Relative cluster entropy for power-law correlated sequences
Аннотация
We propose an information-theoretical measure, the relative cluster entropy \mathcal{D_C}[P\|Q] <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="script"> <mml:msub> <mml:mi>𝒟</mml:mi> <mml:mi>𝒞</mml:mi> </mml:msub> </mml:mstyle> <mml:mo stretchy="false" form="prefix">[</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false" form="postfix">∥</mml:mo> <mml:mi>Q</mml:mi> <mml:mo stretchy="false" form="postfix">]</mml:mo> </mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and H_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> respectively. For subdiffusive, normal and superdiffusive sequences, the relative entropy sensibly depends on the difference between H_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and H_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . 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_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> =0.55, H_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> =0.57, and H_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> =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.