Bayesian estimation of ordinary differential equation models when the likelihood has multiple local modes
-
Baisen Liu
Abstract
Ordinary differential equations (ODEs) are popularly used to model complex dynamic systems by scientists; however, the parameters in ODE models are often unknown and have to be inferred from noisy measurements of the dynamic system. One conventional method is to maximize the likelihood function, but the likelihood function often has many local modes due to the complexity of ODEs, which makes the optimizing algorithm be vulnerable to trap in local modes. In this paper, we solve the global optimization issue of ODE parameters with the help of the Stochastic Approximation Monte Carlo (SAMC) algorithm which is shown to be self-adjusted and escape efficiently from the “local-trapping” problem. Our simulation studies indicate that the SAMC method is a powerful tool to estimate ODE parameters globally. The efficiency of SAMC method is demonstrated by estimating a predator-prey ODEs model from real experimental data.
Funding source: Natural Sciences and Engineering Research Council of Canada
Award Identifier / Grant number: RGPIN 356044-2013
Award Identifier / Grant number: 435713-2013
Funding statement: The research was supported by the Liaoning Provincial Education Department (No. LN2017ZD001) to Baisen Liu and the discovery grants (RGPIN 356044-2013, 435713-2013) from the Natural Science and Engineering Research Council of Canada to Jiguo Cao and Liangliang Wang.
Acknowledgements
We thank Professor Gregor F. Fussmann for providing us the predator-prey data set. The authors are also very grateful for the suggestions of Professor Faming Liang.
References
[1] Y. Bard, Nonlinear Parameter Estimation, Academic Press, New York, 1974. Search in Google Scholar
[2] L. Becks, F. M. Hilker, H. Malchow, K. Jürgens and H. Arndt, Experimental demonstration of chaos in a microbial food web, Nature 435 (2005), 1226–1229. 10.1038/nature03627Search in Google Scholar PubMed
[3] L. T. Biegler, J. J. Damiano and G. E. Blau, Nonlinear parameter estimation: A case study comparison, AIChE J. 32 (1986), 29–45. 10.1002/aic.690320105Search in Google Scholar
[4] N. J.-B. Brunel, Parameter estimation of ODE’s via nonparametric estimators, Electron. J. Stat. 2 (2008), 1242–1267. 10.1214/07-EJS132Search in Google Scholar
[5] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed., John Wiley & Sons, Chichester, 2008. 10.1002/9780470753767Search in Google Scholar
[6] B. Calderhead and M. Girolami, Estimating Bayes factors via thermodynamic integration and population MCMC, Comput. Statist. Data Anal. 53 (2009), no. 12, 4028–4045. 10.1016/j.csda.2009.07.025Search in Google Scholar
[7] D. Campbell and R. J. Steele, Smooth functional tempering for nonlinear differential equation models, Stat. Comput. 22 (2012), no. 2, 429–443. 10.1007/s11222-011-9234-3Search in Google Scholar
[8] J. Cao, G. F. Fussmann and J. O. Ramsay, Estimating a predator-prey dynamical model with the parameter cascades method, Biometrics 64 (2008), no. 3, 959–967. 10.1111/j.1541-0420.2007.00942.xSearch in Google Scholar PubMed
[9] J. Cao, J. Z. Huang and H. Wu, Penalized nonlinear least squares estimation of time-varying parameters in ordinary differential equations, J. Comput. Graph. Statist. 21 (2012), no. 1, 42–56. 10.1198/jcgs.2011.10021Search in Google Scholar PubMed PubMed Central
[10] J. Cao, L. Wang and J. Xu, Robust estimation for ordinary differential equation models, Biometrics 67 (2011), no. 4, 1305–1313. 10.1111/j.1541-0420.2011.01577.xSearch in Google Scholar PubMed
[11] J. Chen and H. Wu, Efficient local estimation for time-varying coefficients in deterministic dynamic models with applications to HIV-1 dynamics, J. Amer. Statist. Assoc. 103 (2008), no. 481, 369–384. 10.1198/016214507000001382Search in Google Scholar
[12] G. F. Fussmann, S. P. Ellner, K. W. Shertzer and N. G. Hairston Jr., Crossing the Hopf bifurcation in a live predator-prey system, Science 290 (2000), 1358–1360. 10.1126/science.290.5495.1358Search in Google Scholar
[13] A. Gelman, F. Bois and J. Jiang, Physiological pharmacokinetic analysis using population modeling and informative prior distributions, J. Amer. Statist. Assoc. 91 (1996), 1400–1412. 10.1080/01621459.1996.10476708Search in Google Scholar
[14] A. Gelman, J. B. Carlin, H. S. Stern and D. B. Rubin, Bayesian Data Analysis, 2nd ed., Texts Statist. Sci. Ser., Chapman & Hall/CRC, Boca, 2004. 10.1201/9780429258480Search in Google Scholar
[15] C. J. Geyer, Estimation and optimization of functions, Markov Chain Monte Carlo in Practice, Chapman & Hall, London (1996), 241–258. Search in Google Scholar
[16] B. Goodwin, Oscillatory behavior in enzymatic control processes, Adv. Enzyme Regul. 3 (1965), 425–438. 10.1016/0065-2571(65)90067-1Search in Google Scholar
[17] Y. Huang, D. Liu and H. Wu, Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system, Biometrics 62 (2006), no. 2, 413–423. 10.1111/j.1541-0420.2005.00447.xSearch in Google Scholar
[18] B. E. Kendall, C. J. Briggs, W. W. Murdoch, P. Turchin, S. P. Ellner, E. McCauley, R. M. Nisbet and S. N. Wood, Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches, Ecology 80 (1999), 1789–1805. 10.1890/0012-9658(1999)080[1789:WDPCAS]2.0.CO;2Search in Google Scholar
[19] F. Liang, C. Liu and R. J. Carroll, Stochastic approximation in Monte Carlo computation, J. Amer. Statist. Assoc. 102 (2007), no. 477, 305–320. 10.1198/016214506000001202Search in Google Scholar
[20] W. W. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-Resource Dynamics, Princeton University Press, New York, 2003. Search in Google Scholar
[21] A. Poyton, Application of principal differential analysis to parameter estimation in fundamental dynamics models, Master’s thesis, Queen’s University, Kingston, 2005. Search in Google Scholar
[22] X. Qi and H. Zhao, Asymptotic efficiency and finite-sample properties of the generalized profiling estimation of parameters in ordinary differential equations, Ann. Statist. 38 (2010), no. 1, 435–481. 10.1214/09-AOS724Search in Google Scholar
[23] J. O. Ramsay, G. Hooker, D. Campbell and J. Cao, Parameter estimation for differential equations: A generalized smoothing approach, J. R. Stat. Soc. Ser. B Stat. Methodol. 69 (2007), no. 5, 741–796. 10.1111/j.1467-9868.2007.00610.xSearch in Google Scholar
[24] J. O. Ramsay and B. W. Silverman, Functional Data Analysis, 2nd ed., Springer Ser. Statist., Springer, New York, 2005. 10.1007/b98888Search in Google Scholar
[25] H. Robbins and S. Monro, A stochastic approximation method, Ann. Math. Statistics 22 (1951), 400–407. 10.1214/aoms/1177729586Search in Google Scholar
[26] C. P. Robert and G. Casella, Monte Carlo statistical methods, 2nd ed., Springer Ser. Statist., Springer, New York, 2004. 10.1007/978-1-4757-4145-2Search in Google Scholar
[27] K. W. Shertzer, S. P. Ellner, G. F. Fussmann and N. G. Hairston, Predator-prey cycles in an aquatic microcosm: Testing hypotheses of mechanism, J. Animal Ecol. 71 (2002), 802–815. 10.1046/j.1365-2656.2002.00645.xSearch in Google Scholar
[28] P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Monogr. Population Biol. 35, Princeton University Press, Princeton, 2003. Search in Google Scholar
[29] J. M. Varah, A spline least squares method for numerical parameter estimation in differential equations, SIAM J. Sci. Statist. Comput. 3 (1982), no. 1, 28–46. 10.1137/0903003Search in Google Scholar
[30] T. Yoshida, L. E. Jones, S. P. Ellner, G. F. Fussmann and N. G. Hairston, Rapid evolution drives ecological dynamics in a predator-prey system, Nature 424 (2003), 303–306. 10.1038/nature01767Search in Google Scholar PubMed
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Probability distribution of the life time of a drift-diffusion-reaction process inside a sphere with applications to transient cathodoluminescence imaging
- A quasi-Monte Carlo implementation of the ziggurat method
- On the efficient simulation of the left-tail of the sum of correlated log-normal variates
- Bayesian estimation of ordinary differential equation models when the likelihood has multiple local modes
- On the modeling of linear system input stochastic processes with given accuracy and reliability
- Remarks on randomization of quasi-random numbers
- On average dimensions of particle transport estimators
Articles in the same Issue
- Frontmatter
- Probability distribution of the life time of a drift-diffusion-reaction process inside a sphere with applications to transient cathodoluminescence imaging
- A quasi-Monte Carlo implementation of the ziggurat method
- On the efficient simulation of the left-tail of the sum of correlated log-normal variates
- Bayesian estimation of ordinary differential equation models when the likelihood has multiple local modes
- On the modeling of linear system input stochastic processes with given accuracy and reliability
- Remarks on randomization of quasi-random numbers
- On average dimensions of particle transport estimators
