Convergence of numerical box-counting and correlation integral multifractal analysis techniques

摘要

A systematic study of the rate of convergence for a numerical box-counting and a numerical correlation integral algorithm for determining the generalized fractal dimension D(q) are described. The algorithms are applied to Euclidean point sets, Koch constructions, and a symmetric chaotic mapping. The results provide a basis for estimating the size of a fractal subset needed for measurement of the generalized dimension D(q). In particular, the number of points N5 required to assure 5% convergence of the algorithms is given within a factor of 4 by log10(N5) ≈ 2.54D(q) - 0.11 for the fractal sets studied here. Approximately 25 times as many points are needed for 1 % convergence. Approximately 0.1 times as many points are needed for 25% convergence. The box-based correlation integral algorithm employed in the present studies, which is well suited to the analysis of large data sets, is also described.