Statistical error in a chord estimator of correlation dimension: the ``rule of five''

Abstract

The statistical precision of a chord method for estimating fractal dimension from a correlation integral is derived. The optimal chord length is determined, and a comparison is made to other estimators. These calculations use the approximation that all pairwise distances between the points are statistically independent; the adequacy of this approximation is assessed numerically. The chord method provides a very quick and easy dimension estimate which is only slightly less precise than the optimal estimator. Keywords: correlation dimension, statistical error