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Mathématiqueset
Applications
Directeurs de la collection :J. Garnier et V. Perrier
73
For further volumes:http://www.springer.com/series/2966
Remi ABGRALLInst. Math., Inst. Polytechnique de Bordeaux, FR
Gregoire ALLAIRECMAP, Ecole Polytechnique, Palaiseau, FR
Michel BENAIMInst. Math., Univ. de Neuchatel, CH
Maıtine BERGOUNIOUXMAPMO, Universite d0Orleans, [email protected]
Thierry COLINInst. Math., Universite Bordeaux 1, FR
Marie-Christine COSTAUMA, ENSTA, Paris, FR
Arnaud DEBUSSCHEENS Cachan, Bruz, FR
Isabelle GALLAGHERInst. Math. Jussieu, Univ. Paris 7, FR
Josselin GARNIERLab. Proba. et Mod. Aleatoires, Univ. Paris 7, FR
Stephane GAUBERTINRIA, Saclay - Ile-de-France, Orsay, FR
Emmanuel GOBETCMAP, Ecole Polytechnique, Palaiseau, FR
Raphaele HERBINCMI LATP, Universite d’Aix-Marseille, FR
Marc HOFFMANNCEREMADE, Universite Paris-Dauphine, FR
Claude LE BRISCERMICS, ENPC, Marne la Vallee, FR
Sylvie MELEARDCMAP, Ecole Polytechnique, Palaiseau, FR
Felix OTTOInstitute of Applied Math., Bonn, GE
Valerie PERRIERLab. Jean-Kunztmann, ENSIMAG, Grenoble, FR
Philippe ROBERTINRIA Rocquencourt, Le Chesnay, FR
Pierre ROUCHONAutomatique et Systemes, Ecole Mines, Paris, FR
Bruno SALVYINRIA, LIP - ENS Lyon, FR
Annick SARTENAERDept. Mathematiques, Univ. Namur, Namur, BE
Eric SONNENDRUCKERIRMA, Strasbourg, [email protected]
Alain TROUVECMLA, ENS Cachan, [email protected]
Cedric VILLANIIHP, Paris, FR
Enrique ZUAZUABCAM, Bilbao, ES
MATHÉMATIQUES & APPLICATIONSComité de Lecture 2012–2015/Editorial Board 2012–2015
Directeurs de la collection:J. GARNIER et V. PERRIER
Serge Cohen • Jacques Istas
Fractional Fieldsand Applications
123
Serge CohenInstitut de Mathématiques de ToulouseUniversité Paul SabatierToulouseFrance
Jacques IstasLaboratoire Jean KuntzmannUniversité de Grenoble et CNRSGrenobleFrance
ISSN 1154-483XISBN 978-3-642-36738-0 ISBN 978-3-642-36739-7 (eBook)DOI 10.1007/978-3-642-36739-7Springer Heidelberg New York Dordrecht London
Library of Congress Control Number: 2013933026
Mathematics Subject Classification (2010): 60G18, 60G22, 62M40, 65C99
� Springer-Verlag Berlin Heidelberg 2013This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed. Exempted from this legal reservation are briefexcerpts in connection with reviews or scholarly analysis or material supplied specifically for thepurpose of being entered and executed on a computer system, for exclusive use by the purchaser of thework. Duplication of this publication or parts thereof is permitted only under the provisions ofthe Copyright Law of the Publisher’s location, in its current version, and permission for use mustalways be obtained from Springer. Permissions for use may be obtained through RightsLink at theCopyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility forany errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained herein.
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Foreword
In May 1990, I attended the annual day of the French Mathematical Society(SMF), which, that year, had chosen the crisp new subject of wavelet analysis. Theflow of wavelets had not yet spread among large areas of Science, but a few topresearchers were already actively working in this field. The SMF was aware of itsgreat potential, both inside mathematics and for other sciences, and had decided topresent it to its members.
At the end of the afternoon, a short man with a brushy moustache addressed me.He explained that he was a probabilist studying Gaussian fields, had heard that Iwas a student of Yves Meyer (the best possible recommendation…) and wished toenroll me in his research project. I had a very light background in probability, andwas rather taken aback. But, if I had learned anything from my short mathematicallife as a student in the starting subject of wavelets, it certainly was that big stepsforward are the consequence of chance and unexpected encounters… and I jumpedin without second thoughts. The next step was a couple of weeks at Clermont-Ferrand, where Albert Benassi explained his program. Albert was not a Bourbaki-style mathematician, but he was far-sighted, and had caught sight of a rich land,which would later become a fruitful field of interactions for mathematicians, signalanalysts, and image processors. When Albert was doing mathematics, he waswearing seven-league boots, and did not clear all the way between each step; sofollowing him in this adventure was a challenge… and a very rewarding one!When I joined, Daniel Roux, then Albert’s Ph.D. student, was already part of theteam. Serge Cohen and Jacques Istas would join a little later, and would quicklybecome prominent contributors in the development of this new area of mathe-matics. So that the present book can certainly be described as the final, polishedoutput of the adventure that we lived as young scientists.
One should not infer that it only is an account of 20 years of exciting mathe-matics. It is much more: An introduction to a subject which now is extremelyactive, and flourishing in many directions. The conjunction of recent high reso-lution data acquisition techniques and Internet has for consequence that largecollections of high resolution signals and images, coming from many differentareas of science, are now available. The challenge for signal and image processingis to store, transmit, treat, and classify these data, which requires the introductionof wider and wider classes of sophisticated models. Their mathematical properties
v
have to be investigated, simulations are needed in order to confront visuallymodels with data, and statistical tools need to be developed so that the corre-sponding key parameters of the models can be identified. This ambitious scientificprogram precisely is the purpose of this book.
A first originality is that it deals with fields, and not processes. If the literatureconcerning stochastic processes is extremely rich, it is much less the case forfields; this may be due to the fact that several key probabilistic tools have beendeveloped specifically in dimension one, as a consequence of the natural inter-pretation of the one-dimensional variable as the time axis, and have less naturalextensions in several dimensions. However, the need for a similar treatment offields increased recently, with many motivations raising from image processing (in2D), or for simulations of three-dimensional phenomena. A second originality isthat the book encompasses the three facets of the same scientific field: A proba-bilistic study of classes of random fields, the development of statistical methods ofidentification of their parameters, and finally simulation techniques. These topicsrequire diverse skills and usually are not met in the same book. However, each ofthem enriches the other two: Questions raised in one part motivate developmentsin another, and their conjunction will make this book an extremely valuable tool,both for mathematicians interested in understanding possible applications, and forscientists working in signal an image processing, and who want to master themathematical background behind the models that they use; let us stress the factthat the very detailed and pedagogical chapter dealing with preliminaries make thebook really accessible to scientists with a light background in probability andanalysis.
This rich mixing of different aspects of the same subject certainly is in the spiritof the new way of performing scientific investigations which was initiated byBenoît Mandelbrot, half a century ago: His motivations to develop mathematicalmodels rose from the inspection of data picked in a wide range of different sci-ences, and the mathematical properties of these models would often follow fromobserving their simulations; it is by no means a surprise that fractal analysis is arecurrent theme in this book. If the reader will allow me a bold comparison, thecomposition of the book with three views of the same scientific topics is remi-niscent of some of the most famous Picasso portraits, where the juxtaposition ofslightly different perspectives give a much deeper insight of the subject.
October 2011 Stéphane Jaffard,Professor of Mathematics,Université Paris Est Créteil
vi Foreword
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Stochastic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Kolmogorov’s Consistency Theorem . . . . . . . . . . . . . . 62.1.3 Gaussian Fields and Non-Negative Definite
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Orthonormal Expansions of Gaussian Fields . . . . . . . . . 132.1.5 Orthogonality Between Gaussian Processes. . . . . . . . . . 172.1.6 Gaussian Random Measure . . . . . . . . . . . . . . . . . . . . . 222.1.7 Poisson Random Measure . . . . . . . . . . . . . . . . . . . . . . 252.1.8 Lévy Random Measure . . . . . . . . . . . . . . . . . . . . . . . 272.1.9 Stable Random Measure . . . . . . . . . . . . . . . . . . . . . . . 282.1.10 Complex Isotropic Random Measure . . . . . . . . . . . . . . 312.1.11 Stationary Fields and Fields with Stationary
Increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.1.12 Regularity of the Sample Paths . . . . . . . . . . . . . . . . . . 362.1.13 Sequences of Continuous Fields . . . . . . . . . . . . . . . . . 37
2.2 Fractal Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2.1 Hölder Continuity, and Exponents . . . . . . . . . . . . . . . . 422.2.2 Fractional Derivative and Integration . . . . . . . . . . . . . . 432.2.3 Fractional Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 462.2.4 Lemarié-Meyer Basis . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.3.1 Inequality for Anti-Correlated Gaussian
Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.3.2 Tail of Standard Gaussian Random Variables . . . . . . . . 502.3.3 Conditional Independence. . . . . . . . . . . . . . . . . . . . . . 512.3.4 Some Properties of Covariance Functions . . . . . . . . . . . 512.3.5 Examples and Counter-Examples of Covariance
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3.6 Covariance Functions on the Sphere . . . . . . . . . . . . . . 52
vii
2.3.7 Gaussian Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.8 Version Versus Modification. . . . . . . . . . . . . . . . . . . . 532.3.9 Sum of Fields with Stationary Increments. . . . . . . . . . . 532.3.10 Equivalence of the Distributions
of Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . 53
3 Self-Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Self-Similarity and Fractional Brownian Motion . . . . . . . . . . . . 56
3.2.1 Deterministic Case. . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.2 Fractional Brownian Motion . . . . . . . . . . . . . . . . . . . . 573.2.3 Semi Self-Similarity. . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Self-Similarity for Multidimensional Fields . . . . . . . . . . . . . . . 793.3.1 Self-Similarity with Linear Stationary Increments . . . . . 803.3.2 Self-Similarity for Sheets . . . . . . . . . . . . . . . . . . . . . . 82
3.4 Stable Self-Similar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.5 Self-Similarity and Regularity of the Sample Paths . . . . . . . . . . 873.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.6.1 Composition of Self-Similar Processes . . . . . . . . . . . . . 893.6.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.6.3 Markov Property for Gaussian Processes . . . . . . . . . . . 893.6.4 Ornstein-Ühlenbeck Process . . . . . . . . . . . . . . . . . . . . 903.6.5 Bifractional Brownian Motion . . . . . . . . . . . . . . . . . . . 903.6.6 Random Measure and Lévy Processes . . . . . . . . . . . . . 913.6.7 Properties of Complex Lévy Random Measure . . . . . . . 913.6.8 Hausdorff Dimension of Graphs of Self-Similar
Processes with Stationary Increments . . . . . . . . . . . . . . 923.6.9 Fractional Brownian Motion and Cantor Set . . . . . . . . . 923.6.10 Fourier Expansion of Fractional Brownian
Motion When 0 \ H B 1/2. . . . . . . . . . . . . . . . . . . . . 933.6.11 Exercise: Self-Similar Process with Smooth
Sample Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4 Asymptotic Self-Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.3 Gaussian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.1 Filtered White Noises. . . . . . . . . . . . . . . . . . . . . . . . . 984.3.2 Multifractional Brownian Field . . . . . . . . . . . . . . . . . . 1034.3.3 Step Fractional Brownian Motion . . . . . . . . . . . . . . . . 1184.3.4 Generalized Multifractional Gaussian Process . . . . . . . . 1244.3.5 Gaussian Random Weierstrass Function . . . . . . . . . . . . 1324.3.6 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.4 Lévy Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
viii Contents
4.4.1 Moving Average Fractional Lévy Fields. . . . . . . . . . . . 1364.4.2 Real Harmonizable Fractional Lévy Fields . . . . . . . . . . 1424.4.3 A Comparison of Lévy Fields . . . . . . . . . . . . . . . . . . . 1524.4.4 Real Harmonizable Multifractional Lévy Fields. . . . . . . 152
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.5.1 Lass and Self-Similarity . . . . . . . . . . . . . . . . . . . . . . . 1544.5.2 Bivariate Lass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.5.3 Multifractional Functions with Jumps. . . . . . . . . . . . . . 1554.5.4 Uniform Convergence of the Series
Expansion of the mBm. . . . . . . . . . . . . . . . . . . . . . . . 155
5 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.1 Unifractional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.1.1 Maximum Likelihood Estimator and Whittle’sApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.1.2 Variations Estimator for the Standard FractionalBrownian Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.1.3 Application to Filtered White Noises . . . . . . . . . . . . . . 1635.1.4 Further Fractional Parameters . . . . . . . . . . . . . . . . . . . 1705.1.5 Singularity Function: Interests and Estimations . . . . . . . 1725.1.6 Higher Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.2.1 Cramer-Rao Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . 1785.2.2 Minimax Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.3 Multifractional Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1795.3.1 Smooth Multifractional Function . . . . . . . . . . . . . . . . . 1805.3.2 Step-Wise Fractional Function. . . . . . . . . . . . . . . . . . . 181
5.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1825.4.1 Harmonizable Fractional Lévy Processes . . . . . . . . . . . 1825.4.2 Intermittency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.5 Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.5.1 A Short Review on Optimal Recovery . . . . . . . . . . . . . 1855.5.2 Approximation of Integral of Fractional Processes . . . . . 186
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.6.1 Ornstein-Ühlenbeck Process . . . . . . . . . . . . . . . . . . . . 1885.6.2 Estimation for Smooth Lass Processes . . . . . . . . . . . . . 1885.6.3 A Strange Estimator. . . . . . . . . . . . . . . . . . . . . . . . . . 1895.6.4 Complex Variations . . . . . . . . . . . . . . . . . . . . . . . . . . 1895.6.5 Optimal Estimation of Integral of Brownian Motion . . . 1905.6.6 Estimation of the Singularity Function . . . . . . . . . . . . . 190
6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.2 Fractional Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . 192
Contents ix
6.2.1 Cholevski Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1926.2.2 Random Midpoint Displacement . . . . . . . . . . . . . . . . . 1936.2.3 Discretization of Integral Representation
in Dimension 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.2.4 Approximate Wavelet Expansion . . . . . . . . . . . . . . . . . 1946.2.5 Multifractional Gaussian Processes . . . . . . . . . . . . . . . 195
6.3 Fractional Gaussian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.3.1 Discretization of Integral Representation
in Dimension d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.3.2 The Procedure Fieldsim for Random Fields . . . . . . . 197
6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.5 Simulation of Real Harmonizable Lévy Fields . . . . . . . . . . . . . 204
6.5.1 Using FracSim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2096.5.2 Instructions for Using FracSim . . . . . . . . . . . . . . . . . . 2156.5.3 Description of R Script . . . . . . . . . . . . . . . . . . . . . . . 216
6.6 Visual Meaning of the Fractional Index . . . . . . . . . . . . . . . . . . 218
Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
x Contents
Notations
• a.s.: almost surely.
• ða:s:Þ= : almost sure equality.
• #X: cardinal of the set X.• BðKÞis the space of bounded functions on a set K:
• CH is the space of Hölder-continuous functions on ½0; 1�d:• Cm is the space of m times differentiable functions.• Cm;n is the space of m times differentiable function in the first variable and n
times differentiable function in the second variable.• detðAÞ: determinant of the matrix A.
• ðdÞ= : equality in distribution.
• Ek: let ZðdÞ¼N 1ð0; 1Þ (notation defined below), then Ek ¼ EjZjk.• i.i.d.: independent and identically distributed.• EX: expectation of the random variable X.
• bf ðnÞ ¼R dR
expðix:nÞf ðxÞ dxð2pÞd=2
• hf ; giL2ðRÞ ¼R
Rf ðnÞ�gðnÞ dn
ð2pÞd=2 :
• We denote by CðtÞ ¼Rþ1
0 xt�1e�xdx the classical Gamma function for t [ 0:• iff : if and only if.• KX is the Reproducing Kernel Hilbert Space (RKHS) associated with a Gaussian
field X:
• log2ðxÞ ¼lnðxÞlnð2Þ :
• K ¼ Z �Z � f1g• Kþ ¼ N
� � Z � f1g [ 0 � Z � f0g• N the set of non-negative integers.• N
� the set of positive integers.• N dðm;RÞ: d-dimensional Gaussian vector of expectation m and covariance
matrix R.• f ðxÞ ¼ OðgðxÞÞ as x! x0 when there exists a finite constant C such thatjf ðxÞj\CjgðxÞj in a neighborhood of x0: The point x0 may be a real number,þ1, or �1:
xi
• Xn ¼ oPðYnÞ if Xn; Yn are sequences of random variables defined on the sameprobability space such that
8�[ 0; limn!1
PðjXnj � �jYnjÞ ¼ 0:
This notation can be generalized to fields XðxÞ; YðxÞas XðxÞ ¼ oPðYðxÞÞ in aneighborhood of some x0:
• Xn ¼ OPðYnÞ if Xn; Yn are sequences of random variables defined on the sameprobability space such that
8�[ 0; 9M [ 0; supn2N
PðjXnj[ MjYnjÞ\�:
This notation can be generalized to fields XðxÞ; YðxÞas XðxÞ ¼ OPðYðxÞÞ in aneighborhood of some x0:
• f ðxÞ ¼ oðgðxÞÞ as x! x0 when limx!x0
f ðxÞgðxÞ ¼ 0;x0 may be a real number or þ1
or �1:• Q is the set of rational numbers.• R the set of real numbers.• RðzÞ; IðzÞ are respectively the real part and the complex part of a complex
number z:• Rðx; yÞ ¼ EXxXy: covariance of a second-order centered field.• r.v.: random variable.• S is the space of fast decreasing functions.• S0 is the space of tempered distributions.• Sd ¼ fx 2 R
dþ1 s. t. kxk ¼ 1g; the d-dimensional sphere where kxk is theEuclidean norm of x 2 R
dþ1:• s.t.: such that.• supp f is the support of the function f :• rðXÞ is a sigma field on the set X:• tA: transpose of the matrix A.• varX: variance of the random variable X.
• un� vn means that limn!þ1
un
vn¼ 1 :
• un � vn means that there exists a constant 0\C\1 such that1C un vnCun 8n 2 N:
• f ðxÞ� gðxÞas x! x0 when limx!x0
f ðxÞgðxÞ ¼ 1;x0may be a real number or þ1 or
�1:• ½x� is the integer such that ½x� x \ ½x� þ 1.
xii Notations
