Although multifractal measures are rooted in probability theory, much of the published literature is found in the physics and mathematics areas. Multifractals: Theory and Applications pulls together ideas from both disciplines to place the material into a probabilistic and statistical context. Using language that makes these ideas accessible and useful to statistical scientists, it provides a framework, in particular, for the evaluation of statistical properties of estimates of the Rényi fractal dimensions.
The book begins with 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, coverings by point-centred spheres, and cascade processes. The final part of the book presents estimators of Rényi dimensions of integer order two and greater, and discusses their properties. It also discusses 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, lime series, and spatial point patterns. With its careful yet practical blend of theory, estimation methods, and applications, Multifractals: Theory and Applications provides a unique opportunity, to explore this subject from a statistical perspective.