Résumé
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.
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
- 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
- Local Behaviour
- Global Averaging and Legendre Transforms
- Fractal Dimensions
- Point Centred Construction
LATTICE BASED MULTIFRACTALS
- Large Deviation Formalism
- Uniform Spatial Sampling Measure
- A Family of Sampling Measures
- Hausdorff Dimensions
- Large Deviation Formalism
- A Family of Sampling Measures
- Hausdorff Dimensions
- Relationship Between Lattice and Point Centred Constructions
- Moran Cascades Processes
- Random Cascades
- Other Cascade Processes
INTERPOINT DISTANCES OF ORDER q AND INTRINSIC BIAS
- Boundary Effect
- Multiplicity of Boundaries
- Decomposition of FY(y)
- Differentiable Distribution
- Generalised Grassberger-Procaccia Algorithm
- Takens Estimator
- Hill Estimator
- Bootstrap Estimation Procedure
- Discussion and Examples
- Imposed Boundary Effect
- Rounding Effect
- Effect of Noise
- More on Estimation and Interpretation
- Spatial and Temporal Point Patterns
- Dynamical Systems
- Is a Process Stochastic or Deterministic?
- Stochastic Processes with Powerlaw Properties
- Sources of Data
- Effects Causing Bias
- Results
- Comparison of Results and Conclusions
- Properties and Dimensions of Sets
- Large Deviations
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Chapman and Hall / CRC |
| Auteur(s) | David Harte |
| Parution | 01/09/2001 |
| Nb. de pages | 248 |
| Format | 16 x 24 |
| Couverture | Relié |
| Poids | 222g |
| Intérieur | Noir et Blanc |
| EAN13 | 9781584881544 |
| ISBN13 | 978-1-58488-154-4 |
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