Statistical estimation of the correlation dimension

We introduce a statistical method of estimation of the correlation dimension d2 of fractal sets. The method is based on the maximum likelihood approach and provides an efficient estimation of the correlation dimension even for very limited data sets. We derive an explicit expression for the asymptotic variance of the estimated dimension, which can be used as a physically meaningful error bar. Both the estimate and its variance depend on the upper cutoff r0 of the presumed power-law scaling range. We propose to choose the cutoff scale r0 by comparing, with the help of the standard X2-test, the observed distribution within this scale to a power-law having the estimated exponent. The estimate of the correlation dimension based on the value of r0 which maximizes the associated X2-probability, is then chosen as the optimal one for the given data set. A numerical example illustrates the application of the method to binomial multifractal measures with known theoretical values of fractal dimensions.