IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i1p109-d307336.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/1/109/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/1/109/
    Download Restriction: no
    ---><---

    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ariza-Hernandez, Francisco J. & Martin-Alvarez, Luis M. & Arciga-Alejandre, Martin P. & Sanchez-Ortiz, Jorge, 2021. "Bayesian inversion for a fractional Lotka-Volterra model: An application of Canadian lynx vs. snowshoe hares," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    2. Carrera, Y. & Avila-de la Rosa, G. & Vernon-Carter, E.J. & Alvarez-Ramirez, J., 2017. "A fractional-order Maxwell model for non-Newtonian fluids," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 482(C), pages 276-285.
    3. Lei, Dong & Liang, Yingjie & Xiao, Rui, 2018. "A fractional model with parallel fractional Maxwell elements for amorphous thermoplastics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 465-475.
    4. Zhang, Hui & Jiang, Xiaoyun & Yang, Xiu, 2018. "A time-space spectral method for the time-space fractional Fokker–Planck equation and its inverse problem," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 302-318.
    5. Duan, Jun-Sheng & Qiu, Xiang, 2018. "Stokes’ second problem of viscoelastic fluids with constitutive equation of distributed-order derivative," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 130-139.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:109-:d:307336. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.