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Relative cluster entropy for powerlaw correlated sequences
by A. Carbone, L. Ponta
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Anna Carbone 
Submission information  

Preprint Link:  https://arxiv.org/abs/2206.02685v2 (pdf) 
Date accepted:  20220818 
Date submitted:  20220810 10:25 
Submitted by:  Carbone, Anna 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Computational 
Abstract
We propose an informationtheoretical measure, the \textit{relative cluster entropy} $\mathcal{D_{C}}[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 $H_1$ 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, realworld cluster partitions of market price series are compared to those obtained from fully uncorrelated sequences (simple Browniam motions) assumed as a model. The \textit{minimum relative cluster 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 nonmarkovianity. Finally, we derive the analytical expression of the relative cluster entropy and the outcomes are discussed for arbitrary pairs of powerlaws probability distribution functions of continuous random variables.
List of changes
Change 1: A short text clarifying the meaning of the approach and its relation with the coarsegrained is now added in the Introduction.
Change 2: Notations in Eq. (6) have been improved. For the sake of clarity, Eq. (6) has been split over two lines.
Change 3: Eq. (3) has been moved after Eq. (1) and before Eq. (2). Eq. (3) is now labelled Eq. (2).
Change 4: The arrows in the nine panels of Fig. 3 have been removed. A text linking each colour to the parameter n is now added in the caption.
Change 5: The left hand of equations in Section III are now in “math roman font” instead of "mathcal".
Change 6: The three panels in Fig. 5 have been merged together in one single panel.
Change 7: A short text, clarifying the main motivations of the work, is now included in the Introduction.
Published as SciPost Phys. 13, 076 (2022)