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Parameter estimation for differential equations: a generalized smoothing approach

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  • J. O. Ramsay
  • G. Hooker
  • D. Campbell
  • J. Cao
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    Abstract

    We propose a new method for estimating parameters in models that are defined by a system of non-linear differential equations. Such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself. Current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation. The paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non-linear differential equations. The approach is based on a modification of data smoothing methods along with a generalization of profiled estimation. We derive estimates and confidence intervals, and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology. The performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus. Copyright 2007 Royal Statistical Society.

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    Bibliographic Info

    Article provided by Royal Statistical Society in its journal Journal of the Royal Statistical Society: Series B (Statistical Methodology).

    Volume (Year): 69 (2007)
    Issue (Month): 5 ()
    Pages: 741-796

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    Handle: RePEc:bla:jorssb:v:69:y:2007:i:5:p:741-796

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    Cited by:
    1. Commenges, D. & Jolly, D. & Drylewicz, J. & Putter, H. & ThiƩbaut, R., 2011. "Inference in HIV dynamics models via hierarchical likelihood," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 446-456, January.
    2. Zhou, Jie & Han, Lu & Liu, Sanyang, 2013. "Nonlinear mixed-effects state space models with applications to HIV dynamics," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1448-1456.
    3. Hong, Zhaoping & Lian, Heng, 2012. "Time-varying coefficient estimation in differential equation models with noisy time-varying covariates," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 58-67, January.
    4. Steffen Borchers & Sandro Bosio & Rolf Findeisen & Utz-Uwe Haus & Philipp Rumschinski & Robert Weismantel, 2011. "Graph problems arising from parameter identification of discrete dynamical systems," Computational Statistics, Springer, vol. 73(3), pages 381-400, June.
    5. Nancy Heckman, 2010. "Comments on: Dynamic relations for sparsely sampled Gaussian processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 19(1), pages 46-49, May.
    6. Carey, Michelle & Gath, Eugene G. & Hayes, Kevin, 2014. "Frontiers in financial dynamics," Research in International Business and Finance, Elsevier, vol. 30(C), pages 369-376.
    7. Pascal Deboeck & Steven Boker, 2010. "Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices," Psychometrika, Springer, vol. 75(3), pages 420-437, September.

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