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Preface | p. xiii |
About the Author | p. xv |
Introduction | p. 1 |
About Geometries and Dimensions | p. 11 |
From Euclidean to Fractal Geometry | p. 11 |
Dimensions | p. 16 |
Euclidean, Topological, and Embedding Dimensions | p. 16 |
Euclidean Dimension | p. 16 |
Topological Dimension | p. 16 |
Embedding Dimension | p. 17 |
Fractal Dimension | p. 18 |
Fractal Codimension | p. 22 |
Sampling Dimension | p. 23 |
Self-Similar Fractals | p. 25 |
Self-Similarity, Power Laws, and the Fractal Dimension | p. 25 |
Methods for Self-Similar Fractals | p. 28 |
Divider Dimension, D_{d} | p. 29 |
Theory | p. 29 |
Case Study: Movement Patterns of the Ocean Sunfish, Mola Mola | p. 32 |
Methodological Considerations | p. 35 |
Box Dimension, D_{b} | p. 46 |
Theory | p. 46 |
Case Study: Burrow Morphology of the Grapsid Crab, Helograpsus Haswellianus | p. 47 |
Methodological Considerations | p. 51 |
Theoretical Considerations | p. 52 |
Cluster Dimension, D_{c} | p. 56 |
Theory | p. 56 |
Case Study: The Microscale Distribution of the Amphipod Corophium Arenarium | p. 57 |
Methodological Considerations: Constant Numbers or Constant Radius? | p. 59 |
Mass Dimension, D_{m} | p. 60 |
Theory | p. 60 |
Case Study: Microscale Distribution of Microphytobenthos Biomass | p. 61 |
Comparing the Mass Dimension D_{m} to Other Fractal Dimensions | p. 65 |
Information Dimension, D_{i} | p. 66 |
Theory | p. 66 |
Comparing the Information Dimension D_{i} to Other Fractal Dimensions | p. 67 |
Correlation Dimension, D_{cor} | p. 68 |
Theory | p. 68 |
Comparing the Correlation Dimension D_{cor} to Other Fractal Dimensions | p. 69 |
Area-Perimeter Dimensions | p. 69 |
Perimeter Dimension, D_{p} | p. 70 |
Area Dimension, D_{a} | p. 72 |
Landscape/Seascape Dimension, D_{s} | p. 72 |
Fractal Dimensions, Areas, and Perimeters | p. 73 |
Ramification Dimension, D_{r} | p. 87 |
Theory | p. 87 |
Fractal Nature of Growth Patterns | p. 87 |
Surface Dimensions | p. 92 |
Transect Dimension, D_{t} | p. 93 |
Contour Dimension, D_{co} | p. 94 |
Geostatistical Dimension, D_{g} | p. 95 |
Elevation Dimension, D_{e} | p. 96 |
Self-Affine Fractals | p. 99 |
Several Steps toward Self-Affinity | p. 99 |
Definitions | p. 99 |
Fractional Brownian Motion | p. 99 |
Dimension of Self-Affine Fractals | p. 100 |
1/f Noise, Self-Affinity, and Fractal Dimensions | p. 102 |
Fractional Brownian Motion, Fractional Gaussian Noise, and Fractal Analysis | p. 103 |
Methods for Self-Affine Fractals | p. 106 |
Power Spectrum Analysis | p. 106 |
Theory | p. 106 |
Spectral Analysis in Aquatic Sciences | p. 108 |
Case Study: Eulerian and Lagrangian Scalar Fluctuations in Turbulent Flows | p. 109 |
Detrended Fluctuation Analysis | p. 117 |
Theory | p. 117 |
Case Study: Assessing Stress in Interacting Bird Species | p. 119 |
Scaled Windowed Variance Analysis | p. 124 |
Theory | p. 124 |
Case Study: Temporal Distribution of the Calanoid Copepod, Temora Longicornis | p. 125 |
Signal Summation Conversion Method | p. 128 |
Dispersion Analysis | p. 128 |
Rescaled Range Analysis and the Hurst Dimension, D_{H} | p. 128 |
Theory | p. 128 |
Example: R/S Analysis and River Flushing Rates | p. 131 |
Autocorrelation Analysis | p. 131 |
Semivariogram Analysis | p. 133 |
Theory | p. 133 |
Case Study: Vertical Distribution of Phytoplankton in Tidally Mixed Coastal Waters | p. 134 |
Wavelet Analysis | p. 139 |
Assessment of Self-Affine Methods | p. 140 |
Comparing Self-Affine Methods | p. 140 |
From Self-Affinity to Intermittent Self-Affinity | p. 143 |
Frequency Distribution Dimensions | p. 147 |
Cumulative Distribution Functions and Probability Density Functions | p. 147 |
Theory | p. 147 |
Case Study: Motion Behavior of the Intertidal Gastropod, Littorina Littorea | p. 147 |
The Study Organism | p. 147 |
Experimental Procedures and Data Analysis | p. 148 |
Results | p. 149 |
Ecological Interpretation | p. 150 |
The Patch-Intensity Dimension, D_{pi} | p. 151 |
The Korcak Dimension, D_{K} | p. 153 |
Fragmentation and Mass-Size Dimensions, D_{fr} and D_{ms} | p. 154 |
Rank-Frequency Dimension, D_{rf} | p. 155 |
Zipf's Law, Human Communication, and the Principle of Least Effort | p. 155 |
Zipf's Law, Information, and Entropy | p. 156 |
From the Zipf Law to the Generalized Zipf Law | p. 158 |
Generalized Rank-Frequency Diagram for Ecologists | p. 160 |
Practical Applications of Rank-Frequency Diagrams for Ecologists | p. 161 |
Zipf's Law as a Diagnostic Tool to Assess Ecosystem Complexity | p. 161 |
Case Study: Zipf Laws of Two-Dimensional Patterns | p. 177 |
Distance between Zipf's Laws | p. 188 |
Beyond Zipf's Law and Entropy | p. 189 |
n-Tuple Zipf's Law | p. 189 |
n-Gram Entropy and n-Gram Redundancy | p. 193 |
Fractal-Related Concepts: Some Clarifications | p. 201 |
Fractals and Deterministic Chaos | p. 201 |
Chaos Theory | p. 201 |
Feigenbaum Universal Numbers | p. 205 |
Attractors | p. 205 |
Visualizing Attractors: Packard-Takens Method | p. 206 |
Quantifying Attractors: Diagnostic Methods for Deterministic Chaos | p. 209 |
Case Study: Plankton Distribution in Turbulent Coastal Waters | p. 213 |
Chaos, Attractors, and Fractals | p. 224 |
Chaos in Ecological Sciences | p. 224 |
A Few Misconceptions about Chaos | p. 225 |
Then, What Is Chaos? | p. 225 |
Fractals and Self-Organization | p. 226 |
Fractals and Self-Organized Criticality | p. 226 |
Defining Self-Organized Criticality | p. 226 |
Self-Organized Criticality in Ecology and Aquatic Sciences | p. 229 |
Estimating Dimensions with Confidence | p. 231 |
Scaling or Not Scaling? That Is the Question | p. 231 |
Identifying Scaling Properties | p. 232 |
Procedure 1: R^{2} - SSR Procedure | p. 233 |
Procedure 2: Zero-Slope Procedure | p. 234 |
Procedure 3: Compensated-Slope Procedure | p. 238 |
Scaling, Multiple Scaling, and Multiscaling: Demixing Apples and Oranges | p. 239 |
Errors Affecting Fractal Dimension Estimates | p. 241 |
Geometrical Constraint, Shape Topology, and Digitization Biases | p. 241 |
Isotropy | p. 243 |
Stationarity | p. 243 |
Statistical Stationarity | p. 243 |
Fractal Stationarity | p. 244 |
Defining the Confidence Limits of Fractal Dimension Estimates | p. 246 |
Performing a Correct Analysis | p. 246 |
Self-Similar Case | p. 247 |
Self-Affine Case | p. 247 |
From Fractals to Multifractals | p. 249 |
A Random Walk toward Multifractality | p. 249 |
A Qualitative Approach to Multifractality | p. 249 |
Multifractality So Far | p. 250 |
From Fractality to Multifractality: Intermittency | p. 253 |
A Bit of History | p. 253 |
Intermittency in Ecology and Aquatic Sciences | p. 253 |
Defining Intermittency | p. 253 |
Variability, Inhomogeneity, and Heterogeneity: Terminological Considerations | p. 255 |
Intuitive Multifractals for Ecologists | p. 257 |
Methods for Multifractals | p. 260 |
Generalized Correlation Dimension Function D(q) and the Mass Exponents ¿(q) | p. 260 |
Theory | p. 260 |
Application: Salinity Stress in the Cladoceran Daphniopsis Australis | p. 262 |
Multifractal Spectrum f(¿) | p. 262 |
Theory | p. 262 |
Application: Temperature Stress in the Calanoid Copepod Temora Longicornis | p. 265 |
Codimension Function c(¿) and Scaling Moment Function K(q) | p. 265 |
Structure Function Exponents ¿(q) | p. 268 |
Theory | p. 268 |
Eulerian and Lagrangian Multiscaling Relations for Turbulent Velocity and Passive Scalars | p. 271 |
Cascade Models for Intermittency | p. 276 |
Historical Background | p. 276 |
Cascade Models for Turbulence | p. 278 |
Lognormal Model | p. 278 |
The Log-Lévy Model | p. 279 |
Log-Poisson Model | p. 280 |
Assessment of Cascade Models for Passive Scalars in a Turbulent Flow | p. 280 |
Multifractals: Misconceptions and Ambiguities | p. 282 |
Spikes, Intermittency, and Power Spectral Analysis | p. 282 |
Frequency Distributions and Multifractality | p. 284 |
Joint Multifractals | p. 285 |
Joint Multifractal Measures | p. 285 |
The Generalized Correlation Functions | p. 287 |
Definition | p. 287 |
Applications | p. 290 |
Intermittency and Multifractals: Biological and Ecological Implications | p. 293 |
Intermittency, Local Dissipation Rates, and Zooplankton Swimming Abilities | p. 294 |
Intermittency, Local Dissipation Rates, and Biological Fluxes in the Ocean | p. 296 |
Intermittency, Turbulence, and Nutrient Fluxes toward Phytoplankton Cells | p. 297 |
Intermittency, Turbulence, and Physical Coagulation of Phytoplankton Cells | p. 298 |
Intermittency, Turbulence, and Encounter Rates in the Plankton | p. 299 |
Conclusion | p. 301 |
References | p. 303 |
Index | p. 337 |
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