Generalized dimensions from near-neighbor information

Abstract

The dimension of fractal sets, such as strange attractors, can be derived from near-neighbor information. This gives rise to practical algorithms to estimate the spectrum of Renyi dimensions from an experimental signal. In high-dimensional phase spaces they may also be very efficient. A proof of these methods is given, using a formalism of local scaling indices. We emphasize the restriction on the calculable range of q values imposed by the neighbor order. In connection with the normalization of the distribution of scaling indices we also discuss possible corrections due to the finite size of the set of data points.