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Entropy 2011, 13(6), 1055-1075; doi:10.3390/e13061055
Article

Distances in Probability Space and the Statistical Complexity Setup

1,2
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1,3
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1,4,* , 3,4,5 and 6
1 Instituto de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata (UNLP), C.C. 727, 1900 La Plata, Argentina 2 Comisión de Investigaciones Científicas (CICPBA), Calle 526 entre 10 y 11, 1900 La Plata, Buenos Aires, Argentina 3 Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Rivadavia 1917, Buenos Aires, Argentina 4 Departamento de Física, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Campus Pampulha, 31270-901 Belo Horizonte, MG, Brazil 5 Chaos & Biology Group, Instituto de Cálculo, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón II, Ciudad Universitaria, 1428 Ciudad Autónoma de Buenos Aires, Argentina 6 IFISC-CSIC, Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain
* Author to whom correspondence should be addressed.
Received: 11 April 2011 / Accepted: 27 May 2011 / Published: 3 June 2011
(This article belongs to the Special Issue Distance in Information and Statistical Physics Volume 2)
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Abstract

Statistical complexity measures (SCM) are the composition of two ingredients: (i) entropies and (ii) distances in probability-space. In consequence, SCMs provide a simultaneous quantification of the randomness and the correlational structures present in the system under study. We address in this review important topics underlying the SCM structure, viz., (a) a good choice of probability metric space and (b) how to assess the best distance-choice, which in this context is called a “disequilibrium” and is denoted with the letter Q. Q, indeed the crucial SCM ingredient, is cast in terms of an associated distance D. Since out input data consists of time-series, we also discuss the best way of extracting from the time series a probability distribution P. As an illustration, we show just how these issues affect the description of the classical limit of quantum mechanics.
Keywords: generalized statistical complexity; disequilibrium; information theory; selection of the probability distribution; semiclassical theories; quantum chaos generalized statistical complexity; disequilibrium; information theory; selection of the probability distribution; semiclassical theories; quantum chaos
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kowalski, A.M.; Martín, M.T.; Plastino, A.; Rosso, O.A.; Casas, M. Distances in Probability Space and the Statistical Complexity Setup. Entropy 2011, 13, 1055-1075.

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