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On the average uncertainty for systems with nonlinear coupling

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  • Nelson, Kenric P.
  • Umarov, Sabir R.
  • Kon, Mark A.

Abstract

The increased uncertainty and complexity of nonlinear systems have motivated investigators to consider generalized approaches to defining an entropy function. New insights are achieved by defining the average uncertainty in the probability domain as a transformation of entropy functions. The Shannon entropy when transformed to the probability domain is the weighted geometric mean of the probabilities. For the exponential and Gaussian distributions, we show that the weighted geometric mean of the distribution is equal to the density of the distribution at the location plus the scale (i.e. at the width of the distribution). The average uncertainty is generalized via the weighted generalized mean, in which the moment is a function of the nonlinear source. Both the Rényi and Tsallis entropies transform to this definition of the generalized average uncertainty in the probability domain. For the generalized Pareto and Student’s t-distributions, which are the maximum entropy distributions for these generalized entropies, the appropriate weighted generalized mean also equals the density of the distribution at the location plus scale. A coupled entropy function is proposed, which is equal to the normalized Tsallis entropy divided by one plus the coupling.

Suggested Citation

  • Nelson, Kenric P. & Umarov, Sabir R. & Kon, Mark A., 2017. "On the average uncertainty for systems with nonlinear coupling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 30-43.
    • Handle: RePEc:eee:phsmap:v:468:y:2017:i:c:p:30-43
    DOI: 10.1016/j.physa.2016.09.046
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    References listed on IDEAS

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    1. Nelson, Kenric P. & Umarov, Sabir, 2010. "Nonlinear statistical coupling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(11), pages 2157-2163.
    2. Nelson, Kenric P., 2015. "A definition of the coupled-product for multivariate coupled-exponentials," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 422(C), pages 187-192.
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    Cited by:

    1. Nelson, Kenric P., 2022. "Independent Approximates enable closed-form estimation of heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
    2. Zubillaga, Bernardo J. & Vilela, André L.M. & Wang, Chao & Nelson, Kenric P. & Stanley, H. Eugene, 2022. "A three-state opinion formation model for financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 588(C).
    3. Nelson, Kenric P. & Kon, Mark A. & Umarov, Sabir R., 2019. "Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 248-257.
    4. Bafghi, Seyed Mohammad Amin Tabatabaei & Kamalvand, Mohammad & Morsali, Ali & Bozorgmehr, Mohammad Reza, 2018. "Radial distribution function within the framework of the Tsallis statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 857-867.

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