9,849 research outputs found
Closed Contour Fractal Dimension Estimation by the Fourier Transform
This work proposes a novel technique for the numerical calculus of the
fractal dimension of fractal objects which can be represented as a closed
contour. The proposed method maps the fractal contour onto a complex signal and
calculates its fractal dimension using the Fourier transform. The Fourier power
spectrum is obtained and an exponential relation is verified between the power
and the frequency. From the parameter (exponent) of the relation, it is
obtained the fractal dimension. The method is compared to other classical
fractal dimension estimation methods in the literature, e. g.,
Bouligand-Minkowski, box-couting and classical Fourier. The comparison is
achieved by the calculus of the fractal dimension of fractal contours whose
dimensions are well-known analytically. The results showed the high precision
and robustness of the proposed technique
On the Numerical Study of the Complexity and Fractal Dimension of CMB Anisotropies
We consider the problem of numerical computation of the Kolmogorov complexity
and the fractal dimension of the anisotropy spots of Cosmic Microwave
Background (CMB) radiation. Namely, we describe an algorithm of estimation of
the complexity of spots given by certain pixel configuration on a grid and
represent the results of computations for a series of structures of different
complexity. Thus, we demonstrate the calculability of such an abstract
descriptor as the Kolmogorov complexity for CMB digitized maps. The correlation
of complexity of the anisotropy spots with their fractal dimension is revealed
as well. This technique can be especially important while analyzing the data of
the forthcoming space experiments.Comment: LATEX, 3 figure
Optimal rates of convergence for persistence diagrams in Topological Data Analysis
Computational topology has recently known an important development toward
data analysis, giving birth to the field of topological data analysis.
Topological persistence, or persistent homology, appears as a fundamental tool
in this field. In this paper, we study topological persistence in general
metric spaces, with a statistical approach. We show that the use of persistent
homology can be naturally considered in general statistical frameworks and
persistence diagrams can be used as statistics with interesting convergence
properties. Some numerical experiments are performed in various contexts to
illustrate our results
Extreme value laws in dynamical systems under physical observables
Extreme value theory for chaotic dynamical systems is a rapidly expanding
area of research. Given a system and a real function (observable) defined on
its phase space, extreme value theory studies the limit probabilistic laws
obeyed by large values attained by the observable along orbits of the system.
Based on this theory, the so-called block maximum method is often used in
applications for statistical prediction of large value occurrences. In this
method, one performs inference for the parameters of the Generalised Extreme
Value (GEV) distribution, using maxima over blocks of regularly sampled
observations along an orbit of the system. The observables studied so far in
the theory are expressed as functions of the distance with respect to a point,
which is assumed to be a density point of the system's invariant measure.
However, this is not the structure of the observables typically encountered in
physical applications, such as windspeed or vorticity in atmospheric models. In
this paper we consider extreme value limit laws for observables which are not
functions of the distance from a density point of the dynamical system. In such
cases, the limit laws are no longer determined by the functional form of the
observable and the dimension of the invariant measure: they also depend on the
specific geometry of the underlying attractor and of the observable's level
sets. We present a collection of analytical and numerical results, starting
with a toral hyperbolic automorphism as a simple template to illustrate the
main ideas. We then formulate our main results for a uniformly hyperbolic
system, the solenoid map. We also discuss non-uniformly hyperbolic examples of
maps (H\'enon and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models).
Our purpose is to outline the main ideas and to highlight several serious
problems found in the numerical estimation of the limit laws
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