Multifractals

Multifractals : Theory and Applications

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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.show more

Product details

  • Hardback | 264 pages
  • 155.4 x 246.9 x 20.6mm | 617.08g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • New.
  • 20 black & white illustrations, 3 black & white tables, 7 black & white halftones
  • 1584881542
  • 9781584881544

Review quote

"[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 CHOICEshow more

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 Renyi Lattice Dimensions Generalised Renyi 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 REFERENCESshow more

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