Abstract
Entropy-based tools are commonly used to describe the dynamics of complex systems. In the last few decades, non-extensive statistics, based on Tsallis entropy, and multifractal techniques have shown to be useful to characterize long-range interaction and scaling behavior. In this paper, an approach based on generalized Tsallis dimensions is used for the formulation of mutual-information-related dependence coefficients in the multifractal domain. Different versions according to the normalizing factor, as well as to the inclusion of the non-extensivity correction term are considered and discussed. An application to the assessment of dimensional interaction in the structural dynamics of a seismic real series is carried out to illustrate the usefulness and comparative performance of the measures introduced.
Highlights
The term “complex system” is nowadays commonly present in the main branches of science
This paper is focused on entropy-based approaches for studying the dimensional interaction in the structural dynamics of complex systems in the multifractal domain
This paper is focused on the study of complex systems using entropy-based approaches
Summary
The term “complex system” is nowadays commonly present in the main branches of science. This paper is focused on entropy-based approaches for studying the dimensional interaction in the structural dynamics of complex systems in the multifractal domain. The scaling behavior present in complex systems led us to formulate in [18] a dependence coefficient in the multifractal domain based on Rényi generalized dimensions. We formulate new alternative versions of dependence coefficients considering different normalization factors, as well as the inclusion of the non-extensivity correction term. We recall the preliminary concepts of Shannon, Rényi and Tsallis entropy, related formulations of divergence and mutual information and the definitions of generalized Rényi and Tsallis dimensions for multifractal measures.
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