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Mathematics

 

Review Essays of Academic, Professional & Technical Books in the Humanities & Sciences

 

Fractals

Multifractals: Theory and Applications by David Harte (Chapman & Hall, CRC) Takes a multidisciplinary view of multifractals, pulling ideas together into a probabilistic and statistical context. Builds on the theory of large deviations to focus on constructions based on lattice coverings. Also discusses applications of dimension estimation, with a detailed case study of point patterns of earthquake locations.

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 provides a unique opportunity to explore the estimation methods from a statistical perspective.  

 

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