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Scaling, Fractals and Wavelets
ISBN-13: 9781848210721 / Angielski / Twarda / 2008 / 464 str.
Scaling, Fractals and Wavelets
ISBN-13: 9781848210721 / Angielski / Twarda / 2008 / 464 str.
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Scaling is a mathematical transformation that enlarges or diminishes objects. The technique is used in a variety of areas, including finance and image processing. This book is organized around the notions of scaling phenomena and scale invariance. The various stochastic models commonly used to describe scaling ? self-similarity, long-range dependence and multi-fractals ? are introduced. These models are compared and related to one another. Next, fractional integration, a mathematical tool closely related to the notion of scale invariance, is discussed, and stochastic processes with prescribed scaling properties (self-similar processes, locally self-similar processes, fractionally filtered processes, iterated function systems) are defined. A number of applications where the scaling paradigm proved fruitful are detailed: image processing, financial and stock market fluctuations, geophysics, scale relativity, and fractal time-space.
Preface 17
Chapter 1. Fractal and Multifractal Analysis in Signal Processing 19
Jacques LEVY VEHEL and Claude TRICOT
1.1. Introduction 19
1.2.Dimensions of sets 20
1.2.1.Minkowski–Bouligand dimension 21
1.2.2. Packing dimension 25
1.2.3.Covering dimension 27
1.2.4. Methods for calculating dimensions 29
1.3. Holder exponents 33
1.3.1. Holder exponents related to a measure 33
1.3.2. Theorems on set dimensions 33
1.3.3. Holder exponent related to a function 36
1.3.4. Signal dimension theorem 42
1.3.5. 2–microlocal analysis 45
1.3.6. An example: analysis of stock market price 46
1.4. Multifractal analysis 48
1.4.1. What is the purpose of multifractal analysis? 48
1.4.2. First ingredient: local regularity measures 49
1.4.3. Second ingredient: the size of point sets of the same regularity 50
1.4.4. Practical calculation of spectra 52
1.4.5. Refinements: analysis of the sequence of capacities, mutual analysis and multisingularity 60
1.4.6. The multifractal spectra of certain simple signals 62
1.4.7.Two applications 66
1.5.Bibliography 68
Chapter 2. Scale Invariance and Wavelets 71
Patrick FLANDRIN, Paulo GONCALVES and Patrice ABRY
2.1. Introduction 71
2.2. Models for scale invariance 72
2.2.1. Intuition 72
2.2.2. Self–similarity 73
2.2.3. Long–range dependence 75
2.2.4. Local regularity 76
2.2.5. Fractional Brownian motion: paradigm of scale invariance 77
2.2.6. Beyond the paradigm of scale invariance 79
2.3.Wavelet transform 81
2.3.1. Continuous wavelet transform 81
2.3.2.Discretewavelet transform 82
2.4. Wavelet analysis of scale invariant processes 85
2.4.1. Self–similarity 86
2.4.2. Long–range dependence 88
2.4.3. Local regularity 90
2.4.4. Beyond second order 92
2.5. Implementation: analysis, detection and estimation 92
2.5.1. Estimation of the parameters of scale invariance 93
2.5.2. Emphasis on scaling laws and determination of the scaling range 96
2.5.3. Robustness of the wavelet approach 98
2.6. Conclusion 100
2.7.Bibliography 101
Chapter 3.Wavelet Methods for Multifractal Analysis of Functions 103
Stephane JAFFARD
3.1. Introduction 103
3.2. General points regarding multifractal functions 104
3.2.1. Important definitions 104
3.2.2. Wavelets and pointwise regularity 107
3.2.3. Local oscillations 112
3.2.4. Complements 116
3.3. Random multifractal processes 117
3.3.1. Levy processes 117
3.3.2. Burgers equation and Brownian motion 120
3.3.3. Random wavelet series 122
3.4. Multifractal formalisms 123
3.4.1. Besov spaces and lacunarity 123
3.4.2. Construction of formalisms 126
3.5. Bounds of the spectrum 129
3.5.1. Bounds according to the Besov domain 129
3.5.2. Bounds deduced from histograms 132
3.6. The grand–canonical multifractal formalism 132
3.7.Bibliography 134
Chapter 4. Multifractal Scaling: General Theory and Approach by Wavelets 139
Rudolf RIEDI
4.1. Introduction and summary 139
4.2. Singularity exponents 140
4.2.1.Holder continuity 140
4.2.2. Scaling of wavelet coefficients 142
4.2.3. Other scaling exponents 144
4.3. Multifractal analysis 145
4.3.1. Dimension based spectra 145
4.3.2. Grain based spectra 146
4.3.3. Partition function and Legendre spectrum 147
4.3.4. Deterministic envelopes 149
4.4. Multifractal formalism 151
4.5. Binomial multifractals 154
4.5.1.Construction 154
4.5.2. Wavelet decomposition 157
4.5.3. Multifractal analysis of the binomial measure 158
4.5.4. Examples 160
4.5.5. Beyond dyadic structure 162
4.6. Wavelet based analysis 163
4.6.1. The binomial revisited with wavelets 163
4.6.2. Multifractal properties of the derivative 165
4.7. Self–similarity and LRD 167
4.8. Multifractal processes 168
4.8.1.Construction and simulation 169
4.8.2. Global analysis 170
4.8.3. Local analysis of warped FBM 170
4.8.4.LRDand estimation ofwarped FBM 173
4.9.Bibliography 173
Chapter 5. Self–similar Processes 179
Albert BENASSI and Jacques ISTAS
5.1. Introduction 179
5.1.1.Motivations 179
5.1.2. Scalings 182
5.1.3. Distributions of scale invariant masses 184
5.1.4. Weierstrass functions 185
5.1.5. Renormalization of sums of random variables 186
5.1.6. A common structure for a stochastic (semi–)self–similar process 187
5.1.7. Identifying Weierstrass functions 188
5.2. The Gaussian case 189
5.2.1. Self–similar Gaussian processes with r–stationary increments 189
5.2.2. Elliptic processes 190
5.2.3. Hyperbolic processes 191
5.2.4. Parabolic processes 192
5.2.5. Wavelet decomposition 192
5.2.6. Renormalization of sums of correlated random variable 193
5.2.7. Convergence towards fractional Brownian motion 193
5.3. Non–Gaussian case 195
5.3.1. Introduction 195
5.3.2. Symmetric –stable processes 196
5.3.3. Censov and Takenaka processes 198
5.3.4. Wavelet decomposition 198
5.3.5. Process subordinated to Brownian measure 199
5.4. Regularity and long–range dependence 200
5.4.1. Introduction 200
5.4.2. Two examples 201
5.5.Bibliography 202
Chapter 6. Locally Self–similar Fields 205
Serge COHEN
6.1. Introduction 205
6.2. Recap of two representations of fractional Brownian motion 207
6.2.1. Reproducing kernel Hilbert space 207
6.2.2. Harmonizable representation 208
6.3. Two examples of locally self–similar fields 213
6.3.1. Definition of the local asymptotic self–similarity (LASS) 213
6.3.2. Filtered white noise (FWN) 214
6.3.3. Elliptic Gaussian random fields (EGRP) 215
6.4. Multifractional fields and trajectorial regularity 218
6.4.1.Two representations of theMBM 219
6.4.2. Study of the regularity of the trajectories of the MBM 221
6.4.3. Towards more irregularities: generalized multifractional Brownian motion (GMBM) and step fractional Brownian motion (SFBM) 222
6.5. Estimate of regularity 226
6.5.1. General method: generalized quadratic variation 226
6.5.2. Application to the examples 228
6.6.Bibliography 235
Chapter 7. An Introduction to Fractional Calculus 237
Denis MATIGNON
7.1. Introduction 237
7.1.1.Motivations 237
7.1.2. Problems 238
7.1.3. Outline 239
7.2. Definitions 240
7.2.1. Fractional integration 240
7.2.2. Fractional derivatives within the framework of causal distributions 242
7.2.3. Mild fractional derivatives, in the Caputo sense 246
7.3. Fractional differential equations 251
7.3.1. Example 251
7.3.2. Framework of causal distributions 254
7.3.3. Framework of functions expandable into fractional power series ( –FPSE) 255
7.3.4. Asymptotic behavior of fundamental solutions 257
7.3.5. Controlled–and–observed linear dynamic systems of fractional order 261
7.4. Diffusive structure of fractional differential systems 262
7.4.1. Introduction to diffusive representations of pseudo–differential operators 263
7.4.2. General decomposition result 264
7.4.3. Connection with the concept of long memory 265
7.4.4. Particular case of fractional differential systems of commensurate orders 265
7.5. Example of a fractional partial differential equation 266
7.5.1. Physical problem considered 267
7.5.2. Spectral consequences 268
7.5.3. Time–domain consequences 268
7.5.4. Free problem 272
7.6. Conclusion 273
7.7.Bibliography 273
Chapter 8. Fractional Synthesis, Fractional Filters 279
Liliane BEL, Georges OPPENHEIM, Luc ROBBIANO and Marie–Claude VIANO
8.1. Traditional and less traditional questions about fractionals 279
8.1.1.Notes on terminology 279
8.1.2. Short and long memory 279
8.1.3. From integer to non–integer powers: filter based sample path design 280
8.1.4. Local and global properties 281
8.2. Fractional filters 282
8.2.1. Desired general properties: association 282
8.2.2. Construction and approximation techniques 282
8.3. Discrete time fractional processes 284
8.3.1. Filters: impulse responses and corresponding processes 284
8.3.2. Mixing and memory properties 286
8.3.3. Parameter estimation 287
8.3.4. Simulated example 289
8.4. Continuous time fractional processes 291
8.4.1. A non–self–similar family: fractional processes designed from fractional filters 291
8.4.2. Sample path properties: local and global regularity, memory 293
8.5. Distribution processes 294
8.5.1. Motivation and generalization of distribution processes 294
8.5.2. The family of linear distribution processes 294
8.5.3. Fractional distribution processes 295
8.5.4. Mixing and memory properties 296
8.6.Bibliography 297
Chapter 9. Iterated Function Systems and Some Generalizations: Local Regularity Analysis and Multifractal Modeling of Signals 301
Khalid DAOUDI
9.1. Introduction 301
9.2. Definition of the Holder exponent 303
9.3. Iterated function systems (IFS) 304
9.4. Generalization of iterated function systems 306
9.4.1. Semi–generalized iterated function systems 307
9.4.2. Generalized iterated function systems 308
9.5. Estimation of pointwise Holder exponent by GIFS 311
9.5.1. Principles of themethod 312
9.5.2. Algorithm 314
9.5.3.Application 315
9.6. Weak self–similar functions and multifractal formalism 318
9.7. Signal representation by WSA functions 320
9.8. Segmentation of signals by weak self–similar functions 324
9.9. Estimation of the multifractal spectrum 326
9.10. Experiments 327
9.11.Bibliography 329
Chapter 10. Iterated Function Systems and Applications in Image Processing 333
Franck DAVOINE and Jean–Marc CHASSERY
10.1. Introduction 333
10.2. Iterated transformation systems 333
10.2.1. Contracting transformations and iterated transformation systems 334
10.2.2.Attractor of an iterated transformation system 335
10.2.3. Collage theorem 336
10.2.4. Finally contracting transformation 338
10.2.5. Attractor and invariant measures 339
10.2.6. Inverse problem 340
10.3. Application to natural image processing: image coding 340
10.3.1. Introduction 340
10.3.2. Coding of natural images by fractals 342
10.3.3. Algebraic formulation of the fractal transformation 345
10.3.4. Experimentation on triangular partitions 351
10.3.5. Coding and decoding acceleration 352
10.3.6. Other optimization diagrams: hybrid methods 360
10.4.Bibliography 362
Chapter 11. Local Regularity and Multifractal Methods for Image and Signal Analysis 367
Pierrick LEGRAND
11.1. Introduction 367
11.2.Basic tools 368
11.2.1. Holder regularity analysis 368
11.2.2. Reminders on multifractal analysis 369
11.3. Holderian regularity estimation 371
11.3.1. Oscillations (OSC) 371
11.3.2. Wavelet coefficient regression (WCR) 372
11.3.3. Wavelet leaders regression (WL) 372
11.3.4.Limit inf and limit sup regressions 373
11.3.5. Numerical experiments 374
11.4. Denoising 376
11.4.1. Introduction 376
11.4.2. Minimax risk, optimal convergence rate and adaptivity 377
11.4.3. Wavelet based denoising 378
11.4.4. Non–linear wavelet coefficients pumping 380
11.4.5. Denoising using exponent between scales 383
11.4.6. Bayesian multifractal denoising 386
11.5. Holderian regularity based interpolation 393
11.5.1. Introduction 393
11.5.2.Themethod 393
11.5.3. Regularity and asymptotic properties 394
11.5.4. Numerical experiments 394
11.6. Biomedical signal analysis 394
11.7. Texture segmentation 401
11.8. Edge detection 403
11.8.1. Introduction 403
11.8.1.1. Edge detection 406
11.9. Change detection in image sequences using multifractal analysis 407
11.10. Image reconstruction 408
11.11.Bibliography 409
Chapter 12. Scale Invariance in Computer Network Traffic 413
Darryl VEITCH
12.1. Teletraffic a new natural phenomenon 413
12.1.1. A phenomenon of scales 413
12.1.2. An experimental science of man–made atoms 415
12.1.3. A random current 416
12.1.4. Two fundamental approaches 417
12.2. From a wealth of scales arise scaling laws 419
12.2.1. First discoveries 419
12.2.2.Laws reign 420
12.2.3. Beyond the revolution 424
12.3. Sources as the source of the laws 426
12.3.1.The sumor its parts 426
12.3.2.The on/off paradigm 427
12.3.3. Chemistry 428
12.3.4. Mechanisms 429
12.4. New models, new behaviors 430
12.4.1. Character of a model 430
12.4.2. The fractional Brownian motion family 431
12.4.3. Greedy sources 432
12.4.4. Never–ending calls 432
12.5. Perspectives 433
12.6.Bibliography 434
Chapter 13. Research of Scaling Law on Stock Market Variations 437
Christian WALTER
13.1. Introduction: fractals in finance 437
13.2. Presence of scales in the study of stock market variations 439
13.2.1. Modeling of stock market variations 439
13.2.2. Time scales in financial modeling 445
13.3. Modeling postulating independence on stock market returns 446
13.3.1. 1960–1970: from Pareto s law to Levy s distributions 446
13.3.2. 1970 1990: experimental difficulties of iid– –stable model 448
13.3.3. Unstable iid models in partial scaling invariance 452
13.4. Research of dependency and memory of markets 454
13.4.1. Linear dependence: testing of H–correlative models on returns 454
13.4.2. Non–linear dependence: validating H–correlative model on volatilities 456
13.5. Towards a rediscovery of scaling laws in finance 457
13.6.Bibliography 458
Chapter 14. Scale Relativity, Non–differentiability and Fractal Space–time 465
Laurent NOTTALE
14.1. Introduction 465
14.2. Abandonment of the hypothesis of space–time differentiability 466
14.3. Towards a fractal space–time 466
14.3.1. Explicit dependence of coordinates on spatio–temporal resolutions 467
14.3.2. From continuity and non–differentiability to fractality 467
14.3.3. Description of non–differentiable process by differential equations 469
14.3.4. Differential dilation operator 471
14.4. Relativity and scale covariance 472
14.5. Scale differential equations 472
14.5.1. Constant fractal dimension: Galilean scale relativity 473
14.5.2. Breaking scale invariance: transition scales 474
14.5.3. Non–linear scale laws: second order equations, discrete scale invariance, log–periodic laws 475
14.5.4. Variable fractal dimension: Euler–Lagrange scale equations 476
14.5.5. Scale dynamics and scale force 478
14.5.6. Special scale relativity log–Lorentzian dilation laws, invariant scale limit under dilations 481
14.5.7. Generalized scale relativity and scale–motion coupling 482
14.6. Quantum–like induced dynamics 488
14.6.1. Generalized Schrodinger equation 488
14.6.2. Application in gravitational structure formation 492
14.7. Conclusion 493
14.8.Bibliography 495
List of Authors 499
Index 503
Patrice Abry is a Professor in the Laboratoire de Physique at the Ecole Normale Superieure de Lyon, France. His current research interests include wavelet–based analysis and modelling of scaling phenomena and related topics, stable processes, multi–fractal, long–range dependence, local regularity of processes, infinitely divisible cascades and departures from exact scale invariance.
Paulo Goncalves graduated from the Signal Processing Department of ICPI, Lyon, France in 1993. He received the Masters (DEA) and Ph.D. degrees in signal processing from the Institut National Polytechnique, Grenoble, France, in 1990 and 1993 respectively. While working toward his Ph.D. degree, he was with Ecole Normale Superieure, Lyon. In 1994–96, he was a Postdoctoral Fellow at Rice University, Houston, TX. Since 1996, he is associate researcher at INRIA, first with Fractales (1996–99), and then with a research team at INRIA Rhone–Alpes (2000–2003). His research interests are in multiscale signal and image analysis, in wavelet–based statistical inference, with application to cardiovascular research and to remote sensing for land cover classification.
Jacques Levy Vehel graduated from Ecole Polytechnique in 1983 and from Ecole Nationale Superieure des Telecommuncations in 1985. He holds a Ph.D in Applied Mathematics from Universite d′Orsay. He is currently a research director at INRIA, Rocquencourt, where he created the Fractales team, a research group devoted to the study of fractal analysis and its applications to signal/image processing. He also leads a research team at IRCCYN, Nantes, with the same scientific focus. His current research interests include (multi)fractal processes, 2–microlocal analysis and wavelets, with application to Internet traffic, image processing and financial data modelling.
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