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Parameter estimation and inference in dynamic systems described by linear partial differential equations

Author

Listed:
  • Gianluca Frasso

    (Université de Liège)

  • Jonathan Jaeger

    (Nestlé Research Center)

  • Philippe Lambert

    (Université de Liège
    Université Catholique de Louvain)

Abstract

Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications, the parameters involved in the DE models are usually unknown and need to be estimated from the available measurements together with the state function. In this paper, we present frequentist and Bayesian approaches for the joint estimation of the parameters and of the state functions involved in linear PDEs. We also propose two strategies to include state (initial and/or boundary) conditions in the estimation procedure. We evaluate the performances of the proposed strategy through simulated examples and a real data analysis involving (known and necessary) state conditions.

Suggested Citation

  • Gianluca Frasso & Jonathan Jaeger & Philippe Lambert, 2016. "Parameter estimation and inference in dynamic systems described by linear partial differential equations," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 100(3), pages 259-287, July.
    • Handle: RePEc:spr:alstar:v:100:y:2016:i:3:d:10.1007_s10182-015-0257-5
    DOI: 10.1007/s10182-015-0257-5
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    References listed on IDEAS

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    1. Ma, Shuangge & Kosorok, Michael R., 2005. "Robust semiparametric M-estimation and the weighted bootstrap," Journal of Multivariate Analysis, Elsevier, vol. 96(1), pages 190-217, September.
    2. D. G. T. Denison & B. K. Mallick & A. F. M. Smith, 1998. "Automatic Bayesian curve fitting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(2), pages 333-350.
    3. J. O. Ramsay & G. Hooker & D. Campbell & J. Cao, 2007. "Parameter estimation for differential equations: a generalized smoothing approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(5), pages 741-796, November.
    4. Xiaolei Xun & Jiguo Cao & Bani Mallick & Arnab Maity & Raymond J. Carroll, 2013. "Parameter Estimation of Partial Differential Equation Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(503), pages 1009-1020, September.
    5. Jaeger, Jonathan & Lambert, Philippe, 2013. "Bayesian P-spline estimation in hierarchical models specified by systems of affine differential equations," LIDAM Reprints ISBA 2013016, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Jonathan Jaeger & Philippe Lambert, 2014. "Bayesian penalized smoothing approaches in models specified using differential equations with unknown error distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(12), pages 2709-2726, December.
    7. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    8. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    9. Jaeger, Jonathan & Lambert, Philippe, 2014. "Bayesian penalized smoothing approaches in models specified using differential equations with unknown error distributions," LIDAM Reprints ISBA 2014033, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
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    Cited by:

    1. Xinyu Zhang & Jiguo Cao & Raymond J. Carroll, 2017. "Estimating varying coefficients for partial differential equation models," Biometrics, The International Biometric Society, vol. 73(3), pages 949-959, September.
    2. Li, Jiayang & Zhang, Zhikun & Dai, Min & Ming, Ju & Wang, Xiangjun, 2023. "Diffusion equations with Markovian switching: Well-posedness, numerical generation and parameter inference," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).

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