Multifractals: Theory and Applications, 1st Edition (Hardback) book cover

Multifractals

Theory and Applications, 1st Edition

By David Harte

Chapman and Hall/CRC

264 pages | 20 B/W Illus.

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Hardback: 9781584881544
pub: 2001-06-26
$165.00
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Description

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.

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

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

Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis