeger, 1968) and for-

gotten, then independently suggested by Simes (Simes, 1986). At both

points in time it went out of favor because it does not control the FWE.

Benjamini and Hochberg, 1995, showed that the procedure does control

the FDR, raising the interest in this procedure. Hence it is now referred

to as the Benjamini and Hochb erg procedure (BH procedure), or (unfor-

tunately) the FDR procedure (e.g. in SAS). Here, we use the descriptive

term, i.e. the linear step-up procedure (for a detailed historical review

see Benjamini and Hochberg, 2000).

For the purpose of practical interpretation and ﬂexibility in use, the

results of the linear step-up procedure can also b e reported in terms of

the FDR adjusted p-values. Formally, the FDR adjusted p-value of H

(i)

is p

LSU

(i)

= min{

mp

(j)

j

| j ≥ i }. Thus the linear step-up procedure at level

q is equivalent to rejecting all hypotheses whose FDR adjusted p-value

is ≤ q.

It should also be noted that the dual linear step-down procedure,

which uses the same constants but starts with the smallest p-value and

stops at the last {P

(i)

≤ iq/m}, also controls the FDR (Sarkar, 2002).

Even though it is obviously less powerful, it is sometimes easier to cal-

culate in very large problems.

The linear step-up procedure is quite striking in its ability to control

the FDR at precisely q · m

0

/m, regardless of the distributions of the test

statistics corresponding to false null hypotheses (when the distributions

under the simple null hypotheses are independent and continuous).

18

Benjamini and Yekutieli (Benjamini and Yekutieli, 2001) studied the

procedure under dependency. For some type of positive dependency they

showed that the above remains an upper bound. Even under the most

general dependence structure, where the FDR is controlled merely at

level q(1 + 1/2 + 1/3 + . . . + 1/m), it is again conservative by the same

factor m

0

/m (Benjamini and Yekutieli, 2001).

Knowledge of m

0

can therefore be very useful in this setting to improve

upon the performance of the FDR controlling procedure. Were this

information to be given to us by an ”oracle”, the linear step-up procedure

with q

0

= q ·m/m

0

would control the FDR at precisely the desired level q

in the independent and continuous case. It would then be more powerful

in rejecting many of the hypotheses for which the alternative holds. In

some precise asymptotic sense, Genovese and Wasserman (Genovese and

Wasserman, 2002a) showed it to be the best possible procedure.

Schweder and Spjotvoll (Schweder and Spjotvoll, 1982) were the ﬁrst

to try and estimate this factor, albeit informally. Hochberg and Ben-

jamini (Hochberg and Benjamini, 1990) formalized the approach. Ben-

jamini and Hochberg (Benjamini and Hochberg, 2000) incorporated it

into the linear step-up procedure, and other adaptive FDR controlling

procedures make use of other estimators (see Efron et al., 2001; Storey,

2002, and Storey, Taylor and Siegmund, 2004). Benjamini, Krieger and

Yekutieli, 2001, oﬀer a very simple and intuitive two-stage procedure

based on the idea that the value of m

0

can be estimated from the results

of the linear step-up procedure itself, and prove it controls the FDR at

level q.

Deﬁnition 4.2 Two-Stage Linear Step-Up Procedure (TST):

1 Use the linear step-up procedure at level q

0

=

q

1+

q

. Let r

1

be the

number of rejected hypotheses. If r

1

= 0 reject no hypotheses and

stop; if r

1

= m reject all m hypotheses and stop; or otherwise

2 Let ˆm

0

= (m − r

1

).

3 Use the linear step-up procedure with q

∗

= q

0

· m/ ˆm

0

Recent papers have illuminated the FDR from many diﬀerent points

of view: asymptotic, Bayesian, empirical Bayes, as the limit of empirical

processes, and in the context of penalized model selection (Efron et al.,

2001; Storey, 2002; Genovese and Wasserman, 2002a; Abramovich et al.,

2001). Some of the studies have emphasized variants of the FDR, such

as its conditional value given some discovery is made (the positive FDR

in Storey, 2002), or the distribution of the proportion of false discover-

ies itself (the FDR in Genovese and Wasserman, 2002a; Genovese and

Wasserman, 2002b).

Statistical Methods for Data Mining 19

Studies on FDR methodologies have become a very active area of re-

search in statistics, many of them making use of the large dimension of

the problems faced, and in that respect relying on the blessing of di-

mensionality. FDR methodologies have not yet found their way into the

practice and theory of DM, though it is our opinion that they have a lot

to oﬀer there, as the following example on variable selection shows

Example: Zytkov and Zembowicz, 1997; Zembowicz and Zytkov, 1996,

developed the 49er software to mine association rules using chi-square

tests of signiﬁcance for the independence assumption, i.e. by testing

whether the lift is signiﬁcantly > 1. Finding that too many of the m

potential rules are usually signiﬁcant, they used 1/m as a threshold for

signiﬁcance, comparing each p-value to the threshold, and choosing only

the rules that pass the threshold. Note that this is a Bonferroni-like

treatment of the multiplicity problem, controlling the FWE at α = 1.

Still, they further suggest increasing the threshold if a few hypotheses are

rejected. In particular they note that the performance of the threshold is

especially good if the largest p-value of the selected k rules is smaller than

k times the original 1/m threshold. This is exactly the BH procedure

used at level q = 1 and they arrived at it by merely checking the actual

performance on a speciﬁc problem. In spite of this remarkable success,

theory further tells us that it is important to use q < 1/2, and not 1,

to always get good performance. The preferable values for q are, as far

as we know, between 0.05 and 0.2. Such values for q further allow us to

conclude that only approximately q of the discovered association rules

are not real ones. With q = 1 such a statement is meaningless.

5. Model (Variables or Features) Selection using

FDR Penalization in GLM

Most of commonly used variable selection procedures in linear models

choose the appropriate subset by minimizing a model selection criterion

of the form: RSS

k

+σ

2

kλ, where RSS

k

is the residual sum of squares for

a model with k parameters as deﬁned in the previous section, and λ is

the penalization parameter. For the generalized linear models discussed

above twice the logarithm of the likelihoo d of the model takes on the

role of RSS

k

, but for simplicity of exposition we shall continue with the

simple linear model. This penalized sum of squares might ideally be

minimized over all k and all subsets of variables of size k, but practically

in larger problems it is usually minimized either by forward selection

or backward elimination, adding or dropping one variable at a time.

The diﬀerent selection criteria can be identiﬁed by the value of λ they

20

use. Most traditional model selection criteria make use of a ﬁxed λ

and can also be described as ﬁxed level testing. The Akaike Information

Criterion (AIC) and the C

p

criterion of Mallows both make use of λ = 2,

and are equivalent to testing at level 0.16 whether the coeﬃcient of each

newly included variable in the model is diﬀerent than 0. Usual backward

and forward algorithms use similar testing at the .05 level, which is

approximately equivalent to using λ = 4.

Note that when the selection of the model is conducted over a large

number of potential variables m, the implications of the above approach

can be disastrous. Take for example m = 500 variables, not an unlikely

situation in DM. Even if there is no connection whatsoever between

the predicted variable and the potential set of predicting variables, you

should expect to get 65 variables into the selected model - an unaccept-

able situation.

More recently model selection approaches have been examined in the

statistical literature in settings where the number of variables is large,

even tending to inﬁnity. Such studies, usually held under an assumption

of orthogonality of the variables, have brought new insight into the choice

of λ. Donoho and Jonhstone (Donoho and Johnstone, 1995) suggested

using λ, where λ = 2log(m), whose square root is called the ”universal

threshold” in wavelet analysis. Note that the larger the pool over which

the model is searched, the larger is the penalty per variable included.

This threshold can also be viewed as a multiple testing Bonferroni pro-

cedure at the level α

m

, with .2 ≤ α

m

≤ .4 for 10 ≤ m ≤ 10000. More

recent studies have emphasized that the penalty should also depend on

the size of the already selected model k, λ = λ

k,m

, increasing in m and

decreasing in k. They include Abramovich and Benjamini, 1996; Birge

and Massart, 2001; Abramovich, Bailey and Sapatinas , 2000; Tibshirani

and Knight, 1999; George and Foster, 2000, and Foster and Stine, 2004.

As full review is beyond our scope, we shall focus on the suggestion that

is directly related to FDR testing.

In the context of wavelet analysis Abramovich and Benjamini, 1996

suggested to using FDR testing, thereby introducing a threshold that

increases in m and decreases with k. Abramovich, Bailey and Sapatinas

, 2000, were able to prove in an asymptotic setup, where m tends to

inﬁnity and the model is sparse, that using FDR testing is asymptotically

minimax in a very wide sense. Their argument hinges on expressing the

FDR testing as a penalized RSS as follows:

RSS

k

+ σ

2

i=k

X

i=1

z

2

i

m

·

q

2

, (1.15)

Statistical Methods for Data Mining 21

where z

α

is the 1−α percentile of a standard normal distribution. This

is equivalent to using λ

k,m

=

1

k

P

i=k

i=1

z

2

i

m

·

q

2

in the general form of penalty.

When the models considered are sparse, the penalty is approximately

2σ

2

log(

m

k

·

2

q

). The FDR level controlled is q, which should be kept at a

level strictly less than 1/2.

In a followup study Gavrilov, 2003, investigated the properties of

such penalty functions using simulations, in setups where the number

of variables is large but ﬁnite, and where the potential variables are

correlated rather than orthogonal. The results show the dramatic failure

of all traditional ”ﬁxed penalty per-parameter” approaches. She found

the FDR-penalized selection procedure to have the best performance in

terms of minimax behavior over a large numb er of situations likely to

arise in practice, when the number of potential variables was more than

32 (and a close second in smaller cases). Interestingly they recommend

using q = .05, which turned out to be well calibrated value for q for

problems with up to 200 variables (the largest investigated).

Example: Foster and Stine, 2004, developed these ideas for the case

when the predicted variable is 0-1, demonstrating their usefulness in DM,

in developing a prediction model for loan default. They started with

approximately 200 potential variables for the model, but then added all

pairwise interactions to reach a set of some 50,000 potential variables.

Their article discusses in detail some of the issues reviewed above, and

has a very nice and useful discussion of important computational aspects

of the application of the ideas in real a large DM problem.

6. Concluding Remarks

KDD and DM are a vaguely deﬁned ﬁeld in the sense that the deﬁni-

tion largely depends on the background and views of the deﬁner. Fayyad

deﬁned DM as the nontrivial pro cess of identifying valid, novel, poten-

tially useful, and ultimately understandable patterns in data. Some def-

initions of DM emphasize the connection of DM to databases containing

ample of data. Another deﬁnitions of KDD and DM is the following

deﬁnition: ”Nontrivial extraction of implicit, previously unknown and

potentially useful information from data, or the search for relationships

and global patterns that exist in databases”. Although mathematics,

like computing is a tool for statistics, statistics has developed over a

long time as a subdiscipline of mathematics. Statisticians have devel-

oped mathematical theories to support their methods and a mathemati-

cal formulation based on probability theory to quantify the uncertainty.

Traditional statistics emphasizes a mathematical formulation and vali-

dation of its methodology rather than empirical or practical validation.

22

The emphasis on rigor has required a proof that a proposed method

will work prior to the use of the metho d. In contrast, computer sci-

ence and machine learning use experimental validation methods. Statis-

tics has developed into a closed discipline, with its own scientiﬁc jargon

and academic objectives that favor analytic proofs rather than practi-

cal methods for learning from data. We need to distinguish between the

theoretical mathematical background of statistics and its use as a tool in

many experimental scientiﬁc research studies. We believe that comput-

ing methodology and many of the other related issues in DM should be

incorporated into traditional statistics. An eﬀort has to be made to cor-

rect the negative connotations that have long surrounded data mining in

the statistics literature (Chatﬁeld, 1995) and the statistical community

will have to recognize that empirical validation does constitute a form

of validation (Friedman, 1998).

Although the terminology used in DM and statistics may diﬀer, in

many cases the concepts are the same. For example, in neural net-

works we use terms like ”learning”, ”weights” and ”knowledge” while in

statistics we use ”estimation”, ”parameters” and ”value of parameters”,

respectively. Not all statistical themes are relevant to DM. For example,

as DM analyzes existing databases, experimental design is not relevant

to DM. However, many of them, including those covered in this chapter,

are highly relevant to DM and any data miner should be familiar with

them.

In summary, there is a need to increase the interaction and collabo-

ration between data miners and statistics. This can be done by over-

coming the terminology barriers, working on problems stemming from

large databases. A question that has often been raised among statisti-

cians is whether DM is not merely part of statistics. The point of this

chapter was to show how each can beneﬁt from the other, making the

inquiry from data a more successful endeavor, rather than dwelling on

theoretical issues of dubious value.

References

Abramovich F. and Benjamini Y., (1996). Adaptive thresholding of wavelet

coeﬃcients. Computational Statistics & Data Analysis, 22:351–361.

Abramovich F., Bailey T .C. and Sapatinas T., (2000). Wavelet analysis

and its statistical applications. Journal of the Royal Statistical Society

Series D-The Statistician, 49:1–29.

Abramovich F., Benjamini Y., Donoho D. and Johnstone I., (2000).

Adapting to unknown sparsity by controlling the false discovery rate.

Technical Report 2000-19, Department of Statistics, Stanford Univer-

sity.

Benjamini Y. and Ho chberg Y., (1995). Controlling the false discover

rate: A practical and powerful approach to multiple testing. J. R.

Statist. Soc. B, 57:289–300.

Benjamini Y. and Hochberg Y., (2000). On the adaptive control of the

false discovery fate in multiple testing with independent statistics.

Journal of Educational and Behavioral Statistics, 25:60–83.

Benjamini Y., Krieger A.M. and Yekutieli D., (2001). Two staged linear

step up for controlling procedure. Technical report, Department of

Statistics and O.R., Tel Aviv University.

Benjamini Y. and Yekutieli D., (2001). The control of the false discov-

ery rate in multiple testing under dependency. Annals of Statistics,

29:1165–1188.

Berthold M. and Hand D., (1999). Intelligent Data Analysis: An Intro-

duction. Springer.

Birge L. and Massart P., (2001). Gaussian model selection. Journal of

the European Mathematical Society, 3:203–268.

Chatﬁeld C., (1995). Model uncertainty, data mining and statistical in-

ference. Journal of the Royal Statistical Society A, 158:419–466.

Cochran W.G., (1977). Sampling Techniques. Wiley.

Cox D.R., (1972). Regressio models and life-tables. Journal of the Royal

Statistical Society B, 34:187–220.

Dell’Aquila R. and Ronchetti E.M., (2004). Introduction to Robust Statis-

tics with Economic and Financial Applications. Wiley.

23

24

Donoho D.L. and Johnstone I.M., (1995). Adapting to unknown smooth-

ness via wavelet shrinkage. Journal of the American Statistical Asso-

ciation, 90:1200–1224.

Donoho D., (2000). American math. society: Math challenges of the 21st

century: High-dimensional data analysis: The curses and blessings of

dimensionality.

Efron B., Tibshirani R.J., Storey J.D. and Tusher V., (2001). Empirical

Bayes analysis of a microarray experiment. Journal of the American

Statistical Association, 96:1151–1160.

Friedman J.H., (1998). Data Mining and Statistics: What’s the connec-

tions?, Proc. 29th Symposium on the Interface (D. Scott, editor).

Foster D.P. and Stine R.A., (2004). Variable selection in data mining:

Building a predictive model for bankruptcy. Journal of the American

Statistical Association, 99:303–313.

Gavrilov Y., (2003). Using the falls discovery rate criteria for model

selection in linear regression. M.Sc. Thesis, Department of Statistics,

Tel Aviv University.

Genovese C. and Wasserman L., (2002a). Operating characteristics and

extensions of the false discovery rate procedure. Journal of the Royal

Statistical Society Series B, 64:499–517.

Genovese C. and Wasserman L., (2002b). A stochastic process approach

to false discovery rates. Technical Report 762, Department of Statis-

tics, Carnegie Mellon University.

George E.I. and Foster D.P., (2000). Calibration and empirical Bayes

variable selection. Biometrika, 87:731–748.

Hand D., (1998). Data mining: Statistics and more? The American Statis-

tician, 52:112–118.

Hand D., Mannila H. and Smyth P., (2001). Principles of Data Mining.

MIT Press.

Han J. and Kamber M., (2001). Data Mining: Concepts and Techniques.

Morgan Kaufmann Publisher.

Hastie T., Tibshirani R. and Friedman J., (2001). The Elements of Sta-

tistical Learning: Data Mining, Inference, and Prediction. Springer.

Hochberg Y. and Benjamini Y., (1990). More powerful procedures for

multiple signiﬁcance testing. Statistics in Medicine, 9:811–818.

Leshno M., Lin V.Y., Pinkus A. and Schocken S., (1993). Multilayer

feedforward networks with a non polynomial activation function can

approximate any function. Neural Networks, 6:861–867.

McCullagh P. and Nelder J.A., (1991). Generalized Linear Model. Chap-

man & Hall.

REFERENCES 25

Meilijson I., (1991). The expected value of some functions of the convex

hull of a random set of points sampled in r

d

. Isr. J. of Math., 72:341–

352.

Mosteller F. and Tukey J.W., (1977). Data Analysis and Regression : A

Second Course in Statistics. Wiley.

Roberts S. and Everson R. (editors), (2001). Independent Component

Analysis : Principles and Practice. Cambridge University Press.

Ronchetti E.M., Hampel F.R., Rousseeuw P.J. and Stahel W.A., (1986).

Robust Statistics : The Approach Based on Inﬂuence Functions. Wiley.

Sarkar S.K., (2002). Some results on false discovery rate in stepwise

multiple testing procedures. Annals of Statistics, 30:239–257.

Schweder T. and Spjotvoll E., (1982). Plots of p-values to evaluate many

tests simultaneously. Biometrika, 69:493–502.

Seeger P., (1968). A note on a method for the analysis of signiﬁcances

en mass. Technometrics, 10:586–593.

Simes R.J., (1986). An improved Bonferroni procedure for multiple tests

of signiﬁcance. Biometrika, 73:751–754.

Storey J.D., (2002). A direct approach to false discovery rates. Journal

of the Royal Statistical Society Series B, 64:479–498.

Storey J.D., Taylor J.E. and Siegmund D., (2004). Strong control, con-

servative point estimation, and simultaneous conservative consistency

of false discovery rates: A uniﬁed approach. Journal of the Royal Sta-

tistical Society Series B, 66:187–205.

Therneau T.M. and Grambsch P.M., (2000). Modeling Survival Data,

Extending the Cox Model. Springer.

Tibshirani R. and Knight K., (1999). The covariance inﬂation criterion

for adaptive model selection. Journal of the Royal Statistical Society

Series B, 61:Part 3 529–546.

Zembowicz R. and Zytkov J.M., (1996). From contingency tables to vari-

ous froms of knowledge in databases. In U.M. Fayyad, R. Uthurusamy,

G. Piatetsky-Shapiro and P. Smyth (editors) Advances in Knowledge

Discovery and Data Mining (pp. 329-349). MIT Press.

Zytkov J.M. and Zembowicz R., (1997). Contingency tables as the foun-

dation for concepts, concept hierarchies and rules: The 49er system

approach. Fundamenta Informaticae, 30:383–399.