Multifractals: theory and applications

MULTIFRACTALS: Theory and Applications......Page 1
Preface......Page 3
List of Figures......Page 5
List of Notation......Page 7
Contents......Page 9
PART I: Introduction and Preliminaries......Page 12
1.2 Fractal Sets and MultifractalMeasures......Page 13
1.2.1 Example - Cantor Measure......Page 14
1.3 Dynamical Systems......Page 18
1.3.1 The Cantor Map......Page 19
1.3.2 Logistic Map......Page 20
1.3.3 Lorenz Attractor (Lorenz, 1963)......Page 22
1.4 Turbulence......Page 24
1.5.2 Example - Brownian Multiplier......Page 26
1.6 Earthquake Modelling......Page 27
1.6.1 Wellington Earthquake Catalogue......Page 30
1.8 Concept of Multifractals......Page 32
Part I - Introduction and Preliminaries......Page 35
Part III - Estimation of the R ï enyi Dimensions......Page 36
References......Page 0
2.1 Introduction......Page 38
2.2 Historical Development of Generalised R ´ enyi Dimensions......Page 39
2.2.1 Example......Page 41
2.3 Generalised R´enyi Lattice Dimensions......Page 42
2.3.3 Theorem (Beck, 1990)......Page 43
2.4 Generalised R´enyi Point Centred Dimensions......Page 44
2.4.2 Definition......Page 45
2.4.5 Theorem - Distribution of qth Order Interpoint Distance......Page 46
2.5.1 Definition......Page 47
2.5.4 Definition......Page 48
2.5.5 Definition......Page 49
2.6.2 Example (Riedi, 1995, 4.1)......Page 50
2.6.3 Overlapping Boxes (Riedi, 1995)......Page 51
2.7.1 Theorem (Young, 1982; Pesin, 1993)......Page 52
2.7.2 Definitions (Cutler, 1991)......Page 53
2.7.6 Average Local Behaviour......Page 54
2.7.7 Theorem (Cutler, 1991, Page 659)......Page 55
3.1.1 Construction of the Measure......Page 56
3.2 Local Behaviour......Page 57
3.2.1 Multifractal Spectrum......Page 58
3.3.1 Global Averaging......Page 59
3.3.2 Example......Page 60
3.3.3 Legendre Transform......Page 61
3.4 Fractal Dimensions......Page 63
3.5 Point Centred Construction......Page 65
3.5.1 Example......Page 66
PART II: Multifractal Formalism Using Large Deviations......Page 68
4.1 Introduction......Page 69
4.2 Large Deviation Formalism......Page 70
4.2.2 Extended Hypotheses......Page 71
4.2.4 Theorem - Large Deviation Bounds......Page 72
4.3.2 Corollary - Exponential Convergence......Page 73
4.3.4 Corollary - Large Deviation Bounds......Page 74
4.3.6 Example (Holley & Waymire, 1992)......Page 75
4.4.2 Theorem - Rescaled Cumulant Generating Function......Page 77
4.4.6 Note......Page 78
4.4.9 Example......Page 79
4.5.1 Definition......Page 80
4.5.4 Lemma......Page 81
4.5.7 Corollary (Billingsley, 1965; Cutler, 1986, Page 1477)......Page 82
4.5.11 Example - Multinomial Measures......Page 83
5.2 Large Deviation Formalism......Page 85
5.2.2 Extended Hypotheses......Page 86
5.3.2 Theorem - Rescaled Cumulant Generating Function......Page 87
5.3.6 Corollary - Large Deviation Bounds......Page 88
5.4.1 Definition......Page 89
5.4.3 Theorem (Young, 1982)......Page 90
5.5 Relationships Between Lattice and Point Centred Constructions......Page 91
5.5.1 Theorem (Cawley & Mauldin, 1992)......Page 92
5.5.2 Theorem (Cawley & Mauldin, 1992)......Page 93
5.5.3 Corollary......Page 94
5.5.4 Theorem......Page 95
6.1 Introduction......Page 97
6.1.1 Definition......Page 99
6.1.2 Definition - Global Averaging......Page 100
6.1.3 Definition - Multifractal Spectrum......Page 101
6.2.2 Theorem......Page 102
6.2.4 Moran Cascade Measure......Page 103
6.2.6 Note......Page 104
6.2.8 Theorem - Multifractal Spectrum......Page 105
6.2.10 Theorem (Cawley & Mauldin, 1992)......Page 106
6.3.1 Construction of Random Measures......Page 107
6.3.4 Corollary (Holley & Waymire, 1992, Corollary 2.5)......Page 108
6.3.7 Example - Log-L ï e vy Generators......Page 109
6.3.8 Rescaled Cumulant Generating Function......Page 110
6.3.10 Corollary......Page 111
6.3.12 Theorem (Holley & Waymire, 1992, Theorem 2.8)......Page 112
6.3.14 Theorem (Holley & Waymire, 1992, Theorem 2.6)......Page 113
6.3.16 Example - Log-Normal Distribution......Page 115
6.4 Other Cascade Processes......Page 117
6.4.1 Other Definitions of theta*(q)......Page 118
PART III: Estimation of the R´enyi Dimensions......Page 119
7.1 Introduction to Part III......Page 120
7.1.1 Overview of Chapters in Part III......Page 121
7.1.2 Review of Notation......Page 122
7.2.1 Distribution of L to the power of infiniti Norm......Page 123
7.2.2 Example - Gaussian Distribution......Page 124
7.2.3 Example - Uniform Distribution......Page 125
7.2.4 Example - ‘Wrap-Around’ Metric......Page 126
7.3 Multiplicity of Boundaries......Page 127
7.4.2 Definitions - Self-Similar Type Behaviour......Page 129
7.4.4 Proposition......Page 130
7.4.7 Proposition......Page 131
7.4.8 Example......Page 132
7.5 Differentiable Distributions......Page 133
8.1.2 Methods of Estimation......Page 134
8.2.3 The Exponent v......Page 136
8.2.5 Consistency for Ergodic Processes......Page 137
8.3.1 Proposition (Theiler, 1988)......Page 138
8.3.2 Example......Page 139
8.4 Hill Estimator......Page 140
8.4.2 Maximum Likelihood Estimator......Page 141
8.4.3 Proposition......Page 142
8.4.4 Proposition......Page 144
8.4.5 Lemma......Page 145
8.4.8 Corollary......Page 146
8.5.1 Example - Cantor Measure......Page 147
8.5.2 Example - Exponential Distribution......Page 148
8.5.3 Bootstrapped Hill Estimate......Page 149
8.5.4 Standard Errors......Page 150
8.6.1 Example - Uniform Distribution......Page 151
8.6.3 Where to \'Draw\' the Line......Page 153
8.6.4 Cantor Example Continued......Page 156
8.6.5 Bias Reduction......Page 159
8.6.6 Discussion......Page 160
9.1 Introduction......Page 162
9.2.1 Boundary Effect Correction......Page 163
9.2.2 Example - Uniform Distribution......Page 164
9.3.1 Truncated Exponential Approximation......Page 165
9.3.3 Further Problems......Page 166
9.3.4 Example......Page 167
9.3.5 Example......Page 168
9.4.1 Cantor Measure......Page 170
10.1 Introduction......Page 171
10.2.1 Example - Multinomial Measures......Page 172
10.2.2 Example - Cantor Measure......Page 174
10.3 Spatial and Temporal Point Patterns......Page 176
10.3.1 Example - Moran Cascade Process......Page 177
10.3.2 Beta Distribution......Page 179
10.4.1 Logistic Map......Page 181
10.4.3 Lorenz Attractor......Page 184
10.4.4 Embedding and Reconstruction......Page 185
10.4.5 Example - Lorenz Attractor Continued......Page 186
10.5.1 Definitions (Cutler, 1997)......Page 190
10.5.4 Theorem (Cutler, 1997, Corollary 2.26)......Page 191
10.5.5 Example - Gaussian Time Series......Page 192
10.6.1 Historical Sketch......Page 193
10.6.3 Fractional Brownian Motion......Page 195
10.6.4 Other ‘Powerlaw’ Processes......Page 198
10.6.5 ‘Multifractal’ Stochastic Processes......Page 199
11.1 Introduction......Page 201
11.1.1 Point Process Perspective......Page 202
11.2.1 Determination of Earthquake Locations......Page 203
11.2.2 Kanto Earthquake Catalogue......Page 205
11.3.2 Boundaries and Lacunarity......Page 208
11.3.3 Data Rounding and Transformation......Page 209
11.4 Results......Page 210
11.4.1 Wellington Catalogue......Page 211
11.4.2 Kanto Catalogue......Page 214
11.5 Comparison of Results and Conclusions......Page 215
11.5.1 Point Process Setting......Page 217
11.5.2 Summary of Related Studies......Page 218
PART IV: Appendices......Page 220
A.1.1 Definitions (Falconer, 1990, Chapter 9)......Page 221
A.2 Hausdorff Dimension......Page 222
A.2.2 Hausdorff Measure Properties (Falconer, 1990, 2.1)......Page 223
A.2.4 Properties of Hausdorff Dimension (Falconer, 1990, 2.2)......Page 224
A.3.1 Definition (Falconer, 1990, Page 38)......Page 225
A.3.6 Comparison with Hausdorff Dimension......Page 226
A.4.1 Packing Measure Definition (Falconer, 1990, Page 47)......Page 227
A.4.4 Lemma (Falconer, 1990, Pages 43 & 48)......Page 228
B.1 Introduction......Page 229
B.2.2 Cramïer\'s Theorem (Bucklew, 1990, Page 7)......Page 230
B.2.4 Example......Page 231
B.2.6 Example......Page 232
B.3.2 Note......Page 233
B.3.4 Definition (Ellis, 1985, Page 35)......Page 234
B.3.8 Hypotheses (Ellis, 1984, Hypothesis II.1)......Page 235
B.3.12 Lemma (Ellis, 1985, Page 214)......Page 236
B.3.16 Example......Page 237
B.3.17 Theorem (Ellis, 1984, Theorem V.1)......Page 238
References......Page 239