Relative cluster entropy for power-law correlated sequences

We propose an information-theoretical 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 fraction...

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Published in:arXiv.org
Main Authors: Carbone, A, Ponta, L
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 09.08.2022
Subjects:
Clustering
Continuity (mathematics)
Distribution functions
Divergence
Entropy
Exponents
Pricing
Probability distribution
Probability distribution functions
ISSN:2331-8422
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Abstract We propose an information-theoretical 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, 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 \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 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.
AbstractList We propose an information-theoretical 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, 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 \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 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.
Author Carbone, A
Ponta, L
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Copyright 2022. This work is published under http://creativecommons.org/licenses/by-nc-nd/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
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SubjectTerms Clustering
Continuity (mathematics)
Distribution functions
Divergence
Entropy
Exponents
Pricing
Probability distribution
Probability distribution functions
Title Relative cluster entropy for power-law correlated sequences
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