Multifractals: Theory and Applications
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Summary
Although multifractals are rooted in probability, much of the related literature comes from the physics and mathematics arena. Multifractals: Theory and Applications pulls together ideas from both these areas using a language that makes them accessible and useful to statistical scientists. It provides a framework, in particular, for the evaluation of statistical properties of estimates of the Renyi fractal dimensions.
The first section provides introductory material and different definitions of a multifractal measure. The author then examines some of the various constructions for describing multifractal measures. Building from the theory of large deviations, he focuses on constructions based on lattice coverings, covering by point-centered spheres, and cascades processes. The final section presents estimators of Renyi dimensions of integer order two and greater and discusses their properties. It also explores various applications of dimension estimation and provides a detailed case study of spatial point patterns of earthquake locations.
Estimating fractal dimensions holds particular value in studies of nonlinear dynamical systems, time series, and spatial point patterns. With its careful yet practical blend of multifractals, estimation methods, and case studies, Multifractals: Theory and Applications provides a unique opportunity to explore the estimation methods from a statistical perspective.
Table of Contents
SECTION I: INTRODUCTION AND PRELIMINARIES
MOTIVATION AND BACKGROUND
Fractal Sets and Multifractal Measures
Dynamical Systems
Turbulence
Rainfall Fields
Earthquake Modelling
Other Applications
Concept of Multifractals
Overview of Book
THE MULTIFRACTAL FORMALISM
Historical Development of Generalised Renyi Dimensions
Generlised Rényi Lattice Dimensions
Generalised Rényi Point Centred Dimensions
Multifractal Spectrum and Formalism
Review of Related Lattice Based Results
Review of Related Point Centred Results
THE MULTINOMIAL MEASURE
Local Behaviour
Global Averaging and Legendre Transforms
Fractal Dimensions
Point Centred Construction
SECTION II: MULTIFRACTAL FORMALISM USING LARGE DEVIATIONS
LATTICE BASED MULTIFRACTALS
Large Deviation Formalism
Uniform Spatial Sampling Measure
A Family of Sampling Measures
Hausdorff Dimensions
POINT CENTERED MULTIFRACTALS
Large Deviation Formalism
A Family of Sampling Measures
Hausdorff Dimensions
Relationship Between Lattice and Point Centred Constructions
MULTIPLICATIVE CASCADE PROCESSES
Moran Cascades Processes
Random Cascades
Other Cascade Processes
SECTION III: ESTIMATION OF THE RENYI DIMENSIONS
INTERPOINT DISTANCES OF ORDER q AND INTRINSIC BIAS
Boundary Effect
Multiplicity of Boundaries
Decomposition of FY(y)
Differentiable Distribution
ESTIMATION OF POINT CENTRED RENYI DIMENSIONS WITH q=2
Generalised Grassberger-Procaccia Algorithm
Takens Estimator
Hill Estimator
Bootstrap Estimation Procedure
Discussion and Examples
EXTRINSIC SOURCES OF BIAS
Imposed Boundary Effect
Rounding Effect
Effect of Noise
APPLICATIONS OF DIMENSION ESTIMATION
More on Estimation and Interpretation
Spatial and Temporal Point Patterns
Dynamical Systems
Is a Process Stochastic or Deterministic?
Stochastic Processes with Powerlaw Properties
EARTHQUAKE ANALYSES
Sources of Data
Effects Causing Bias
Results
Comparison of Results and Conclusions
APPENDICES
Properties and Dimensions of Sets
Large Deviations
REFERENCES
Editorial Reviews
"… [This book] is best suited for applied scientists who want to use multifractals in their research, but it can also be recommended for theorists as an easily accessible introduction to the field."
- Mathematical Reviews, Issue 2005c
"Many references; nice discussion of the relevant literature…"
- J.D, Fehribach, Worcester Polytechnic Institute, in CHOICE
