Relative cluster entropy for power-law correlated sequences

We propose an information-theoretical measure, the relative cluster entropy DC[PkQ], 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 H1 and H2 respectively. For subdiffusive, normal and superdiffusive sequences, the relative entropy sensibly depends on the difference between H1 and H2. 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 H1 = 0.55, H1 = 0.57, and H1 = 0.63 respectively for the prices of DJIA, S&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.