Advertisement

Statistics and Computing

, Volume 27, Issue 6, pp 1677–1692 | Cite as

A second-order iterated smoothing algorithm

  • Dao Nguyen
  • Edward L. Ionides
Article
  • 357 Downloads

Abstract

Simulation-based inference for partially observed stochastic dynamic models is currently receiving much attention due to the fact that direct computation of the likelihood is not possible in many practical situations. Iterated filtering methodologies enable maximization of the likelihood function using simulation-based sequential Monte Carlo filters. Doucet et al. (2013) developed an approximation for the first and second derivatives of the log likelihood via simulation-based sequential Monte Carlo smoothing and proved that the approximation has some attractive theoretical properties. We investigated an iterated smoothing algorithm carrying out likelihood maximization using these derivative approximations. Further, we developed a new iterated smoothing algorithm, using a modification of these derivative estimates, for which we establish both theoretical results and effective practical performance. On benchmark computational challenges, this method beat the first-order iterated filtering algorithm. The method’s performance was comparable to a recently developed iterated filtering algorithm based on an iterated Bayes map. Our iterated smoothing algorithm and its theoretical justification provide new directions for future developments in simulation-based inference for latent variable models such as partially observed Markov process models.

Keywords

Iterated smoothing Sequential Monte Carlo State space model Hidden Markov model Parameter estimation 

Notes

Acknowledgments

This research was funded in part by National Science Foundation Grant DMS-1308919 and National Institutes of Health Grants 1-U54-GM111274 and 1-U01-GM110712.

Supplementary material

11222_2016_9711_MOESM_ESM.pdf (603 kb)
Supplementary material 1 (pdf 603 KB)

References

  1. Andrieu, C., Doucet, A., Holenstein, R.: Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B 72(3), 269–342 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. Bhadra, A., Ionides, E.L., Laneri, K., Pascual, M., Bouma, M., Dhiman, R.C.: Malaria in Northwest India: data analysis via partially observed stochastic differential equation models driven by Lévy noise. J. Am. Stat. Assoc. 106, 440–451 (2011)CrossRefMATHGoogle Scholar
  3. Bjørnstad, O.N., Grenfell, B.T.: Noisy clockwork: time series analysis of population fluctuations in animals. Science 293, 638–643 (2001)CrossRefGoogle Scholar
  4. Blackwood, J.C., Cummings, D.A.T., Broutin, H., Iamsirithaworn, S., Rohani, P.: Deciphering the impacts of vaccination and immunity on pertussis epidemiology in Thailand. Proc. Natl. Acad. Sci. USA 110, 9595–9600 (2013a)CrossRefGoogle Scholar
  5. Blackwood, J.C., Streicker, D.G., Altizer, S., Rohani, P.: Resolving the roles of immunity, pathogenesis, and immigration for rabies persistence in vampire bats. Proc. Natl. Acad. Sci. USA 110, 2083720842 (2013b)Google Scholar
  6. Blake, I.M., Martin, R., Goel, A., Khetsuriani, N., Everts, J., Wolff, C., Wassilak, S., Aylward, R.B., Grassly, N.C.: The role of older children and adults in wild poliovirus transmission. Proc. Natl. Acad. Sci. USA 111(29), 10604–10609 (2014)CrossRefGoogle Scholar
  7. Bretó, C., He, D., Ionides, E.L., King, A.A.: Time series analysis via mechanistic models. Ann. Appl. Stat. 3, 319–348 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. Camacho, A., Ballesteros, S., Graham, A.L., Carrat, F., Ratmann, O., Cazelles, B.: Explaining rapid reinfections in multiple-wave influenza outbreaks: Tristan da Cunha 1971 epidemic as a case study. Proc. R. Soc. Lond. Ser. B 278(1725), 3635–3643 (2011)CrossRefGoogle Scholar
  9. Chopin, N., Jacob, P.E., Papaspiliopoulos, O.: SMC\(^2\): an efficient algorithm for sequential analysis of state space models. J. R. Stat. Soc. Ser. B 75(3), 397–426 (2013)MathSciNetCrossRefGoogle Scholar
  10. Dahlin, J., Lindsten, F., Schön, T.B.: Particle Metropolis-Hastings using gradient and Hessian information. Stat. Comput. 25(1), 81–92 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. Douc, R., Cappé, O., Moulines, E.: Comparison of resampling schemes for particle filtering. In: Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005, pp 64–69. IEEE, New York (2005)Google Scholar
  12. Doucet, A., Jacob, P. E., and Rubenthaler, S.: Derivative-free estimation of the score vector and observed information matrix with application to state-space models (version 2). arXiv:1304.5768v2 (2013)
  13. Earn, D.J., He, D., Loeb, M.B., Fonseca, K., Lee, B.E., Dushoff, J.: Effects of school closure on incidence of pandemic influenza in Alberta. Ann. Int. Med. 156(3), 173–181 (2012)CrossRefGoogle Scholar
  14. Gordon, N.J., Salmond, D.J., Smith, A.F.: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F. Radar Signal Process. 140, 107–113 (1993)CrossRefGoogle Scholar
  15. He, D., Dushoff, J., Day, T., Ma, J., Earn, D.J.D.: Inferring the causes of the three waves of the 1918 influenza pandemic in England and Wales. Proc. R. Soc. Lond. Ser. B 280(1766), 20131345 (2013)CrossRefGoogle Scholar
  16. He, D., Ionides, E.L., King, A.A.: Plug-and-play inference for disease dynamics: measles in large and small populations as a case study. J. R. Soc. Interface 7(43), 271–283 (2010)CrossRefGoogle Scholar
  17. Ionides, E.L., Bhadra, A., Atchadé, Y., King, A.: Iterated filtering. Ann. Stat. 39, 1776–1802 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. Ionides, E.L., Bretó, C., King, A.A.: Inference for nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 103, 18438–18443 (2006)CrossRefGoogle Scholar
  19. Ionides, E.L., Nguyen, D., Atchadé, Y., Stoev, S., King, A.A.: Inference for dynamic and latent variable models via iterated, perturbed Bayes maps. P. Natl. Acad. Sci. USA 112(3), 719–724 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. Kevrekidis, I.G., Gear, C.W., Hummer, G.: Equation-free: the computer-assisted analysis of complex, multiscale systems. Am. Inst. Chem. Eng. J. 50, 1346–1354 (2004)CrossRefGoogle Scholar
  21. King, A.A., Domenech de Celle, M., Magpantay, F.M.G., Rohani, P.: Avoidable errors in the modelling of outbreaks of emerging pathogens, with special reference to Ebola. Proc. R. Soc. Lond. Ser. B 282, 20150347 (2015)CrossRefGoogle Scholar
  22. King, A.A., Ionides, E.L., Pascual, M., Bouma, M.J.: Inapparent infections and cholera dynamics. Nature 454, 877–880 (2008)CrossRefGoogle Scholar
  23. King, A.A., Nguyen, D., Ionides, E.L.: Statistical inference for partially observed Markov processes via the R package pomp. J. Stat. Softw 69, 1–43 (2016)CrossRefGoogle Scholar
  24. Kloeden, P.E., Platen, E.: Numerical Soluion of Stochastic Differential Equations, 3rd edn. Springer, New York (1999)Google Scholar
  25. Kushner, H.J., Clark, D.S.: Stochastic Approximation Methods for Constrained and Unconstrained Systems. Springer, New York (1978)CrossRefMATHGoogle Scholar
  26. Laneri, K., Bhadra, A., Ionides, E.L., Bouma, M., Dhiman, R.C., Yadav, R.S., Pascual, M.: Forcing versus feedback: epidemic malaria and monsoon rains in Northwest India. PLoS Comput. Biol. 6(9), e1000898 (2010)CrossRefGoogle Scholar
  27. Laneri, K., Paul, R.E., Tall, A., Faye, J., Diene-Sarr, F., Sokhna, C., Trape, J.-F., Rodó, X.: Dynamical malaria models reveal how immunity buffers effect of climate variability. Proc. Natl. Acad. Sci. USA 112(28), 8786–8791 (2015)CrossRefGoogle Scholar
  28. Lavine, J.S., King, A.A., Andreasen, V., Bjrnstad, O.N.: Immune boosting explains regime-shifts in prevaccine-era pertussis dynamics. PLoS ONE 8(8), e72086 (2013)CrossRefGoogle Scholar
  29. Lavine, J.S., Rohani, P.: Resolving pertussis immunity and vaccine effectiveness using incidence time series. Expert Rev. Vaccines 11, 1319–1329 (2012)CrossRefGoogle Scholar
  30. Macdonald, G.: The Epidemiology and Control of Malaria. Oxford University Press, Oxford (1957)Google Scholar
  31. Martinez-Bakker, M., King, A.A., Rohani, P.: Unraveling the transmission ecology of polio. PLoS Biol. 13(6), e1002172 (2015)CrossRefGoogle Scholar
  32. Nemeth, C., Fearnhead, P., Mihaylova, L.: Particle approximations of the score and observed information matrix for parameter estimation in state space models with linear computational cost. arXiv:1306.0735 (2013)
  33. Nguyen, D. (2015). Iterated smoothing r package, is2. https://r-forge.r-project.org/projects/is2
  34. Olsson, J., Cappé, O., Douc, R., Moulines, E.: Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models. Bernoulli 14(1), 155–179 (2008)MathSciNetCrossRefMATHGoogle Scholar
  35. Poyiadjis, G., Doucet, A., Singh, S.S.: Particle approximations of the score and observed information matrix in state space models with application to parameter estimation. Biometrika 98(1), 65–80 (2011)MathSciNetCrossRefMATHGoogle Scholar
  36. Romero-Severson, E., Volz, E., Koopman, J., Leitner, T., Ionides, E.: Dynamic variation in sexual contact rates in a cohort of HIV-negative gay men. Am. J. Epidemiol. 182, 255–262 (2015)CrossRefGoogle Scholar
  37. Ross, R.: The Prevention of Malaria. Dutton, Boston (1910)Google Scholar
  38. Roy, M., Bouma, M.J., Ionides, E.L., Dhiman, R.C., Pascual, M.: The potential elimination of plasmodium vivax malaria by relapse treatment: Insights from a transmission model and surveillance data from NW India. PLoS Negl. Trop. Dis. 7, e1979 (2013)CrossRefGoogle Scholar
  39. Shrestha, S., Foxman, B., Weinberger, D.M., Steiner, C., Viboud, C., Rohani, P.: Identifying the interaction between influenza and pneumococcal pneumonia using incidence data. Sci. Transl. Med. 5(191), 191ra84 (2013)CrossRefGoogle Scholar
  40. Shrestha, S., King, A.A., Rohani, P.: Statistical inference for multi-pathogen systems. PLoS Comput. Biol. 7(8), e1002135 (2011)Google Scholar
  41. Sisson, S.A., Fan, Y., Tanaka, M.M.: Sequential Monte Carlo without likelihoods. Proc. Natl. Acad. Sci. USA 104(6), 1760–1765 (2007)MathSciNetCrossRefMATHGoogle Scholar
  42. Spall, J.C.: Introduction to Stochastic Search and Optimization. Wiley, Hoboken (2003)CrossRefMATHGoogle Scholar
  43. Toni, T., Welch, D., Strelkowa, N., Ipsen, A., Stumpf, M.P.: Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface 6, 187–202 (2009)Google Scholar
  44. Wood, S.N.: Statistical inference for noisy nonlinear ecological dynamic systems. Nature 466(7310), 1102–1104 (2010)CrossRefGoogle Scholar
  45. Yıldırım, S., Singh, S.S., Dean, T., Jasra, A.: Parameter estimation in hidden Markov models with intractable likelihoods using sequential Monte Carlo. J. Comput. Graph. Stat. 24, 846–865 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of MichiganAnn ArborUSA

Personalised recommendations