Generalized Statistical Complexity: A New Tool for Dynamical Systems

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A generalized Statistical Complexity Measure (SCM) is a functional of the probability distribution P associated with the time series generated by a given dynamical system. A SCM is the composition of two ingredients: i) an entropy and ii) a distance in the probability-space. We address in this review important topics underlying the SCM structure, viz., a) the selection of the information measure I; b) the choice of the probability metric space and associated distance D, which in this context is called a "disequilibrium" (denoted with the letter Q). Q, indeed the crucial SCM ingredient, is cast in terms of an associated distance D. c) The adequate way of picking up the probability distribution P associated with a dynamical system or time series under study, which is indeed a fundamental problem. A good analysis of this topics is essential to get a SCM that quantifies not only randomness but also the presence of correlational structures. In this chapter we specially stress how sensible improvements in the final results can be obtained if the underlying probability distribution is "extracted" via appropriate considerations regarding causal effects in the system's dynamics. As an illustration, we show just how these issues affect the description of the celebrated logistic map.