Scaling, Fractals and Wavelets

Cover......Page 1
Scaling, Fractals and Wavelets......Page 5
Copyright - ISBN: 1848210728......Page 6
Table of Contents......Page 7
Preface......Page 19
1.1. Introduction......Page 21
1.2. Dimensions of sets......Page 22
1.2.1. Minkowski-Bouligand dimension......Page 23
1.2.2. Packing dimension......Page 27
1.2.3. Covering dimension......Page 29
1.2.4. Methods for calculating dimensions......Page 31
1.3.2. Theorems on set dimensions......Page 35
1.3.3. Hölder exponent related to a function......Page 38
1.3.4. Signal dimension theorem......Page 44
1.3.5. 2-microlocal analysis......Page 47
1.3.6. An example: analysis of stock market price......Page 48
1.4.1. What is the purpose of multifractal analysis?......Page 50
1.4.2. First ingredient: local regularity measures......Page 51
1.4.3. Second ingredient: the size of point sets of the same regularity......Page 52
1.4.4. Practical calculation of spectra......Page 54
1.4.5. Refinements: analysis of the sequence of capacities, mutual analysis and multisingularity......Page 62
1.4.6. The multifractal spectra of certain simple signals......Page 64
1.4.7.1. Image segmentation......Page 68
1.4.7.2. Analysis of TCP traffic......Page 69
1.5. Bibliography......Page 70
2.1. Introduction......Page 73
2.2.1. Intuition......Page 74
2.2.2. Self-similarity......Page 75
2.2.3. Long-range dependence......Page 77
2.2.4. Local regularity......Page 78
2.2.5. Fractional Brownian motion: paradigm of scale invariance......Page 79
2.2.6. Beyond the paradigm of scale invariance......Page 81
2.3.1. Continuous wavelet transform......Page 83
2.3.2. Discretewavelet transform......Page 84
2.4. Wavelet analysis of scale invariant processes......Page 87
2.4.1. Self-similarity......Page 88
2.4.2. Long-range dependence......Page 90
2.4.3. Local regularity......Page 92
2.5. Implementation: analysis, detection and estimation......Page 94
2.5.1. Estimation of the parameters of scale invariance......Page 95
2.5.2. Emphasis on scaling laws and determination of the scaling range......Page 98
2.5.3. Robustness of the wavelet approach......Page 100
2.6. Conclusion......Page 102
2.7. Bibliography......Page 103
3.1. Introduction......Page 105
3.2.1. Important definitions......Page 106
3.2.2. Wavelets and pointwise regularity......Page 109
3.2.3. Local oscillations......Page 114
3.2.4. Complements......Page 118
3.3.1. Lévy processes......Page 119
3.3.2. Burgers’ equation and Brownian motion......Page 122
3.3.3. Random wavelet series......Page 124
3.4.1. Besov spaces and lacunarity......Page 125
3.4.2. Construction of formalisms......Page 128
3.5.1. Bounds according to the Besov domain......Page 131
3.6. The grand-canonical multifractal formalism......Page 134
3.7. Bibliography......Page 136
4.1. Introduction and summary......Page 141
4.2.1. Hölder continuity......Page 142
4.2.2. Scaling of wavelet coefficients......Page 144
4.2.3. Other scaling exponents......Page 146
4.3.1. Dimension based spectra......Page 147
4.3.2. Grain based spectra......Page 148
4.3.3. Partition function and Legendre spectrum......Page 149
4.3.4. Deterministic envelopes......Page 151
4.4. Multifractal formalism......Page 153
4.5.1. Construction......Page 156
4.5.2. Wavelet decomposition......Page 159
4.5.3. Multifractal analysis of the binomial measure......Page 160
4.5.4. Examples......Page 162
4.5.5. Beyond dyadic structure......Page 164
4.6.1. The binomial revisited with wavelets......Page 165
4.6.2. Multifractal properties of the derivative......Page 167
4.7. Self-similarity and LRD......Page 169
4.8. Multifractal processes......Page 170
4.8.1. Construction and simulation......Page 171
4.8.3. Local analysis of warped FBM......Page 172
4.9. Bibliography......Page 175
5.1.1. Motivations......Page 181
5.1.2.1.Trees......Page 184
5.1.2.3. Renormalizing Cantor set......Page 185
5.1.3.1. Distribution of masses associated with Poisson measures......Page 186
5.1.4. Weierstrass functions......Page 187
5.1.5. Renormalization of sums of random variables......Page 188
5.1.6. A common structure for a stochastic (semi-)self-similar process......Page 189
5.1.7.1. Pseudo-correlation......Page 190
5.2.1.2. Definitions......Page 191
5.2.2. Elliptic processes......Page 192
5.2.3. Hyperbolic processes......Page 193
5.2.5.1. Gaussian elliptic processes......Page 194
5.2.7.1. Quadratic variations......Page 195
5.2.7.2. Acceleration of convergence......Page 196
5.3.1. Introduction......Page 197
5.3.2.2. Ellipticity......Page 198
5.3.4. Wavelet decomposition......Page 200
5.3.5. Process subordinated to Brownian measure......Page 201
5.4.1. Introduction......Page 202
5.4.2.2. Filtered white noise......Page 203
5.5. Bibliography......Page 204
6.1. Introduction......Page 207
6.2.1. Reproducing kernel Hilbert space......Page 209
6.2.2. Harmonizable representation......Page 210
6.3.1. Definition of the local asymptotic self-similarity (LASS)......Page 215
6.3.2. Filtered white noise (FWN)......Page 216
6.3.3. Elliptic Gaussian random fields (EGRP)......Page 217
6.4. Multifractional fields and trajectorial regularity......Page 220
6.4.1.Two representations of the MBM......Page 221
6.4.2. Study of the regularity of the trajectories of the MBM......Page 223
6.4.3. Towards more irregularities: generalized multifractional Brownian motion (GMBM) and step fractional Brownian motion (SFBM)......Page 224
6.4.3.1. Step fractional Brownian motion......Page 225
6.4.3.2. Generalized multifractional Brownian motion......Page 226
6.5.1. General method: generalized quadratic variation......Page 228
6.5.2.1. Identification of filteredwhite noise......Page 230
6.5.2.2. Identification of elliptic Gaussian random processes......Page 232
6.5.2.3. Identification of MBM......Page 233
6.5.2.4. Identification of SFBMs......Page 235
6.6. Bibliography......Page 237
7.1.1.1. Fields of application......Page 239
7.1.2. Problems......Page 240
7.1.3. Outline......Page 241
7.2.1. Fractional integration......Page 242
7.2.2.1. Motivation......Page 244
7.2.2.2. Fundamental solutions......Page 247
7.2.3.1. Motivation......Page 248
7.2.3.2. Definition......Page 249
7.2.3.3. Mittag-Leffler eigenfunctions......Page 250
7.2.3.4. Fractional power series expansions of order α (α-FPSE)......Page 252
7.3.1.1. Framework of causal distributions......Page 253
7.3.1.2. Framework of fractional power series expansion of order one half......Page 254
7.3.1.3. Notes......Page 255
7.3.2. Framework of causal distributions......Page 256
7.3.3. Framework of functions expandable into fractional power series (α-FPSE)......Page 257
7.3.4.2. Asymptotic behavior at infinity......Page 259
7.3.5. Controlled-and-observed linear dynamic systems of fractional order......Page 263
7.4. Diffusive structure of fractional differential systems......Page 264
7.4.1. Introduction to diffusive representations of pseudo-differential operators......Page 265
7.4.2. General decomposition result......Page 266
7.4.4. Particular case of fractional differential systems of commensurate orders......Page 267
7.5. Example of a fractional partial differential equation......Page 268
7.5.1. Physical problem considered......Page 269
7.5.3. Time-domain consequences......Page 270
7.5.3.1. Decomposition into wavetrains......Page 271
7.5.3.2. Quasi-modal decomposition......Page 272
7.5.3.3. Fractional modal decomposition......Page 273
7.5.4. Free problem......Page 274
7.7. Bibliography......Page 275
8.1.2. Short and long memory......Page 281
8.1.3. From integer to non-integer powers: filter based sample path design......Page 282
8.1.4. Local and global properties......Page 283
8.2.2. Construction and approximation techniques......Page 284
8.3.1. Filters: impulse responses and corresponding processes......Page 286
8.3.2. Mixing and memory properties......Page 288
8.3.3. Parameter estimation......Page 289
8.3.4. Simulated example......Page 291
8.4.1. A non-self-similar family: fractional processes designed from fractional filters......Page 293
8.4.2. Sample path properties: local and global regularity, memory......Page 295
8.5.2. The family of linear distribution processes......Page 296
8.5.3. Fractional distribution processes......Page 297
8.5.4. Mixing and memory properties......Page 298
8.6. Bibliography......Page 299
9.1. Introduction......Page 303
9.2. Definition of the Hölder exponent......Page 305
9.3. Iterated function systems (IFS)......Page 306
9.4. Generalization of iterated function systems......Page 308
9.4.1. Semi-generalized iterated function systems......Page 309
9.4.2. Generalized iterated function systems......Page 310
9.5. Estimation of pointwise Hölder exponent by GIFS......Page 313
9.5.1. Principles of themethod......Page 314
9.5.2. Algorithm......Page 316
9.5.3. Application......Page 317
9.6. Weak self-similar functions and multifractal formalism......Page 320
9.7. Signal representation by WSA functions......Page 322
9.8. Segmentation of signals by weak self-similar functions......Page 326
9.9. Estimation of the multifractal spectrum......Page 328
9.10. Experiments......Page 329
9.11. Bibliography......Page 331
10.2. Iterated transformation systems......Page 335
10.2.1.4. Hausdorff distance......Page 336
10.2.2. Attractor of an iterated transformation system......Page 337
10.2.3. Collage theorem......Page 338
10.2.4. Finally contracting transformation......Page 340
10.2.5. Attractor and invariant measures......Page 341
10.3.1. Introduction......Page 342
10.3.2.1. Collage of a source block onto a destination block......Page 344
10.3.2.2. Hierarchical partitioning......Page 346
10.3.3. Algebraic formulation of the fractal transformation......Page 347
10.3.3.1. Formulation of themass transformation......Page 349
10.3.3.2. Contraction control of the fractal transformation......Page 351
10.3.3.3. Fisher formulation......Page 352
10.3.4. Experimentation on triangular partitions......Page 353
10.3.5.1. Coding simplification suppressing the research for similarities......Page 354
10.3.5.2. Decoding simplification by collage space orthogonalization......Page 360
10.3.6. Other optimization diagrams: hybrid methods......Page 362
10.4. Bibliography......Page 364
11.1. Introduction......Page 369
11.2.1. Hölder regularity analysis......Page 370
11.2.2.1. Hausdorff multifractal spectrum......Page 371
11.2.2.2. Large deviation multifractal spectrum......Page 372
11.3.1. Oscillations (OSC)......Page 373
11.3.3. Wavelet leaders regression (WL)......Page 374
11.3.4. Limit inf and limit sup regressions......Page 375
11.3.5. Numerical experiments......Page 376
11.4.1. Introduction......Page 378
11.4.2. Minimax risk, optimal convergence rate and adaptivity......Page 379
11.4.3. Wavelet based denoising......Page 380
11.4.4.1. Minimax properties......Page 382
11.4.4.2. Regularity control......Page 383
11.4.4.3. Numerical experiments......Page 384
11.4.5.1. Introduction......Page 385
11.4.5.2. Estimating the local regularity of a signal from noisy observations......Page 386
11.4.6.1. Introduction......Page 388
11.4.6.2. The set of parameterized classes S(g,ψ)......Page 389
11.4.6.3. Bayesian denoising in S(g,ψ)......Page 390
11.4.6.4. Numerical experiments......Page 392
11.4.6.5. Denoising of road pro.les......Page 393
11.5.2. The method......Page 395
11.6. Biomedical signal analysis......Page 396
11.7. Texture segmentation......Page 403
11.8.1. Introduction......Page 405
11.8.1.1. Edge detection......Page 408
11.9. Change detection in image sequences using multifractal analysis......Page 409
11.10. Image reconstruction......Page 410
11.11. Bibliography......Page 411
12.1.1. A phenomenon of scales......Page 415
12.1.2. An experimental science of “man-made atoms”......Page 417
12.1.3. A random current......Page 418
12.1.4. Two fundamental approaches......Page 419
12.2.1. First discoveries......Page 421
12.2.2. Laws reign......Page 422
12.2.3. Beyond the revolution......Page 426
12.3.1.The sumor its parts......Page 428
12.3.2.The on/off paradigm......Page 429
12.3.3. Chemistry......Page 430
12.3.4. Mechanisms......Page 431
12.4.1. Character of a model......Page 432
12.4.2. The fractional Brownian motion family......Page 433
12.4.4. Never-ending calls......Page 434
12.5. Perspectives......Page 435
12.6. Bibliography......Page 436
13.1. Introduction: fractals in finance......Page 439
13.2.1.1. Statistical apprehension of stock market fluctuations......Page 441
13.2.1.2. Profit and stock market return operations in different scales......Page 444
13.2.1.3. Traditional financial modeling: Brownian motion......Page 445
13.2.2.1. The existence of characteristic time......Page 447
13.3.1.1. Leptokurtic problem and Mandelbrot’s first model......Page 448
13.3.2.1. Statistical problem of parameter estimation of stable laws......Page 450
13.3.2.2. Non-normality and controversies on scaling invariance......Page 451
13.3.2.3. Scaling anomalies of parameters under iid hypothesis......Page 453
13.3.3.1. Partial scaling invariances by regime switching models......Page 454
13.3.3.2. Partial scaling invariances as compared with extremes......Page 455
13.4.1.1. Question of dependency of stock market returns......Page 456
13.4.1.3. Introduction of fractional differentiation in econometrics......Page 457
13.4.2.1. The 1980s: ARCH modeling and its limits......Page 458
13.5. Towards a rediscovery of scaling laws in finance......Page 459
13.6. Bibliography......Page 460
14.1. Introduction......Page 467
14.3. Towards a fractal space-time......Page 468
14.3.2. From continuity and non-differentiability to fractality......Page 469
14.3.3. Description of non-differentiable process by differential equations......Page 471
14.3.4. Differential dilation operator......Page 473
14.5. Scale differential equations......Page 474
14.5.1. Constant fractal dimension: “Galilean” scale relativity......Page 475
14.5.2. Breaking scale invariance: transition scales......Page 476
14.5.3. Non-linear scale laws: second order equations, discrete scale invariance, log-periodic laws......Page 477
14.5.4. Variable fractal dimension: Euler-Lagrange scale equations......Page 478
14.5.5. Scale dynamics and scale force......Page 480
14.5.5.1. Constant scale force......Page 481
14.5.5.2. Scale harmonic oscillator......Page 482
14.5.6. Special scale relativity – log-Lorentzian dilation laws, invariant scale limit under dilations......Page 483
14.5.7. Generalized scale relativity and scale-motion coupling......Page 484
14.5.7.1. A reminder about gauge invariance......Page 485
14.5.7.2. Nature of gauge fields......Page 486
14.5.7.3. Nature of the charges......Page 488
14.6.1. Generalized Schrödinger equation......Page 490
14.6.2. Application in gravitational structure formation......Page 494
14.7. Conclusion......Page 495
14.8. Bibliography......Page 497
List of Authors......Page 501
Index......Page 505