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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