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Recent advances of stretched Gaussian distribution underlying Hausdorff fractal distance and its applications in fitting stretched Gaussian noise

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  • Xu, Wei
  • Liang, Yingjie
  • Chen, Wen
  • Wang, Fajie

Abstract

This paper summarizes the latest advances of the third author’s research group on stretched Gaussian distribution underlying the Hausdorff fractal theory and its applications in fitting stretched Gaussian noise. Firstly, the Hausdorff fractal metrics are introduced as an extension of non-Euclidean distance. Based on the fractal scaling, the Hausdorff derivative is derived which can describe the problems of interest to construct non-integer differential equations. Secondly, we introduce the Hausdorff derivatives in tackling partial differential equations, and the fundamental solution of Hausdorff derivative diffusion equation is stretched Gaussian distribution. Thirdly, we analyze the feasibility of the least square method for stretched Gaussian noise, which obeys stretched Gaussian distribution. The least square method is inapplicable when the noise level is larger than 5%. Finally, by using the Hausdorff fractal distance, we introduce the stretched least square method, which improves the traditional least square method, to fit stretched Gaussian noise.

Suggested Citation

  • Xu, Wei & Liang, Yingjie & Chen, Wen & Wang, Fajie, 2020. "Recent advances of stretched Gaussian distribution underlying Hausdorff fractal distance and its applications in fitting stretched Gaussian noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    • Handle: RePEc:eee:phsmap:v:539:y:2020:i:c:s0378437119316930
    DOI: 10.1016/j.physa.2019.122996
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    References listed on IDEAS

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    1. Gorenflo, Rudolf & Mainardi, Francesco & Moretti, Daniele & Pagnini, Gianni & Paradisi, Paolo, 2002. "Fractional diffusion: probability distributions and random walk models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(1), pages 106-112.
    2. Yang, Xiao-Jun & Machado, J.A. Tenreiro, 2017. "A new fractional operator of variable order: Application in the description of anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 481(C), pages 276-283.
    3. Hasegawa, Yoshihiko & Arita, Masanori, 2009. "Properties of the maximum q-likelihood estimator for independent random variables," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(17), pages 3399-3412.
    4. Mahmoud, Gamal M. & Aly, Shaban A.H., 2000. "On periodic solutions of parametrically excited complex non-linear dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 278(3), pages 390-404.
    5. A. Antoniadis & D. Leporini & J.–C. Pesquet, 2002. "Wavelet thresholding for some classes of non–Gaussian noise," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(4), pages 434-453, November.
    6. Xu, Wei & Chen, Wen & Liang, Yingjie, 2018. "Feasibility study on the least square method for fitting non-Gaussian noise data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 1917-1930.
    7. Mancilla Canales, M.A. & Leguto, A.J. & Riquelme, B.D. & León, P. Ponce de & Bortolato, S.A. & Korol, A.M., 2017. "Hurst exponent: A Brownian approach to characterize the nonlinear behavior of red blood cells deformability," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 488(C), pages 1-7.
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