The Annals of Applied Statistics

Time series analysis via mechanistic models

Carles Bretó, Daihai He, Edward L. Ionides, and Aaron A. King

Full-text: Open access

Abstract

The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae.

Article information

Source
Ann. Appl. Stat. Volume 3, Number 1 (2009), 319-348.

Dates
First available: 16 April 2009

Permanent link to this document
http://projecteuclid.org/euclid.aoas/1239888373

Digital Object Identifier
doi:10.1214/08-AOAS201

Mathematical Reviews number (MathSciNet)
MR2668710

Zentralblatt MATH identifier
1160.62080

Citation

Bretó, Carles; He, Daihai; Ionides, Edward L.; King, Aaron A. Time series analysis via mechanistic models. The Annals of Applied Statistics 3 (2009), no. 1, 319--348. doi:10.1214/08-AOAS201. http://projecteuclid.org/euclid.aoas/1239888373.


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References

  • Alonso, D., McKane, A. J. and Pascual, M. (2007). Stochastic amplification in epidemics. J. R. Soc. Interface 4 575–582.
  • Anderson, B. D. and Moore, J. B. (1979). Optimal Filtering. Prentice-Hall, Englewood Cliffs, NJ.
  • Anderson, R. M. and May, R. M. (1991). Infectious Diseases of Humans. Oxford Univ. Press.
  • Arulampalam, M. S., Maskell, S., Gordon, N. and Clapp, T. (2002). A tutorial on particle filters for online nonlinear, non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing 50 174–188.
  • Barndorff-Nielsen, O. E. and Cox, D. R. (1994). Inference and Asymptotics. Chapman and Hall, London.
  • Bartlett, M. S. (1960). Stochastic Population Models in Ecology and Epidemiology. Wiley, New York.
  • Basawa, I. V. and Prakasa Rao, B. L. S. (1980). Statistical Inference for Stochastic Processes. Academic Press, London.
  • Bauch, C. T. and Earn, D. J. D. (2003). Transients and attractors in epidemics. Proc. R. Soc. Lond. B 270 1573–1578.
  • Beskos, A., Papaspiliopoulos, O., Roberts, G. and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based inference for discretely observed diffusion processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 333–382.
  • Bjornstad, O. N., Finkenstadt, B. F. and Grenfell, B. T. (2002). Dynamics of measles epidemics: Estimating scaling of transmission rates using a time series SIR model. Ecological Monographs 72 169–184.
  • Bjornstad, O. N. and Grenfell, B. T. (2001). Noisy clockwork: Time series analysis of population fluctuations in animals. Science 293 638–643.
  • Boys, R. J., Wilkinson, D. J. and Kirkwood, T. B. L. (2008). Bayesian inference for a discretely observed stochastic kinetic model. Statist. Comput. 18 125–135.
  • Brémaud, P. (1999). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York.
  • Bretó, C., He, D., Ionides, E. L. and King, A. A. (2009). Supplement to “Time series analysis via mechanistic models.” DOI: 10.1214/08-AOAS201SUPP.
  • Brillinger, D. R. (2008). The 2005 Neyman lecture: Dynamic indeterminism in science. Statist. Sci. 23 48–64.
  • Cai, X. and Xu, Z. (2007). K-leap method for accelerating stochastic simulation of coupled chemical reactions. J. Chem. Phys. 126 074102.
  • Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. Springer, New York.
  • Cauchemez, S. and Ferguson, N. M. (2008). Likelihood-based estimation of continuous-time epidemic models from time-series data: Application to measles transmission in London. J. R. Soc. Interface 5 885–897.
  • Cauchemez, S., Temime, L., Guillemot, D., Varon, E., Valleron, A.-J., Thomas, G. and Boëlle, P.-Y. (2006). Investigating heterogeneity in pneumococcal transmission: A Bayesian MCMC approach applied to a follow-up of schools. J. Amer. Statist. Assoc. 101 946–958.
  • Cauchemez, S., Valleron, A., Boëlle, P., Flahault, A. and Ferguson, N. M. (2008). Estimating the impact of school closure on influenza transmission from sentinel data. Nature 452 750–754.
  • Conlan, A. and Grenfell, B. (2007). Seasonality and the persistence and invasion of measles. Proc. R. Soc. Ser. B Biol. 274 1133–1141.
  • Coulson, T., Rohani, P. and Pascual, M. (2004). Skeletons, noise and population growth: The end of an old debate? Trends in Ecology and Evolution 19 359–364.
  • Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 411–436.
  • Doucet, A., de Freitas, N. and Gordon, N. J., eds. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
  • Durbin, J. and Koopman, S. J. (2001). Time Series Analysis by State Space Methods. Oxford Univ. Press.
  • Dushoff, J., Plotkin, J. B., Levin, S. A. and Earn, D. J. D. (2004). Dynamical resonance can account for seasonality of influenza epidemics. Proc. Natl. Acad. Sci. USA 101 16915–16916.
  • Ferguson, N., Anderson, R. and Gupta, S. (1999). The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens. Proc. Natl. Acad. Sci. USA 96 187–205.
  • Ferguson, N. M., Galvani, A. P. and Bush, R. M. (2003). Ecological and immunological determinants of influenza evolution. Nature 422 428–433.
  • Fernández-Villaverde, J. and Rubio-Ramírez, J. F. (2005). Estimating dynamic equilibrium economies: Linear versus nonlinear likelihood. J. Appl. Econometrics 20 891–910.
  • Fine, P. E. M. and Clarkson, J. A. (1982). Measles in England and Wales—I. An analysis of factors underlying seasonal patterns. Int. J. Epid. 11 5–14.
  • Finkenstädt, B. F. and Grenfell, B. T. (2000). Time series modelling of childhood diseases: A dynamical systems approach. Appl. Statist. 49 187–205.
  • Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81 2340–2361.
  • Gillespie, D. T. (2001). Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115 1716–1733.
  • Glass, K., Xia, Y. and Grenfell, B. T. (2003). Interpreting time-series analyses for continuous-time biological models: Measles as a case study. J. Theoret. Biol. 223 19–25.
  • Grais, R., Ferrari, M., Dubray, C., Bjornstad, O., Grenfell, B., Djibo, A., Fermon, F. and Guerin, P. (2006). Estimating transmission intensity for a measles epidemic in Niamey, Niger: Lessons for intervention. Trans. R. Soc. Trop. Med. Hyg. 100 867–873.
  • Grenfell, B. T., Pybus, O. G., Gog, J. R., Wood, J. L. N., Daly, J. M., Mumford, J. A. and Holmes, E. C. (2004). Unifying the epidemiological and evolutionary dynamics of pathogens. Science 303 327–332.
  • Gupta, S., Trenholme, K., Anderson, R. and Day, K. (1994). Antigenic diversity and the transmission dynamics of Plasmodium falciparum. Science 263 961–963.
  • Houtekamer, P. L. and Mitchell, H. L. (2001). Data assimilation using an ensemble Kalman filter technique. Monthly Weather Rev. 129 123–137.
  • Ionides, E. L. (2005). Maximum smoothed likelihood estimation. Statist. Sinica 15 1003–1014.
  • Ionides, E. L., Bretó, C. and King, A. A. (2006). Inference for nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 103 18438–18443.
  • Ionides, E. L., Bretó, C. and King, A. A. (2008). Modeling disease dynamics: Cholera as a case study. In Statistical Advances in the Biomedical Sciences (A. Biswas, S. Datta, J. Fine and M. Segal, eds.) Chapter 8. Wiley, Hoboken, NJ.
  • Ionides, E. L., Fang, K. S., Isseroff, R. R. and Oster, G. F. (2004). Stochastic models for cell motion and taxis. J. Math. Biol. 48 23–37.
  • Jacquez, J. A. (1996). Comparmental Ananlysis in Biology and Medicine, 3rd ed. BioMedware, Ann Arbor, MI.
  • Kamo, M. and Sasaki, A. (2002). The effect of cross-immunity and seasonal forcing in a multi-strain epidemic model. Phys. D 165 228–241.
  • Kaper, J. B., Morris, J. G. and Levine, M. M. (1995). Cholera. Clin. Microbiol. Rev. 8 48–86.
  • Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.
  • Kendall, B. E., Briggs, C. J., Murdoch, W. W., Turchin, P., Ellner, S. P., McCauley, E., Nisbet, R. M. and Wood, S. N. (1999). Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches. Ecology 80 1789–1805.
  • Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 115 700–721.
  • Kevrekidis, I. G., Gear, C. W. and Hummer, G. (2004). Equation-free: The computer-assisted analysis of complex, multiscale systems. American Institute of Chemical Engineers J. 50 1346–1354.
  • King, A. A., Ionides, E. L. and Bretó, C. M. (2008a). pomp: Statistical inference for partially observed Markov processes. Available at http://cran.r-project.org/web/packages/pomp/.
  • King, A. A., Ionides, E. L., Pascual, M. and Bouma, M. J. (2008b). Inapparent infections and cholera dynamics. Nature 454 877–880.
  • Kitagawa, G. (1998). A self-organising state-space model. J. Amer. Statist. Assoc. 93 1203–1215.
  • Koelle, K., Cobey, S., Grenfell, B. and Pascual, M. (2006a). Epochal evolution shapes the philodynamics of interpandemic influenza A (H5N2) in humans. Science 314 1898–1903.
  • Koelle, K. and Pascual, M. (2004). Disentangling extrinsic from intrinsic factors in disease dynamics: A nonlinear time series approach with an application to cholera. Amer. Nat. 163 901–913.
  • Koelle, K., Pascual, M. and Yunus, M. (2006b). Serotype cycles in cholera dynamics. Proc. R. Soc. Ser. B Biol. 273 2879–2886.
  • Kou, S. C., Xie, S. and Liu, J. S. (2005). Bayesian analysis of single-molecule experimental data. Appl. Statist. 54 469–506.
  • Liu, J. and West, M. (2001). Combining parameter and state estimation in simulation-based filtering. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. J. Gordon, eds.) 197–224. Springer, New York.
  • Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York.
  • Lunn, D. J., Thomas, A., Best, N. and Speigelhalter, D. (2000). Winbugs—a Bayesian modelling framework: Concepts, structure and extensibility. Statist. Comput. 10 325–337.
  • Matis, J. H. and Kiffe, T. R. (2000). Stochastic Population Models. A Compartmental Perspective. Springer, New York.
  • May, R. M. (2004). Uses and abuses of mathematics in biology. Science 303 790–793.
  • McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman and Hall, London.
  • Morton, A. and Finkenstadt, B. F. (2005). Discrete time modelling of disease incidence time series by using Markov chain Monte Carlo methods. J. Roy. Statist. Soc. Ser. C 54 575–594.
  • Newman, K. B., Fernandez, C., Thomas, L. and Buckland, S. T. (2008). Monte Carlo inference for state-space models of wild animal populations. Biometrics. Pre-published Online.
  • Newman, K. B. and Lindley, S. T. (2006). Accounting for demographic and environmental stochasticity, observation error and parameter uncertainty in fish population dynamic models. North American J. Fisheries Management 26 685–701.
  • Øksendal, B. (1998). Stochastic Differential Equations, 5th ed. Springer, New York.
  • Polson, N. G., Stroud, J. R. and Muller, P. (2008). Practical filtering with sequential parameter learning. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 413–428.
  • R Development Core Team (2006). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
  • Ramsay, J. O., Hooker, G., Campbell, D. and Cao, J. (2007). Parameter estimation for differential equations: A generalized smoothing approach. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 741–796.
  • Reinker, S., Altman, R. M. and Timmer, J. (2006). Parameter estimation in stochastic biochemical reactions. IEE Proc. Sys. Biol. 153 168–178.
  • Sack, D. A., Sack, R. B., Nair, G. B. and Siddique, A. K. (2004). Cholera. Lancet 363 223–233.
  • Sato, K. (1999). Levy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
  • Sellke, T. (1983). On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Probab. 20 390–394.
  • Sisson, S. A., Fan, Y. and Tanaka, M. M. (2007). Sequential Monte Carlo without likelihoods. Proc. Natl. Acad. Sci. USA 104 1760–1765.
  • Solomon, S., Qin, D., Manning, M., Chen, Z., Marquis, M., Averyt, K. B., Tignor, M. and Miller, H. L., eds. (2007). Climate change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge Univ. Press, New York.
  • Swishchuk, A. and Wu, J. (2003). Evolution of Biological Systems in Random Media: Limit Theorems and Stability. Kluwer Academic Publishers, Dordrecht.
  • Tian, T. and Burrage, K. (2004). Binomial leap methods for simulating stochastic chemical kinetics. J. Chem. Phys. 121 10356–10364.
  • Toni, T., Welch, D., Strelkowa, N., Ipsen, A. and Stumpf, M. P. (2008). Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface. Pre-published online.
  • Wearing, H. J., Rohani, P. and Keeling, M. J. (2005). Appropriate models for the management of infectious diseases. PLoS Med. 2 e174.
  • Xiu, D., Kevrekidis, I. G. and Ghanem, R. (2005). An equation-free, multiscale approach to uncertainty quantification. Computing in Science and Engineering 7 16–23.

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