Relative cluster entropy for power-law correlated sequences

We propose an information-theoretical measure, the \textit{relative cluster entropy} ${\mathcal{D}}_{\mathcal{C}}[P\mathrm{\parallel }Q]$, 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 $H_1$ and $H_2$ respectively. For subdiffusive, normal and superdiffusive sequences, the relative entropy sensibly depends on the difference between $H_1$ and $H_2$. By using the \textit{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 $\mathit{\text{minimum}}\mathit{\text{relative}}\mathit{\text{cluster}}\mathit{\text{entropy}}$ yields optimal Hurst exponents $H_1=0.55$, $H_1=0.57$, and $H_1=0.63$ 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.