A bootstrap approach to estimating fractal dimensions

This paper outlines a bootstrap approach to estimating the “correlation dimension” of a fractal measure, from a set of observations on that measure. An important example is estimating the dimension of the attractor of a chaotic system from a set of observations on that system. The method reduces the effects of dependence among the observations, is stable to minor variations in the procedure, and provides a preliminary estimate of the variance of the estimate. Examples from chaotic and stochastic systems show a systematic tendency to underestimate the dimensions from stochastic systems. Extensions to other generalised Renyi dimensions appear to be difficult