Correlation Analysis With Scale-local Entropy Measures.pdf

a r X i v : m a t h - p h / 0 3 0 4 0 1 0 v 1 5 A p r 2 0 0 3 Correlation Analysis With Scale-local Entropy Measures J G Reid and T A Trainor CENPA, Box 354290, University of Washington, Seattle, Washington 98195-4290 E-mail: trainor@hausdorf.npl.washington.edu Abstract. A novel method for correlation analysis using scale-dependent Re?nyi entropies is described. The method involves calculating the entropy of a data distribution as an explicit function of the scale of a d-dimensional partition of d-cubes, which is dithered to remove bias. Analytic expressions for dithered scale-local entropy and dimension for a uniform random point set are derived and compared to Monte Carlo results. Simulated nontrivial point-set correlations representing condensation and clustering are similarly analyzed. keywords: scale, entropy, dimension, information, fractal, complex system, phase transition Submitted to: J. Phys. A: Math. Gen. Correlation Analysis With Scale-local Entropy Measures 2 1. Introduction Entropy is a measure of the size of a data distribution contained within a bounded region (distribution support) of some space. In a thermodynamic context this distribution size is interpreted as the number of quantum states accessible to a dynamical system given macroscopic constraints. More generally, if a measure space is partitioned, the measure distribution size is estimated by the effective number of partition elements given the distribution weighting. Definition of the space partition is a central element of entropy calculation. The partition is sometimes defined as a small-scale limiting partition of the space (e.g., thermodynamic limit, limit procedures in classical analysis [3]), sometimes based on properties of the data distribution itself and/or on the analysis goals (as in wavelet analysis [4]). We define entropy as an explicit function of the partition definition, a scaled binning of the measure space. We calculate the entropy and related quantities as functions o