Minimization Problems Based on Relative alpha-Entropy I: Forward Projection

Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative alpha-entropies (denoted I-alpha), arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative alpha-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimizers of these relative alpha-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum Renyi or Tsallis entropy principle. The minimizing probability distribution (termed forward I-alpha-projection) for a linear family is shown to obey a power-law. Other results in connection with statistical inference, namely subspace transitivity and iterated projections, are also established. In a companion paper, a related minimization problem of interest in robust statistics that leads to a reverse I-alpha-projection is studied.

Nonparametric entropy-based tests of independence between stochastic processes

This paper develops nonparametric tests of independence between two stationary stochastic processes. The testing strategy boils down to gauging the closeness between the joint and the product of the marginal stationary densities. For that purpose, I take advantage of a generalized entropic measure so as to build a class of nonparametric tests of independence. Asymptotic normality and local power are derived using the functional delta method for kernels, whereas finite sample properties are investigated through Monte Carlo simulations.

Entropy effect in quantum computing and information: an open-source environment simulation

The technologies are rapidly developing, but some of them present in the computers, as for instance their processing capacity, are reaching their physical limits. It is up to quantum computation offer solutions to these limitations and issues that may arise. In the field of information security, encryption is of paramount importance, being then the development of quantum methods instead of the classics, given the computational power offered by quantum computing. In the quantum world, the physical states are interrelated, thus occurring phenomenon called entanglement. This study presents both a theoretical essay on the merits of quantum mechanics, computing, information, cryptography and quantum entropy, and some simulations, implementing in C language the effects of entropy of entanglement of photons in a data transmission, using Von Neumann entropy and Tsallis entropy.

Agregação e viés em medidas de diversidade: uma abordagem computacional intensiva

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Ensaios sobre computação e informação quânticas: fundamentação e simulações sobre o efeito da entropia

Pós-graduação em Ciência da Computação - IBILCE

Espaço de Hilbert e quantificação de emaranhamento via entropia não extensiva

Pós-graduação em Matemática - IBILCE

Multi-q pattern analysis: A case study in image classification

This paper compares the effectiveness of the Tsallis entropy over the classic Boltzmann-Gibbs-Shannon entropy for general pattern recognition, and proposes a multi-q approach to improve pattern analysis using entropy. A series of experiments were carried out for the problem of classifying image patterns. Given a dataset of 40 pattern classes, the goal of our image case study is to assess how well the different entropies can be used to determine the class of a newly given image sample. Our experiments show that the Tsallis entropy using the proposed multi-q approach has great advantages over the Boltzmann-Gibbs-Shannon entropy for pattern classification, boosting image recognition rates by a factor of 3. We discuss the reasons behind this success, shedding light on the usefulness of the Tsallis entropy and the multi-q approach. (C) 2012 Elsevier B.V. All rights reserved.

Coupled symplectic maps as models for subdiffusive processes in disordered Hamiltonian lattices

Acknowledgements One of us (T. B.) acknowledges many interesting discussions on coupled maps with Professor C. Tsallis. We are also grateful to the anonymous referees for their constructive feedback that helped us improve the manuscript and to the HPCS Laboratory of the TEI of Western Greece for providing the computer facilities where all our simulations were performed. C. G. A. was partially supported by the “EPSRC EP/I032606/1” grant of the University of Aberdeen. This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES - Investing in knowledge society through the European Social Fund.

Coupled symplectic maps as models for subdiffusive processes in disordered Hamiltonian lattices

Acknowledgements One of us (T. B.) acknowledges many interesting discussions on coupled maps with Professor C. Tsallis. We are also grateful to the anonymous referees for their constructive feedback that helped us improve the manuscript and to the HPCS Laboratory of the TEI of Western Greece for providing the computer facilities where all our simulations were performed. C. G. A. was partially supported by the “EPSRC EP/I032606/1” grant of the University of Aberdeen. This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES - Investing in knowledge society through the European Social Fund.

Tsallis statistics and gradually truncated Levy flight - distribution of an economical index

Tsallis postulated a generalized form for entropy and give rise to a new statistics now known as Tsallis statistics. In the present work, we compare the Tsallis statistics with the gradually truncated Levy flight, and discuss the distribution of an economical index-the Standard and Poor's 500-using the values of standard deviation as calculated by our model. We find that both statistics give almost the same distribution. Thus we feel that gradual truncation of Levy distribution, after certain critical step size for describing complex systems, is a requirement of generalized thermodynamics or similar. The gradually truncated Levy flight is based on physical considerations and bring a better physical picture of the dynamics of the whole system. Tsallis statistics gives a theoretical support. Both statistics together can be utilized for the development of a more exact portfolio theory or to understand better the complexities in human and financial behaviors. A comparison of both statistics is made. (C) 2002 Published by Elsevier B.V. B.V.

Size limiting in Tsallis statistics

Power law scaling is observed in many physical, biological and socio-economical complex systems and is now considered an important property of these systems. In general, power law exists in the central part of the distribution. It has deviations from power law for very small and very large variable sizes. Tsallis, through non-extensive thermodynamics, explained power law distribution in many cases including deviation from the power law. In case of very large steps, the used the heuristic crossover approach. In the present we present an alternative model in which we consider that the entropy factor 9 decreases with variable size due to the softening of long range interactions or memory. We apply this model for distribution of citation index of scientists and examination scores and are able to explain the distribution for entire variable range. In the present model, we can have very sharp cut-off without interfering with power law in its central part as observed in many cases. (C) 2008 Elsevier B.V. All rights reserved.

Quantum entanglement and information: The effect of entropy revisited

In non-extensive statistical mechanics [14], it is a nonsense statement to say that the entropy of a system is extensive (or not), without mentioning a law of composition of its elements. In this theory quantum correlations might be perceived through quantum information process. This article, that is an extension of recent work [4], is a comparative study between the entropies of Von Neumann and of Tsallis, with some implementations of the effect of entropy in quantum entanglement, important as a process for transmission of quantum information. We consider two factorized (Fock number) states, which interact through a beam splitter bilinear Hamiltonian with two entries. This comparison showed us that the entropies of Tsallis and Von Neumann behave differently depending on the reflectance of the beam splitter. © 2011 Academic Publications.

Entropia de Tsallis e sua aplicação em ações da Bolsa de Valores

Pós-graduação em Física - IGCE

Evaluation of physiologic complexity in time series using generalized sample entropy and surrogate data analysis

Complexity in time series is an intriguing feature of living dynamical systems, with potential use for identification of system state. Although various methods have been proposed for measuring physiologic complexity, uncorrelated time series are often assigned high values of complexity, errouneously classifying them as a complex physiological signals. Here, we propose and discuss a method for complex system analysis based on generalized statistical formalism and surrogate time series. Sample entropy (SampEn) was rewritten inspired in Tsallis generalized entropy, as function of q parameter (qSampEn). qSDiff curves were calculated, which consist of differences between original and surrogate series qSampEn. We evaluated qSDiff for 125 real heart rate variability (HRV) dynamics, divided into groups of 70 healthy, 44 congestive heart failure (CHF), and 11 atrial fibrillation (AF) subjects, and for simulated series of stochastic and chaotic process. The evaluations showed that, for nonperiodic signals, qSDiff curves have a maximum point (qSDiff(max)) for q not equal 1. Values of q where the maximum point occurs and where qSDiff is zero were also evaluated. Only qSDiff(max) values were capable of distinguish HRV groups (p-values 5.10 x 10(-3); 1.11 x 10(-7), and 5.50 x 10(-7) for healthy vs. CHF, healthy vs. AF, and CHF vs. AF, respectively), consistently with the concept of physiologic complexity, and suggests a potential use for chaotic system analysis. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4758815]

Relative entropy rate based multiple hidden Markov Model Approximation

This paper proposes a novel relative entropy rate (RER) based approach for multiple HMM (MHMM) approximation of a class of discrete-time uncertain processes. Under different uncertainty assumptions, the model design problem is posed either as a min-max optimisation problem or stochastic minimisation problem on the RER between joint laws describing the state and output processes (rather than the more usual RER between output processes). A suitable filter is proposed for which performance results are established which bound conditional mean estimation performance and show that estimation performance improves as the RER is reduced. These filter consistency and convergence bounds are the first results characterising multiple HMM approximation performance and suggest that joint RER concepts provide a useful model selection criteria. The proposed model design process and MHMM filter are demonstrated on an important image processing dim-target detection problem.