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Bayesian Derivative Order Estimation for a Fractional Logistic Model

Author

Listed:
  • Francisco J. Ariza-Hernandez

    (Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N Cd. Universitaria. Chilpancingo, Guerrero C.P. 39087, Mexico)

  • Martin P. Arciga-Alejandre

    (Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N Cd. Universitaria. Chilpancingo, Guerrero C.P. 39087, Mexico)

  • Jorge Sanchez-Ortiz

    (Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N Cd. Universitaria. Chilpancingo, Guerrero C.P. 39087, Mexico)

  • Alberto Fleitas-Imbert

    (Departamento de Matemáticas, Universidad Carlos III de Madrid, Getafe, 28903 Madrid, Spain)

Abstract

In this paper, we consider the inverse problem of derivative order estimation in a fractional logistic model. In order to solve the direct problem, we use the Grünwald-Letnikov fractional derivative, then the inverse problem is tackled within a Bayesian perspective. To construct the likelihood function, we propose an explicit numerical scheme based on the truncated series of the derivative definition. By MCMC samples of the marginal posterior distributions, we estimate the order of the derivative and the growth rate parameter in the dynamic model, as well as the noise in the observations. To evaluate the methodology, a simulation was performed using synthetic data, where the bias and mean square error are calculated, the results give evidence of the effectiveness for the method and the suitable performance of the proposed model. Moreover, an example with real data is presented as evidence of the relevance of using a fractional model.

Suggested Citation

  • Francisco J. Ariza-Hernandez & Martin P. Arciga-Alejandre & Jorge Sanchez-Ortiz & Alberto Fleitas-Imbert, 2020. "Bayesian Derivative Order Estimation for a Fractional Logistic Model," Mathematics, MDPI, vol. 8(1), pages 1-9, January.
  • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:109-:d:307336
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    References listed on IDEAS

    as
    1. Fan, Wenping & Jiang, Xiaoyun & Qi, Haitao, 2015. "Parameter estimation for the generalized fractional element network Zener model based on the Bayesian method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 427(C), pages 40-49.
    2. Francisco J. Ariza-Hernandez & Jorge Sanchez-Ortiz & Martin P. Arciga-Alejandre & Luis X. Vivas-Cruz, 2017. "Bayesian Analysis for a Fractional Population Growth Model," Journal of Applied Mathematics, Hindawi, vol. 2017, pages 1-9, January.
    3. M. H. Heydari & M. R. Hooshmandasl & C. Cattani & Ming Li, 2013. "Legendre Wavelets Method for Solving Fractional Population Growth Model in a Closed System," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-8, September.
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