A swarm intelligence approach to density function reconstruction from moments using entropy optimization

Abstract

This paper presents a bioinspired optimization approach to address a class of inverse problems involving entropy optimization (EOP) from knowledge of the moments of a distribution function. In particular, we study the Hausdorff moment problem, where one seeks to reconstruct a (probability) density distribution by inverting a completely monotonic sequence of moments of the distribution in a bounded interval. It is shown that the resulting EOP can be handled very efficiently using the collective intelligence of a swarm (of optimizers), which provides a robust and accurate solution by effectively incorporating information from up to a thousand moments of the density. The efficacy of the approach is demonstrated by reconstructing the invariant density functions for the logistics map, spectral densities of large real-symmetric random matrices, encountered in the study of physics of disordered solids, and financial time series involving daily price fluctuations of a mutual fund. The agreement between true densities and the corresponding maximum-entropy approximants is examined by comparing the Kullback–Leibler divergence and the Fisher information of the densities.