κ-generalization of Gauss' law of error

Wada, Tatsuaki; Suyari, Hiroki

doi:10.1016/j.physleta.2005.08.086

Cited by 40 publications

(32 citation statements)

References 14 publications

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“…The statistical correlations among events are taken into account by means of a suitable composition law (a deformed product), which arise in a natural way in the framework of the generalized statistical mechanics [9][10][11] and could play a role in the foundations of these theories [12][13][14]. The results presented in this work unify some already known one-parameter generalizations of Gauss' law of error [15,16], together with the original Gauss' derivation which is now recovered as a particular case. The paper is organized as follows.…”

Section: Introductionsupporting

confidence: 58%

“…The corresponding distribution error is provided by a generalized Gaussian, which exhibits an asymptotic power-law behavior different from the exponential one of the standard Gaussian distribution. As a by product, we have presented a unique mechanism which joins, in a unified model, several derivation of generalized distributions' error already presented in literature [15,16]. The original Gauss' derivation is now recovered as a special limit corresponding to statistically independent measurements.…”

Section: Resultsmentioning

confidence: 92%

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Gauss’ law of error revisited in the framework of Sharma-Taneja-Mittal information measure

2009

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Abstract:We reformulate the Gauss' law of error in presence of correlations which are taken into account by means of a deformed product arising in the framework of the Sharma-Taneja-Mittal measure. Having reviewed the main proprieties of the generalized product and its related algebra, we derive, according to the Maximum Likelihood Principle, a family of error distributions with an asymptotic power-law behavior.

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Section: Introductionsupporting

confidence: 58%

Section: Resultsmentioning

confidence: 92%

Gauss’ law of error revisited in the framework of Sharma-Taneja-Mittal information measure

2009

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“…Hence, we specify a probability distribution from a given likelihood function and observed data. Generalizations of Gauss's law of error in non-extensive statistical physics have been obtained in [35,36], etc.…”

Section: Maximum κ-Likelihood Estimatorsmentioning

confidence: 99%

Deformed Algebras and Generalizations of Independence on Deformed Exponential Families

Matsuzoe

Wada

2015

Entropy

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A deformed exponential family is a generalization of exponential families. Since the useful classes of power law tailed distributions are described by the deformed exponential families, they are important objects in the theory of complex systems. Though the deformed exponential families are defined by deformed exponential functions, these functions do not satisfy the law of exponents in general. The deformed algebras have been introduced based on the deformed exponential functions. In this paper, after summarizing such deformed algebraic structures, it is clarified how deformed algebras work on deformed exponential families. In fact, deformed algebras cause generalization of expectations. The three kinds of expectations for random variables are introduced in this paper, and it is discussed why these generalized expectations are natural from the viewpoint of information geometry. In addition, deformed algebras cause generalization of independences. Whereas it is difficult to check the well-definedness of deformed independence in general, the κ-independence is always well-defined on κ-exponential families. This is one of advantages of κ-exponential families in complex systems. Consequently, we can well generalize the maximum likelihood method for the κ-exponential family from the viewpoint of information geometry.

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“…Recently [29,32], a possible generalization of the central limit theorem has been proposed in order to justify the recurrence of non-Gaussian distributions [39][40][41] in the limit of a large number of statistically dependent events.…”

Section: Gauss Law Of Errormentioning

confidence: 99%

Entropic Forms and Related Algebras

Scarfone

2013

Entropy

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Starting from a very general trace-form entropy, we introduce a pair of algebraic structures endowed by a generalized sum and a generalized product. These algebras form, respectively, two Abelian fields in the realm of the complex numbers isomorphic each other. We specify our results to several entropic forms related to distributions recurrently observed in social, economical, biological and physical systems including the stretched exponential, the power-law and the interpolating Bosons-Fermions distributions. Some potential applications in the study of complex systems are advanced.

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κ-generalization of Gauss' law of error

Cited by 40 publications

References 14 publications

Gauss’ law of error revisited in the framework of Sharma-Taneja-Mittal information measure

Gauss’ law of error revisited in the framework of Sharma-Taneja-Mittal information measure

Deformed Algebras and Generalizations of Independence on Deformed Exponential Families

Entropic Forms and Related Algebras

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