Nonlinear statistical coupling

Nelson, Kenric P.; Umarov, Sabir

doi:10.1016/j.physa.2010.01.044

Cited by 29 publications

(31 citation statements)

References 38 publications

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“…Around 2000 [29], q-Gaussians have been conjectured (see details in [30]) to be attractors in the CLT sense whenever the N random variables that are being summed are strongly correlated in a specific manner. The conjecture was recently proved in the presence of q-independent variables [31][32][33][34][35]. The proof presented in [33] is based on a q-generalization of the Fourier transform, denoted as q-Fourier transform, and the theorem is currently referred to as the q-CLT.…”

Section: Probability Distributions That Are Attractors In the Sense Omentioning

confidence: 99%

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

Tsallis

2011

Entropy

171

184

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The nonadditive entropy S q has been introduced in 1988 focusing on a generalization of Boltzmann-Gibbs (BG) statistical mechanics. The aim was to cover a (possibly wide) class of systems among those very many which violate hypothesis such as ergodicity, under which the BG theory is expected to be valid. It is now known that S q has a large applicability; more specifically speaking, even outside Hamiltonian systems and their thermodynamical approach. In the present paper we review and comment some relevant aspects of this entropy, namely (i) Additivity versus extensivity; (ii) Probability distributions that constitute attractors in the sense of Central Limit Theorems; (iii) The analysis of paradigmatic low-dimensional nonlinear dynamical systems near the edge of chaos; and (iv) The analysis of paradigmatic long-range-interacting many-body classical Hamiltonian systems. Finally, we exhibit recent as well as typical predictions, verifications and applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational results.

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Section: Probability Distributions That Are Attractors In the Sense Omentioning

confidence: 99%

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

Tsallis

2011

Entropy

171

184

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“…Theoretical and experimental illustrations in natural systems include long-range-interacting many-body classical Hamiltonian systems [13][14][15][16][17][18][19][20] (see also [21,22] [47,48], chemistry [49], earthquakes [50], biology [51,52], solar wind [53,54], anomalous diffusion in relation to central limit theorems and overdamped systems [55][56][57][58][59][60][61][62][63][64], quantum entangled systems [65,66], quantum chaos [67], astronomical systems [68,69], thermal conductance [70], mathematical structures [71][72][73][74][75][76] and nonlinear quantum mechanics [77][78][79][80][81][82][83][84]…”

Section: Introductionmentioning

confidence: 99%

Economics and Finance: q-Statistical Stylized Features Galore

Tsallis

2017

Entropy

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Abstract:The Boltzmann-Gibbs (BG) entropy and its associated statistical mechanics were generalized, three decades ago, on the basis of the nonadditive entropy S q (q ∈ R), which recovers the BG entropy in the q → 1 limit. The optimization of S q under appropriate simple constraints straightforwardly yields the so-called q-exponential and q-Gaussian distributions, respectively generalizing the exponential and Gaussian ones, recovered for q = 1. These generalized functions ubiquitously emerge in complex systems, especially as economic and financial stylized features. These include price returns and volumes distributions, inter-occurrence times, characterization of wealth distributions and associated inequalities, among others. Here, we briefly review the basic concepts of this q-statistical generalization and focus on its rapidly growing applications in economics and finance.

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“…This function solves the nonlinear differential equation dy dx = y 1−κ . For this and further reasons (explained below) κ is referred to as the nonlinear statistical coupling [12] or simply the coupling. We use a shorthand notation for the power function emphasizing its role as a deformation of the exponential function [13] by writing…”

Section: Review Of the Coupled Exponential Family Of Distributionsmentioning

confidence: 99%

Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions

Nelson

Kon

Umarov

2019

Physica A: Statistical Mechanics and its Applications

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Highlights:1. A functional relationship is established between the scale of a coupled Gaussian distribution and the geometric mean of the distribution.2. Numerical evidence is provided suggesting that an estimator of the scale based on this function is unbiased with diminishing variance as the sample size grows. 3.A relationship between the generalized mean and the scale of the coupled Gaussian is also established, but this relationship does not lead to an effective estimator. 4. Fluctuations due to global correlation are shown to be proportional to the nonlinear statistical coupling. Abstract:The geometric mean is shown to be an appropriate statistic for the scale of a heavy-tailed coupled Gaussian distribution or equivalently the Student's t distribution. The coupled Gaussian is a member of a family of distributions parameterized by the nonlinear statistical coupling which is the reciprocal of the degree of freedom and is proportional to fluctuations in the inverse scale of the Gaussian. Existing estimators of the scale of the coupled Gaussian have relied on estimates of the full distribution, and they suffer from problems related to outliers in heavy-tailed distributions. In this paper, the scale of a coupled Gaussian is proven to be equal to the product of the generalized mean and the square root of the coupling. From our numerical computations of the scales of coupled Gaussians using the generalized mean of random samples, it is indicated that only samples from a Cauchy distribution (with coupling parameter one) form an unbiased estimate with diminishing variance for large samples. Nevertheless, we also prove that the scale is a function of the geometric mean, the coupling term and a harmonic number. Numerical experiments show that this estimator is unbiased with diminishing variance for large samples for a broad range of coupling values.

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Nonlinear statistical coupling

Cited by 29 publications

References 38 publications

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

Economics and Finance: q-Statistical Stylized Features Galore

Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions

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