Relative cluster entropy for power-law correlated sequences

Abstract

We propose an information-theoretical measure, the relative cluster entropy \mathcal{D_C}[P\|Q]𝒟𝒞[P∥Q], to discriminate among cluster partitions characterised by probability distribution functions P and Q. The measure is illustrated with the clusters generated by pairs of fractional Brownian motions with Hurst exponents H_1H1 and H_2H2 respectively. For subdiffusive, normal and superdiffusive sequences, the relative entropy sensibly depends on the difference between H_1H1 and H_2H2. By using the minimum relative entropy principle, cluster sequences characterized by different correlation degrees are distinguished and the optimal Hurst exponent is selected. As a case study, real-world cluster partitions of market price series are compared to those obtained from fully uncorrelated sequences (simple Browniam motions) assumed as a model. The minimum relative cluster entropy yields optimal Hurst exponents H_1H1=0.55, H_1H1=0.57, and H_1H1=0.63 respectively for the prices of DJIA, S and P500, NASDAQ: a clear indication of non-markovianity. Finally, we derive the analytical expression of the relative cluster entropy and the outcomes are discussed for arbitrary pairs of power-laws probability distribution functions of continuous random variables.