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Some results on the behavior and estimation of the fractal dimensions of distributions on attractors
Authors:C D Cutler
Institution:(1) Department of Statistics and Actuarial Science, University of Waterloo, N2L 3G1 Waterloo, Ontario, Canada
Abstract:The strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation. In this paper we first provide a rigorous mathematical framework for the study of dimension, focusing on pointwise dimensionsgr(x) and the generalized Renyi dimensionsD(q), and give a rigorous proof of inequalities first derived by Grassberger and Procaccia and Hentschel and Procaccia. We then specialize to the problem of statistical estimation of the correlation dimension ngr and information dimensionsgr. It has been recognized for some time that the error estimates accompanying the usual procedures (which generally involve least squares methods and nearest neighbor calculations) grossly underestimate the true statistical error involved. In least squares analyses of ngr andsgr we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates. We then develop an estimation procedure forsgr which corrects for an important bias term (the local measure density) and provides confidence intervals forsgr. The general applicability of this method is illustrated with various numerical examples.
Keywords:Information dimension  correlation dimension  fractal dimension  fractal measures  dynamical systems  attractors
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