tai lieu tham khao ve thuat toan ICA

Independent component analysis ICA is a statistical and computational techniquefor revealing hidden factors that underlie sets of random variables, measurements, orsignals.. Chapter 18 d

Independent Component Analysis

Independent Component

Analysis

Final version of 7 March 2001

Aapo Hyv¨arinen, Juha Karhunen, and Erkki Oja

A Wiley-Interscience Publication

JOHN WILEY & SONS, INC.

New York / Chichester / Weinheim / Brisbane / Singapore / Toronto

1.1.3 Independence as a guiding principle 3

1.2.1 Observing mixtures of unknown signals 4 1.2.2 Source separation based on independence 5

vi CONTENTS

Part I MATHEMATICAL PRELIMINARIES

2.1 Probability distributions and densities 15 2.1.1 Distribution of a random variable 15 2.1.2 Distribution of a random vector 17 2.1.3 Joint and marginal distributions 18

2.2.1 Definition and general properties 19 2.2.2 Mean vector and correlation matrix 20

2.3.1 Uncorrelatedness and whiteness 24

2.4 Conditional densities and Bayes’ rule 28

2.5.1 Properties of the gaussian density 32

2.8.2 Stationarity, mean, and autocorrelation 45 2.8.3 Wide-sense stationary processes 46

CONTENTS vii

3.2 Learning rules for unconstrained optimization 63

3.2.3 The natural gradient and relative gradient 67

3.2.5 Convergence of stochastic on-line algorithms * 71 3.3 Learning rules for constrained optimization 73

4.4.2 Nonlinear and generalized least squares * 88

4.6.1 Minimum mean-square error estimator 94

4.6.3 Maximum a posteriori (MAP) estimator 97

5.2.2 Definition using Kullback-Leibler divergence 110

6.1.2 PCA by minimum MSE compression 128 6.1.3 Choosing the number of principal components 129 6.1.4 Closed-form computation of PCA 131

6.2.1 The stochastic gradient ascent algorithm 133 6.2.2 The subspace learning algorithm 134

6.2.4 PCA and back-propagation learning * 136 6.2.5 Extensions of PCA to nonquadratic criteria * 137

CONTENTS ix

Part II BASIC INDEPENDENT COMPONENT ANALYSIS

7 What is Independent Component Analysis? 147

7.4.1 Uncorrelatedness and whitening 158

7.5 Why gaussian variables are forbidden 161

8.5.1 Searching for interesting directions 197

x CONTENTS

9.2 Algorithms for maximum likelihood estimation 207

10.6 Concluding remarks and references 225

11.2 Tensor eigenvalues give independent components 230 11.3 Tensor decomposition by a power method 232 11.4 Joint approximate diagonalization of eigenmatrices 234 11.5 Weighted correlation matrix approach 235

11.6 Concluding remarks and references 236

CONTENTS xi

12 ICA by Nonlinear Decorrelation and Nonlinear PCA 239 12.1 Nonlinear correlations and independence 240

12.4 The estimating functions approach * 245 12.5 Equivariant adaptive separation via independence 247

12.7 The nonlinear PCA criterion and ICA 251 12.8 Learning rules for the nonlinear PCA criterion 254

12.8.2 Convergence of the nonlinear subspace rule * 255 12.8.3 Nonlinear recursive least-squares rule 258 12.9 Concluding remarks and references 261

13.1.1 Why time filtering is possible 264

14.3.3 Practical choice of nonlinearity 279

xii CONTENTS

14.4 Experimental comparison of ICA algorithms 280 14.4.1 Experimental set-up and algorithms 281

14.4.3 Comparisons with real-world data 286

15.4.2 Higher-order cumulant methods 298

15.5 Estimation of the noise-free independent components 299 15.5.1 Maximum a posteriori estimation 299 15.5.2 Special case of shrinkage estimation 300 15.6 Denoising by sparse code shrinkage 303

16.1 Estimation of the independent components 306 16.1.1 Maximum likelihood estimation 306 16.1.2 The case of supergaussian components 307

16.2.2 Maximizing likelihood approximations 308 16.2.3 Approximate estimation by quasiorthogonality 309

CONTENTS xiii

17.1.1 The nonlinear ICA and BSS problems 315 17.1.2 Existence and uniqueness of nonlinear ICA 317 17.2 Separation of post-nonlinear mixtures 319 17.3 Nonlinear BSS using self-organizing maps 320 17.4 A generative topographic mapping approach * 322

18.1.3 Extension to several time lags 344 18.2 Separation by nonstationarity of variances 346 18.2.1 Using local autocorrelations 347

18.3.1 Comparison of separation principles 351 18.3.2 Kolmogoroff complexity as unifying framework 352

19.2.2 Reformulation as ordinary ICA 363

19.2.5 Spatiotemporal decorrelation methods 367 19.2.6 Other methods for convolutive mixtures 367

Appendix Discrete-time filters and the z -transform 369

20.1.1 Motivation for prior information 371

20.3.6 Relation to independent subspaces 387

CONTENTS xv

Part IV APPLICATIONS OF ICA

21.4 Image denoising by sparse code shrinkage 398

22.1 Electro- and magnetoencephalography 407 22.1.1 Classes of brain imaging techniques 407 22.1.2 Measuring electric activity in the brain 408 22.1.3 Validity of the basic ICA model 409 22.2 Artifact identification from EEG and MEG 410 22.3 Analysis of evoked magnetic fields 411 22.4 ICA applied on other measurement techniques 413

23.1 Multiuser detection and CDMA communications 417

23.4 Blind separation of convolved CDMA mixtures * 430

Independent component analysis (ICA) is a statistical and computational techniquefor revealing hidden factors that underlie sets of random variables, measurements, orsignals ICA defines a generative model for the observed multivariate data, which istypically given as a large database of samples In the model, the data variables areassumed to be linear or nonlinear mixtures of some unknown latent variables, andthe mixing system is also unknown The latent variables are assumed nongaussianand mutually independent, and they are called the independent components of theobserved data These independent components, also called sources or factors, can befound by ICA

ICA can be seen as an extension to principal component analysis and factoranalysis ICA is a much more powerful technique, however, capable of finding theunderlying factors or sources when these classic methods fail completely

The data analyzed by ICA could originate from many different kinds of tion fields, including digital images and document databases, as well as economicindicators and psychometric measurements In many cases, the measurements aregiven as a set of parallel signals or time series; the term blind source separation is used

applica-to characterize this problem Typical examples are mixtures of simultaneous speechsignals that have been picked up by several microphones, brain waves recorded bymultiple sensors, interfering radio signals arriving at a mobile phone, or parallel timeseries obtained from some industrial process

The technique of ICA is a relatively new invention It was for the first time troduced in early 1980s in the context of neural network modeling In mid-1990s,some highly successful new algorithms were introduced by several research groups,

in-xvii

xviii PREFACE

together with impressive demonstrations on problems like the cocktail-party effect,where the individual speech waveforms are found from their mixtures ICA becameone of the exciting new topics, both in the field of neural networks, especially unsu-pervised learning, and more generally in advanced statistics and signal processing.Reported real-world applications of ICA on biomedical signal processing, audio sig-nal separation, telecommunications, fault diagnosis, feature extraction, financial timeseries analysis, and data mining began to appear

Many articles on ICA were published during the past 20 years in a large number

of journals and conference proceedings in the fields of signal processing, artificialneural networks, statistics, information theory, and various application fields Severalspecial sessions and workshops on ICA have been arranged recently [70, 348], andsome edited collections of articles [315, 173, 150] as well as some monographs onICA, blind source separation, and related subjects [105, 267, 149] have appeared.However, while highly useful for their intended readership, these existing texts typ-ically concentrate on some selected aspects of the ICA methods only In the briefscientific papers and book chapters, mathematical and statistical preliminaries areusually not included, which makes it very hard for a wider audience to gain fullunderstanding of this fairly technical topic

A comprehensive and detailed text book has been missing, which would coverboth the mathematical background and principles, algorithmic solutions, and practicalapplications of the present state of the art of ICA The present book is intended to fillthat gap, serving as a fundamental introduction to ICA

It is expected that the readership will be from a variety of disciplines, such

as statistics, signal processing, neural networks, applied mathematics, neural andcognitive sciences, information theory, artificial intelligence, and engineering Bothresearchers, students, and practitioners will be able to use the book We have madeevery effort to make this book self-contained, so that a reader with a basic background

in college calculus, matrix algebra, probability theory, and statistics will be able toread it This book is also suitable for a graduate level university course on ICA,which is facilitated by the exercise problems and computer assignments given inmany chapters

Scope and contents of this book

This book provides a comprehensive introduction to ICA as a statistical and tational technique The emphasis is on the fundamental mathematical principles andbasic algorithms Much of the material is based on the original research conducted

compu-in the authors’ own research group, which is naturally reflected compu-in the weightcompu-ing ofthe different topics We give a wide coverage especially to those algorithms that arescalable to large problems, that is, work even with a large number of observed vari-ables and data points These will be increasingly used in the near future when ICA

is extensively applied in practical real-world problems instead of the toy problems

or small pilot studies that have been predominant until recently Respectively,

we may have overlooked.

For easier reading, the book is divided into four parts

 Part I gives the mathematical preliminaries It introduces the general

math-ematical concepts needed in the rest of the book We start with a crash course

on probability theory in Chapter 2 The reader is assumed to be familiar withmost of the basic material in this chapter, but also some concepts more spe-cific to ICA are introduced, such as higher-order cumulants and multivariateprobability theory Next, Chapter 3 discusses essential concepts in optimiza-tion theory and gradient methods, which are needed when developing ICAalgorithms Estimation theory is reviewed in Chapter 4 A complementarytheoretical framework for ICA is information theory, covered in Chapter 5.Part I is concluded by Chapter 6, which discusses methods related to principalcomponent analysis, factor analysis, and decorrelation

More confident readers may prefer to skip some or all of the introductorychapters in Part I and continue directly to the principles of ICA in Part II

 In Part II, the basic ICA model is covered and solved This is the linear

instantaneous noise-free mixing model that is classic in ICA, and forms the core

of the ICA theory The model is introduced and the question of identifiability ofthe mixing matrix is treated in Chapter 7 The following chapters treat differentmethods of estimating the model A central principle is nongaussianity, whoserelation to ICA is first discussed in Chapter 8 Next, the principles of maximumlikelihood (Chapter 9) and minimum mutual information (Chapter 10) arereviewed, and connections between these three fundamental principles areshown Material that is less suitable for an introductory course is covered

in Chapter 11, which discusses the algebraic approach using higher-ordercumulant tensors, and Chapter 12, which reviews the early work on ICA based

on nonlinear decorrelations, as well as the nonlinear principal componentapproach Practical algorithms for computing the independent componentsand the mixing matrix are discussed in connection with each principle Next,some practical considerations, mainly related to preprocessing and dimensionreduction of the data are discussed in Chapter 13, including hints to practitioners

on how to really apply ICA to their own problem An overview and comparison

of the various ICA methods is presented in Chapter 14, which thus summarizesPart II

 In Part III, different extensions of the basic ICA model are given This part is by

its nature more speculative than Part II, since most of the extensions have beenintroduced very recently, and many open problems remain In an introductory

xx PREFACE

course on ICA, only selected chapters from this part may be covered First,

in Chapter 15, we treat the problem of introducing explicit observational noise

in the ICA model Then the situation where there are more independentcomponents than observed mixtures is treated in Chapter 16 In Chapter 17,the model is widely generalized to the case where the mixing process can be of

a very general nonlinear form Chapter 18 discusses methods that estimate alinear mixing model similar to that of ICA, but with quite different assumptions:the components are not nongaussian but have some time dependencies instead.Chapter 19 discusses the case where the mixing system includes convolutions.Further extensions, in particular models where the components are no longerrequired to be exactly independent, are given in Chapter 20

 Part IV treats some applications of ICA methods Feature extraction

(Chap-ter 21) is relevant to both image processing and vision research Brain imagingapplications (Chapter 22) concentrate on measurements of the electrical andmagnetic activity of the human brain Telecommunications applications aretreated in Chapter 23 Some econometric and audio signal processing applica-tions, together with pointers to miscellaneous other applications, are treated inChapter 24

Throughout the book, we have marked with an asterisk some sections that arerather involved and can be skipped in an introductory course

Several of the algorithms presented in this book are available as public domainsoftware through the World Wide Web, both on our own Web pages and those ofother ICA researchers Also, databases of real-world data can be found there fortesting the methods We have made a special Web page for this book, which containsappropriate pointers The address is

www.cis.hut.fi/projects/ica/book

The reader is advised to consult this page for further information

This book was written in cooperation between the three authors A Hyv ¨arinenwas responsible for the chapters 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 20, 21, and 22;

J Karhunen was responsible for the chapters 2, 4, 17, 19, and 23; while E Oja wasresponsible for the chapters 3, 6, and 12 The Chapters 1 and 24 were written jointly

by the authors

Acknowledgments

We are grateful to the many ICA researchers whose original contributions form thefoundations of ICA and who have made this book possible In particular, we wish toexpress our gratitude to the Series Editor Simon Haykin, whose articles and books onsignal processing and neural networks have been an inspiration to us over the years

on joint work with Harri Valpola and Petteri Pajunen, and Section 24.1 is joint workwith Kimmo Kiviluoto and Simona Malaroiu.

Over various phases of writing this book, several people have kindly agreed toread and comment on parts or all of the text We are grateful for this to Ella Bingham,Jean-Franc¸ois Cardoso, Adrian Flanagan, Mark Girolami, Antti Honkela, JarmoHurri, Petteri Pajunen, Tapani Ristaniemi, and Kari Torkkola Leila Koivisto helped

in technical editing, while Antti Honkela, Mika Ilmoniemi, Merja Oja, and TapaniRaiko helped with some of the figures

Our original research work on ICA as well as writing this book has been mainlyconducted at the Neural Networks Research Centre of the Helsinki University of Tech-nology, Finland The research had been partly financed by the project “BLISS” (Eu-ropean Union) and the project “New Information Processing Principles” (Academy

of Finland), which are gratefully acknowledged Also, A H wishes to thank G¨oteNyman and Jukka H¨akkinen of the Department of Psychology of the University ofHelsinki who hosted his civilian service there and made part of the writing possible

AAPOHYV ¨ ARINEN, JUHAKARHUNEN, ERKKIOJA

Espoo, Finland

March 2001

Introduction

Independent component analysis (ICA) is a method for finding underlying factors orcomponents from multivariate (multidimensional) statistical data What distinguishes

ICA from other methods is that it looks for components that are both statistically

independent, and nongaussian Here we briefly introduce the basic concepts,

appli-cations, and estimation principles of ICA

1.1 LINEAR REPRESENTATION OF MULTIVARIATE DATA

1.1.1 The general statistical setting

A long-standing problem in statistics and related areas is how to find a suitablerepresentation of multivariate data Representation here means that we somehowtransform the data so that its essential structure is made more visible or accessible

In neural computation, this fundamental problem belongs to the area of vised learning, since the representation must be learned from the data itself withoutany external input from a supervising “teacher” A good representation is also acentral goal of many techniques in data mining and exploratory data analysis Insignal processing, the same problem can be found in feature extraction, and also inthe source separation problem that will be considered below

unsuper-Let us assume that the data consists of a number of variables that we have observedtogether Let us denote the number of variables bymand the number of observations

by T We can then denote the data by x i ( t ) where the indices take the values

i = 1 ;:::;mandt = 1 ;:::;T The dimensionsmandTcan be very large

1

2 INTRODUCTION

A very general formulation of the problem can be stated as follows: What could

be a function from anm-dimensional space to ann-dimensional space such that thetransformed variables give information on the data that is otherwise hidden in the

large data set That is, the transformed variables should be the underlying factors or

components that describe the essential structure of the data It is hoped that these

components correspond to some physical causes that were involved in the processthat generated the data in the first place

In most cases, we consider linear functions only, because then the interpretation

of the representation is simpler, and so is its computation Thus, every component,sayy i, is expressed as a linear combination of the observed variables:

y i ( t ) =X

j w ij x j ( t ) ; fori = 1 ;:::;n;j = 1 ;:::;m (1.1)

where the w ij are some coefficients that define the representation The problemcan then be rephrased as the problem of determining the coefficients w ij Usinglinear algebra, we can express the linear transformation in Eq (1.1) as a matrixmultiplication Collecting the coefficientsw ijin a matrixW, the equation becomes

0 B B

0 B B

(1.2)

A basic statistical approach consists of considering thex i ( t )as a set ofT izations ofmrandom variables Thus each set x i ( t ) ;t = 1 ;:::;T is a sample ofone random variable; let us denote the random variable byx i In this framework,

real-we could determine the matrixW by the statistical properties of the transformedcomponentsy i In the following sections, we discuss some statistical properties thatcould be used; one of them will lead to independent component analysis

1.1.2 Dimension reduction methods

One statistical principle for choosing the matrixWis to limit the number of ponentsy i to be quite small, maybe only 1 or 2, and to determine Wso that the

com-y i contain as much information on the data as possible This leads to a family oftechniques called principal component analysis or factor analysis

In a classic paper, Spearman [409] considered data that consisted of school mance rankings given to schoolchildren in different branches of study, complemented

perfor-by some laboratory measurements Spearman then determinedWby finding a singlelinear combination such that it explained the maximum amount of the variation inthe results He claimed to find a general factor of intelligence, thus founding factoranalysis, and at the same time starting a long controversy in psychology

BLIND SOURCE SEPARATION 3

Fig 1.1 The density function of the Laplacian distribution, which is a typical supergaussian distribution For comparison, the gaussian density is given by a dashed line The Laplacian density has a higher peak at zero, and heavier tails Both densities are normalized to unit variance and have zero mean.

1.1.3 Independence as a guiding principle

Another principle that has been used for determiningWis independence: the ponentsy ishould be statistically independent This means that the value of any one

com-of the components gives no information on the values com-of the other components

In fact, in factor analysis it is often claimed that the factors are independent,but this is only partly true, because factor analysis assumes that the data has agaussian distribution If the data is gaussian, it is simple to find components thatare independent, because for gaussian data, uncorrelated components are alwaysindependent

In reality, however, the data often does not follow a gaussian distribution, and thesituation is not as simple as those methods assume For example, many real-world

data sets have supergaussian distributions This means that the random variables

take relatively more often values that are very close to zero or very large In otherwords, the probability density of the data is peaked at zero and has heavy tails (largevalues far from zero), when compared to a gaussian density of the same variance Anexample of such a probability density is shown in Fig 1.1

This is the starting point of ICA We want to find statistically independent ponents, in the general case where the data is nongaussian.

com-1.2 BLIND SOURCE SEPARATION

Let us now look at the same problem of finding a good representation, from adifferent viewpoint This is a problem in signal processing that also shows thehistorical background for ICA

4 INTRODUCTION

1.2.1 Observing mixtures of unknown signals

Consider a situation where there are a number of signals emitted by some physicalobjects or sources These physical sources could be, for example, different brainareas emitting electric signals; people speaking in the same room, thus emittingspeech signals; or mobile phones emitting their radio waves Assume further thatthere are several sensors or receivers These sensors are in different posit ions, so thateach records a mixture of the original source signals with slightly different weights.For the sake of simplicity of exposition, let us say there are three underlyingsource signals, and also three observed signals Denote byx1( t ) ;x2( t )andx3( t )theobserved signals, which are the amplitudes of the recorded signals at time pointt,and bys1( t ) ;s2( t )ands3( t )the original signals Thex i ( t )are then weighted sums

of thes i ( t ), where the coefficients depend on the distances between the sources andthe sensors:

x1( t ) = a11s1( t ) + a12s2( t ) + a13s3( t ) (1.3)

x2( t ) = a21s1( t ) + a22s2( t ) + a23s3( t )

x3( t ) = a31s1( t ) + a32s2( t ) + a33s3( t )

Thea ij are constant coefficients that give the mixing weights They are assumed

of the physical mixing system, which can be extremely difficult in general Thesource signals s i are unknown as well, since the very problem is that we cannot

record them directly

As an illustration, consider the waveforms in Fig 1.2 These are three linearmixturesx i of some original source signals They look as if they were completelynoise, but actually, there are some quite structured underlying source signals hidden

in these observed signals

What we would like to do is to find the original signals from the mixtures

x1( t ) ;x2( t )andx3( t ) This is the blind source separation (BSS) problem Blind

means that we know very little if anything about the original sources

We can safely assume that the mixing coefficientsa ijare different enough to makethe matrix that they form invertible Thus there exists a matrixWwith coefficients

w ij, such that we can separate thes ias

Now we see that in fact this problem is mathematically similar to the one where

we wanted to find a good representation for the random data in x i ( t ), as in (1.2).Indeed, we could consider each signalx i ( t ) ;t = 1 ;:::;T as a sample of a randomvariablex i, so that the value of the random variable is given by the amplitudes ofthat signal at the time points recorded

BLIND SOURCE SEPARATION 5

1.2.2 Source separation based on independence

The question now is: How can we estimate the coefficientsw ij in (1.4)? We want

to obtain a general method that works in many different circumstances, and in factprovides one answer to the very general problem that we started with: finding agood representation of multivariate data Therefore, we use very general statisticalproperties All we observe is the signalsx1;x2andx3, and we want to find a matrix

Wso that the representation is given by the original source signalss1;s2, ands3

A surprisingly simple solution to the problem can be found by considering just

the statistical independence of the signals In fact, if the signals are not gaussian, it

is enough to determine the coefficientsw ij, so that the signals

Using just this information on the statistical independence, we can in fact estimatethe coefficient matrixWfor the signals in Fig 1.2 What we obtain are the sourcesignals in Fig 1.3 (These signals were estimated by the FastICA algorithm that

we shall meet in several chapters of this book.) We see that from a data set thatseemed to be just noise, we were able to estimate the original source signals, using

an algorithm that used the information on the independence only These estimatedsignals are indeed equal to those that were used in creating the mixtures in Fig 1.2

1.3 INDEPENDENT COMPONENT ANALYSIS

1.3.1 Definition

We have now seen that the problem of blind source separation boils down to finding

a linear representation in which the components are statistically independent Inpractical situations, we cannot in general find a representation where the componentsare really independent, but we can at least find components that are as independent

as possible

This leads us to the following definition of ICA, which will be considered

in more detail in Chapter 7 Given a set of observations of random variables

( x1( t ) ;x2( t ) ;:::;x n ( t )), wheretis the time or sample index, assume that they aregenerated as a linear mixture of independent components:

0 B B

C= A

0 B B

whereAis some unknown matrix Independent component analysis now consists ofestimating both the matrixAand thes i ( t ), when we only observe thex i ( t ) Note

INDEPENDENT COMPONENT ANALYSIS 7

that we assumed here that the number of independent componentss iis equal to thenumber of observed variables; this is a simplifying assumption that is not completelynecessary

Alternatively, we could define ICA as follows: find a linear transformation given by

a matrixWas in (1.2), so that the random variablesy i ;i = 1 ;:::;nare as independent

as possible This formulation is not really very different from the previous one, sinceafter estimatingA, its inverse givesW

It can be shown (see Section 7.5) that the problem is well-defined, that is, themodel in (1.6) can be estimated if and only if the componentss i are nongaussian.

This is a fundamental requirement that also explains the main difference betweenICA and factor analysis, in which the nongaussianity of the data is not taken into

account In fact, ICA could be considered as nongaussian factor analysis, since in

factor analysis, we are also modeling the data as linear mixtures of some underlyingfactors

1.3.2 Applications

Due to its generality the ICA model has applications in many different areas, some

of which are treated in Part IV Some examples are:

 In brain imaging, we often have different sources in the brain emit signals thatare mixed up in the sensors outside of the head, just like in the basic blindsource separation model (Chapter 22)

 In econometrics, we often have parallel time series, and ICA could decomposethem into independent components that would give an insight to the structure

of the data set (Section 24.1)

 A somewhat different application is in image feature extraction, where we want

to find features that are as independent as possible (Chapter 21)

1.3.3 How to find the independent components

It may be very surprising that the independent components can be estimated fromlinear mixtures with no more assumptions than their independence Now we will try

to explain briefly why and how this is possible; of course, this is the main subject ofthe book (especially of Part II)

Uncorrelatedness is not enough The first thing to note is that independence

is a much stronger property than uncorrelatedness Considering the blind source aration problem, we could actually find many different uncorrelated representations

sep-of the signals that would not be independent and would not separate the sources.Uncorrelatedness in itself is not enough to separate the components This is also thereason why principal component analysis (PCA) or factor analysis cannot separatethe signals: they give components that are uncorrelated, but little more

8 INTRODUCTION

Fig 1.4 A sample of independent

compo-nents s1 and s2 with uniform distributions.

Horizontal axis:s1 ; vertical axis:s2

Fig 1.5 Uncorrelated mixturesx1 andx2 Horizontal axis: x1 ; vertical axis:x2

Let us illustrate this with a simple example using two independent componentswith uniform distributions, that is, the components can have any values inside acertain interval with equal probability Data from two such components are plotted

in Fig 1.4 The data is uniformly distributed inside a square due to the independence

of the components

Now, Fig 1.5 shows two uncorrelated mixtures of those independent components.

Although the mixtures are uncorrelated, one sees clearly that the distributions are notthe same The independent components are still mixed, using an orthogonal mixingmatrix, which corresponds to a rotation of the plane One can also see that in Fig 1.5the components are not independent: if the component on the horizontal axis has avalue that is near the corner of the square that is in the extreme right, this clearlyrestricts the possible values that the components on the vertical axis can have

In fact, by using the well-known decorrelation methods, we can transform anylinear mixture of the independent components into uncorrelated components, in whichcase the mixing is orthogonal (this will be proven in Section 7.4.2) Thus, the trick

in ICA is to estimate the orthogonal transformation that is left after decorrelation.This is something that classic methods cannot estimate because they are based onessentially the same covariance information as decorrelation

Figure 1.5 also gives a hint as to why ICA is possible By locating the edges ofthe square, we could compute the rotation that gives the original components In thefollowing, we consider a couple more sophisticated methods for estimating ICA

Nonlinear decorrelation is the basic ICA method One way of stating howindependence is stronger than uncorrelatedness is to say that independence implies

trans-formationsg ( s ) andh ( s )are uncorrelated (in the sense that their covariance is

INDEPENDENT COMPONENT ANALYSIS 9

zero) In contrast, for two random variables that are merely uncorrelated, suchnonlinear transformations do not have zero covariance in general

Thus, we could attempt to perform ICA by a stronger form of decorrelation, byfinding a representation where the y i are uncorrelated even after some nonlineartransformations This gives a simple principle of estimating the matrixW:

ICA estimation principle 1: Nonlinear decorrelation Find the matrixW so that for anyi6=j, the componentsyi andyjare uncorrelated, and the transformed

componentsg(yi) andh(yj are uncorrelated, wheregandhare some suitable nonlinear functions.

This is a valid approach to estimating ICA: If the nonlinearities are properly chosen,the method does find the independent components In fact, computing nonlinearcorrelations between the two mixtures in Fig 1.5, one would immediately see thatthe mixtures are not independent

Although this principle is very intuitive, it leaves open an important question:How should the nonlinearitiesgandhbe chosen? Answers to this question can befound be using principles from estimation theory and information theory Estimationtheory provides the most classic method of estimating any statistical model: the

maximum likelihood method (Chapter 9) Information theory provides exact measures

of independence, such as mutual information (Chapter 10) Using either one of these

theories, we can determine the nonlinear functionsgandhin a satisfactory way

Independent components are the maximally nongaussian components

Another very intuitive and important principle of ICA estimation is maximum gaussianity (Chapter 8) The idea is that according to the central limit theorem,sums of nongaussian random variables are closer to gaussian that the original ones.Therefore, if we take a linear combinationy = P

non-i b i x i of the observed mixturevariables (which, because of the linear mixing model, is a linear combination of theindependent components as well), this will be maximally nongaussian if it equalsone of the independent components This is because if it were a real mixture of two

or more components, it would be closer to a gaussian distribution, due to the centrallimit theorem

Thus, the principle can be stated as follows

ICA estimation principle 2: Maximum nongaussianity Find the local maxima

of nongaussianity of a linear combination y =

P

ibixi under the constraint that the variance ofyis constant Each local maximum gives one independent component.

To measure nongaussianity in practice, we could use, for example, the kurtosis.

Kurtosis is a higher-order cumulant, which are some kind of generalizations ofvariance using higher-order polynomials Cumulants have interesting algebraic andstatistical properties which is why they have an important part in the theory of ICA.For example, comparing the nongaussianities of the components given by the axes

in Figs 1.4 and 1.5, we see that in Fig 1.5 they are smaller, and thus Fig 1.5 cannotgive the independent components (see Chapter 8)

An interesting point is that this principle of maximum nongaussianity showsthe very close connection between ICA and an independently developed technique

10 INTRODUCTION

called projection pursuit In projection pursuit, we are actually looking for maximally

nongaussian linear combinations, which are used for visualization and other purposes.Thus, the independent components can be interpreted as projection pursuit directions.When ICA is used to extract features, this principle of maximum nongaussianity

also shows an important connection to sparse coding that has been used in

neuro-scientific theories of feature extraction (Chapter 21) The idea in sparse coding is

to represent data with components so that only a small number of them are “active”

at the same time It turns out that this is equivalent, in some situations, to findingcomponents that are maximally nongaussian

The projection pursuit and sparse coding connections are related to a deep result

that says that ICA gives a linear representation that is as structured as possible.

This statement can be given a rigorous meaning by information-theoretic concepts(Chapter 10), and shows that the independent components are in many ways easier

to process than the original random variables In particular, independent componentsare easier to code (compress) than the original variables

ICA estimation needs more than covariances There are many other ods for estimating the ICA model as well Many of them will be treated in thisbook What they all have in common is that they consider some statistics that are notcontained in the covariance matrix (the matrix that contains the covariances betweenall pairs of thex i)

meth-Using the covariance matrix, we can decorrelate the components in the ordinarylinear sense, but not any stronger Thus, all the ICA methods use some form of

higher-order statistics, which specifically means information not contained in the

covariance matrix Earlier, we encountered two kinds of higher-order info rmation:the nonlinear correlations and kurtosis Many other kinds can be used as well

Numerical methods are important In addition to the estimation principle, onehas to find an algorithm for implementing the computations needed Because theestimation principles use nonquadratic functions, the computations needed usuallycannot be expressed using simple linear algebra, and therefore they can be quite de-manding Numerical algorithms are thus an integral part of ICA estimation methods.The numerical methods are typically based on optimization of some objectivefunctions The basic optimization method is the gradient method Of particularinterest is a fixed-point algorithm called FastICA that has been tailored to exploit theparticular structure of the ICA problem For example, we could use both of thesemethods to find the maxima of the nongaussianity as measured by the absolute value

of kurtosis

HISTORY OF ICA 11

1.4 HISTORY OF ICA

The technique of ICA, although not yet the name, was introduced in the early 1980s

by J H´erault, C Jutten, and B Ans [178, 179, 16] As recently reviewed by Jutten[227], the problem first came up in 1982 in a neurophysiological setting In asimplified model of motion coding in muscle contraction, the outputsx1( t )andx2( t )

were two types of sensory signals measuring muscle contraction, ands1( t )ands2( t )

were the angular position and velocity of a moving joint Then it is not unreasonable

to assume that the ICA model holds between these signals The nervous systemmust be somehow able to infer the position and velocity signalss1( t ) ;s2( t )from themeasured responsesx1( t ) ;x2( t ) One possibility for this is to learn the inverse modelusing the nonlinear decorrelation principle in a simple neural network H´erault andJutten proposed a specific feedback circuit to solve the problem This approach iscovered in Chapter 12

All through the 1980s, ICA was mostly known among French researchers, withlimited influence internationally The few ICA presentations in international neuralnetwork conferences in the mid-1980s were largely buried under the deluge of in-terest in back-propagation, Hopfield networks, and Kohonen’s Self-Organizing Map(SOM), which were actively propagated in those times Another related field washigher-order spectral analysis, on which the first international workshop was orga-nized in 1989 In this workshop, early papers on ICA by J.-F Cardoso [60] and

P Comon [88] were given Cardoso used algebraic methods, especially higher-ordercumulant tensors, which eventually led to the JADE algorithm [72] The use offourth-order cumulants has been earlier proposed by J.-L Lacoume [254] In signalprocessing literature, classic early papers by the French group are [228, 93, 408, 89]

A good source with historical accounts and a more complete list of references is[227]

In signal processing, there had been earlier approaches in the related problem ofblind signal deconvolution [114, 398] In particular, the results used in multichannelblind deconvolution are very similar to ICA techniques

The work of the scientists in the 1980’s was extended by, among others, A chocki and R Unbehauen, who were the first to propose one of the presently mostpopular ICA algorithms [82, 85, 84] Some other papers on ICA and signal separationfrom early 1990s are [57, 314] The “nonlinear PCA” approach was introduced by thepresent authors in [332, 232] However, until the mid-1990s, ICA remained a rathersmall and narrow research effort Several algorithms were proposed that wor ked,usually in somewhat restricted problems, but it was not until later that the rigorousconnections of these to statistical optimization criteria were exposed

Ci-ICA attained wider attention and growing interest after A.J Bell and T.J Sejnowskipublished their approach based on the infomax principle [35, 36] in the mid-90’s.This algorithm was further refined by S.-I Amari and his co-workers using the naturalgradient [12], and its fundamental connections to maximum likelihood estimation, aswell as to the Cichocki-Unbehauen algorithm, were established A couple of yearslater, the present authors presented the fixed-point or FastICA algorithm, [210, 192,

in Helsinki, Finland Both gathered more than 100 researchers working on ICA andblind signal separation, and contributed to the transformation of ICA to an establishedand mature field of research.

Part I

MATHEMATICAL PRELIMINARIES

to have basic knowledge on single variable probability theory, so that fundamentaldefinitions such as probability, elementary events, and random variables are familiar.Readers who already have a good knowledge of multivariate statistics can skip most

of this chapter For those who need a more extensive review or more information onadvanced matters, many good textbooks ranging from elementary ones to advancedtreatments exist A widely used textbook covering probability, random variables, andstochastic processes is [353]

2.1 PROBABILITY DISTRIBUTIONS AND DENSITIES

2.1.1 Distribution of a random variable

In this book, we assume that random variables are continuous-valued unless stated

otherwise The cumulative distribution function (cdf)F xof a random variablexatpointx = x0is defined as the probability thatxx0:

F x ( x0) = P ( xx0) (2.1)Allowingx0to change from?1to1defines the whole cdf for all values ofx.Clearly, for continuous random variables the cdf is a nonnegative, nondecreasing(often monotonically increasing) continuous function whose values lie in the interval

15

16 RANDOM VECTORS AND INDEPENDENCE

σ σ

m

Fig 2.1 A gaussian probability density function with meanmand standard deviation.

0F x ( x )1 From the definition, it also follows directly thatF x (?1) = 0, and

Usually a probability distribution is characterized in terms of its density function

rather than cdf Formally, the probability density function (pdf)p x ( x )of a continuousrandom variablexis obtained as the derivative of its cumulative distribution function:

p x ( x0) = dF x ( x )

dx

Example 2.1 The gaussian (or normal) probability distribution is used in numerous

models and applications, for example to describe additive noise Its density function

PROBABILITY DISTRIBUTIONS AND DENSITIES 17

Here the parameterm(mean) determines the peak point of the symmetric densityfunction, and(standard deviation), its effective width (flatness or sharpness of thepeak) See Figure 2.1 for an illustration

Generally, the cdf of the gaussian density cannot be evaluated in closed form using(2.3) The term1 =p

2.1.2 Distribution of a random vector

Assume now thatxis ann-dimensional random vector

whereTdenotes the transpose (We take the transpose because all vectors in this bookare column vectors Note that vectors are denoted by boldface lowercase letters.) Thecomponentsx1;x2;::: ;x nof the column vectorxare continuous random variables.The concept of probability distribution generalizes easily to such a random vector

In particular, the cumulative distribution function ofxis defined by

where P ( : ) again denotes the probability of the event in parentheses, and x0 issome constant value of the random vectorx The notationxx0means that eachcomponent of the vectorxis less than or equal to the respective component of thevectorx0 The multivariate cdf in Eq (2.7) has similar properties to that of a singlerandom variable It is a nondecreasing function of each component, with values lying

in the interval0 Fx( x ) 1 When all the components ofxapproach infinity,

Fx( x )achieves its upper limit1; when any componentx i! ?1,Fx( x ) = 0.The multivariate probability density functionpx( x )ofxis defined as the derivative

of the cumulative distribution functionFx( x )with respect to all components of therandom vectorx:

18 RANDOM VECTORS AND INDEPENDENCE

wherex0;iis theith component of the vectorx0 Clearly,

Z +1

?1

This provides the appropriate normalization condition that a true multivariate bility densitypx( x )must satisfy

proba-In many cases, random variables have nonzero probability density functions only

on certain finite intervals An illustrative example of such a case is presented below

Example 2.2 Assume that the probability density function of a two-dimensional

random vectorz=( x;y ) T is

pz( z ) = p x;y ( x;y ) =

( 3

7(2?x )( x + y ) ; x2[0 ; 2] ; y2[0 ; 1]

Let us now compute the cumulative distribution function of z It is obtained byintegrating over bothxandy, taking into account the limits of the regions where thedensity is nonzero When eitherx0ory0, the densitypz( z )and consequentlyalso the cdf is zero In the region where0 < x2and0 < y1, the cdf is givenby

4xy ) ; 0 < x2 ; 0 < y1

3

7x (1 +3

4x? 1

2.1.3 Joint and marginal distributions

The joint distribution of two different random vectors can be handled in a similarmanner In particular, letybe another random vector having in general a dimension

mdifferent from the dimensionnofx The vectorsxandycan be concatenated to

Assume now thatxis ann-dimensional random vector

whereTdenotes the transpose (We take the transpose because all vectors in... column vectors Note that vectors are denoted by boldface lowercase letters.) Thecomponentsx1;x2;::: ;x nof the column vectorxare