Statistics and Computing

, Volume 25, Issue 2, pp 427–437 | Cite as

Posterior inference on parameters of stochastic differential equations via non-linear Gaussian filtering and adaptive MCMC

  • Simo Särkkä
  • Jouni Hartikainen
  • Isambi Sailon Mbalawata
  • Heikki Haario
Article

Abstract

This article is concerned with Bayesian estimation of parameters in non-linear multivariate stochastic differential equation (SDE) models occurring, for example, in physics, engineering, and financial applications. In particular, we study the use of adaptive Markov chain Monte Carlo (AMCMC) based numerical integration methods with non-linear Kalman-type approximate Gaussian filters for parameter estimation in non-linear SDEs. We study the accuracy and computational efficiency of gradient-free sigma-point approximations (Gaussian quadratures) in the context of parameter estimation, and compare them with Taylor series and particle MCMC approximations. The results indicate that the sigma-point based Gaussian approximations lead to better approximations of the parameter posterior distribution than the Taylor series, and the accuracy of the approximations is comparable to that of the computationally significantly heavier particle MCMC approximations.

Keywords

Stochastic differential equation Parameter estimation Gaussian approximation Non-linear Kalman filter Adaptive Markov chain Monte Carlo 

Supplementary material

11222_2013_9441_MOESM1_ESM.ps (366 kb)
Supplementary Material for “Posterior Inference on Parameters of Stochastic Differential Equations via Non-Linear Gaussian Filtering and Adaptive MCMC” (PS 366 kB)

References

  1. Aït-Sahalia, Y.: Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica 70(1), 223–262 (2002) CrossRefMATHMathSciNetGoogle Scholar
  2. Aït-Sahalia, Y.: Closed-form expansion for multivariate diffusions. Ann. Stat. 36(2), 906–937 (2008) CrossRefMATHGoogle Scholar
  3. Andrieu, C., Moulines, E.: On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16(3), 1462–1505 (2006) CrossRefMATHMathSciNetGoogle Scholar
  4. Andrieu, C., Thoms, J.: A tutorial on adaptive MCMC. Stat. Comput. 18(4), 343–373 (2008) CrossRefMathSciNetGoogle Scholar
  5. Andrieu, C., Doucet, A., Holenstein, R.: Particle Markov chain Monte Carlo methods. J. R. Stat. Soc., Ser. B, Stat. Methodol. 72(3), 269–342 (2010) CrossRefMathSciNetGoogle Scholar
  6. Arasaratnam, I., Haykin, S., Hurd, T.R.: Cubature Kalman filtering for continuous-discrete systems: theory and simulations. IEEE Trans. Signal Process. 58(10), 4977–4993 (2010) CrossRefMathSciNetGoogle Scholar
  7. Archambeau, C., Opper, M.: Approximate inference for continuous-time Markov processes. In: Inference and Learning in Dynamic Models. Cambridge University Press, Cambridge (2011) Google Scholar
  8. Atchade, Y.F., Rosenthal, J.S.: On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11(5), 815–828 (2005) CrossRefMATHMathSciNetGoogle Scholar
  9. Bar-Shalom, Y., Li, X.R., Kirubarajan, T.: Estimation with Applications to Tracking and Navigation. Wiley, New York (2001) CrossRefGoogle Scholar
  10. Bernardo, J.M., Smith, A.F.M.: Bayesian Theory. Wiley, New York (1994) CrossRefMATHGoogle Scholar
  11. Beskos, A., Papaspiliopoulos, O., Roberts, G., Fearnhead, P.: Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion). J. R. Stat. Soc., B 68(3), 333–382 (2006) CrossRefMATHMathSciNetGoogle Scholar
  12. Beskos, A., Papaspiliopoulos, O., Roberts, G.: Monte Carlo maximum likelihood estimation for discretely observed diffusion processes. Ann. Stat. 37(1), 223–245 (2009) CrossRefMATHMathSciNetGoogle Scholar
  13. Brandt, M., Santa-Clara, P.: Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets. J. Financ. Econ. 63(2), 161–210 (2002) CrossRefGoogle Scholar
  14. Doucet, A., Pitt, M., Kohn, R.: Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator (2012). arXiv:1210.1871
  15. Elerian, O., Chib, S., Shephard, N.: Likelihood inference for discretely observed non-linear diffusions. Econometrica 69(4), 959–993 (2001) CrossRefMATHMathSciNetGoogle Scholar
  16. Eraker, B.: MCMC analysis of diffusion models with application to finance. J. Bus. Econ. Stat. 19(2), 177–191 (2001) CrossRefMathSciNetGoogle Scholar
  17. Ferm, L., Lötstedt, P., Hellander, A.: A hierarchy of approximations of the master equation scaled by a size parameter. J. Sci. Comput. 34(2), 127–151 (2008) CrossRefMATHMathSciNetGoogle Scholar
  18. Fort, G., Moulines, E., Priouret, P.: Convergence of adaptive MCMC algorithms: ergodicity and law of large number. Tech. Rep., LTCI (2011) Google Scholar
  19. Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian Data Analysis, 2nd edn. CRC Press/Chapman & Hall, Boca Raton/London (2004) MATHGoogle Scholar
  20. Godsill, S.J., Doucet, A., West, M.: Monte Carlo smoothing for nonlinear time series. J. Am. Stat. Assoc. 99(465), 156–168 (2004) CrossRefMATHMathSciNetGoogle Scholar
  21. Golightly, A., Wilkinson, D.J.: Bayesian sequential inference for nonlinear multivariate diffusions. Stat. Comput. 16(4), 323–338 (2006) CrossRefMathSciNetGoogle Scholar
  22. Golightly, A., Wilkinson, D.J.: Bayesian inference for nonlinear multivariate diffusion models observed with error. Comput. Stat. Data Anal. 52(3), 1674–1693 (2008) CrossRefMATHMathSciNetGoogle Scholar
  23. Golightly, A., Wilkinson, D.J.: Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo. Interface Focus 1(6), 807–820 (2011) CrossRefGoogle Scholar
  24. Haario, H., Saksman, E., Tamminen, J.: Adaptive proposal distribution for random walk Metropolis algorithm. Comput. Stat. 14(3), 375–395 (1999) CrossRefMATHGoogle Scholar
  25. Haario, H., Saksman, E., Tamminen, J.: An adaptive Metropolis algorithm. Bernoulli 7(2), 223–242 (2001) CrossRefMATHMathSciNetGoogle Scholar
  26. Haario, H., Laine, M., Mira, A., Saksman, E.: DRAM: efficient adaptive MCMC. Stat. Comput. 16(4), 339–354 (2006) CrossRefMathSciNetGoogle Scholar
  27. Hurn, A.S., Lindsay, K.A.: Estimating the parameters of stochastic differential equations. Math. Comput. Simul. 48(4–6), 373–384 (1999) CrossRefMATHMathSciNetGoogle Scholar
  28. Hurn, A.S., Lindsay, K.A., Martin, V.L.: On the efficacy of simulated maximum likelihood for estimating the parameters of stochastic differential equations. J. Time Ser. Anal. 24(1), 45–63 (2003) CrossRefMATHMathSciNetGoogle Scholar
  29. Hurn, A.S., Lindsay, K.A., McClelland, A.J.: A quasi-maximum likelihood method for estimating the parameters of multivariate diffusions. J. Econom. 172, 106–126 (2013) CrossRefMathSciNetGoogle Scholar
  30. Ito, K., Xiong, K.: Gaussian filters for nonlinear filtering problems. IEEE Trans. Autom. Control 45(5), 910–927 (2000) CrossRefMATHMathSciNetGoogle Scholar
  31. Jazwinski, A.H.: Stochastic Processes and Filtering Theory. Academic Press, San Diego (1970) MATHGoogle Scholar
  32. Jeisman, J.: Estimation of the Parameters of Stochastic Differential Equations. Doctoral dissertation, Queensland University of Technology (2005) Google Scholar
  33. Jensen, B., Poulsen, R.: Transition densities of diffusion processes: numerical comparison of approximation techniques. J. Deriv. 9(4), 18–32 (2002) CrossRefGoogle Scholar
  34. Jones, C.S.: A simple Bayesian method for the analysis of diffusion processes. Working paper, University of Pennsylvania (1998) Google Scholar
  35. Julier, S.J., Uhlmann, J.K., Durrant-Whyte, H.F.: A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Trans. Autom. Control 45(3), 477–482 (2000) CrossRefMATHMathSciNetGoogle Scholar
  36. Kandepu, R., Foss, B., Imsland, L.: Applying the unscented Kalman filter for nonlinear state estimation. J. Process Control 18(7–8), 753–768 (2008) CrossRefGoogle Scholar
  37. Kitagawa, G.: Non-Gaussian state-space modeling of nonstationary time series. J. Am. Stat. Assoc. 82(400), 1032–1041 (1987) MATHMathSciNetGoogle Scholar
  38. Kloeden, P.E., Platen, E.: Numerical Solution to Stochastic Differential Equations. Springer, Berlin (1999) Google Scholar
  39. Komorowski, M., Finkenstädt, B., Harper, C., Rand, D.: Bayesian inference of biochemical kinetic parameters using the linear noise approximation. BMC Bioinform. 10(1), 343 (2009) CrossRefGoogle Scholar
  40. Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 7(1), 49–58 (1970) CrossRefMATHMathSciNetGoogle Scholar
  41. Kurtz, T.G.: Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 8(2), 344–356 (1971) CrossRefMATHMathSciNetGoogle Scholar
  42. Kushner, H.J.: Approximations to optimal nonlinear filters. IEEE Trans. Autom. Control 12(5), 546–556 (1967) CrossRefGoogle Scholar
  43. Liang, F., Liu, C., Carroll, R.J.: Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples. Wiley, Chichester (2010) CrossRefGoogle Scholar
  44. Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, Berlin (2001) MATHGoogle Scholar
  45. Maybeck, P.: Stochastic Models, Estimation and Control vol. 2. Academic Press, San Diego (1982) MATHGoogle Scholar
  46. Mbalawata, I.S., Särkkä, S., Haario, H.: Parameter estimation in stochastic differential equations with Markov chain Monte Carlo and non-linear Kalman filtering. Comput. Stat. 28(3), 1195–1223 (2013) CrossRefGoogle Scholar
  47. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2003) CrossRefGoogle Scholar
  48. Pedersen, A.R.: A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Stat. 22(1), 55–71 (1995) MATHGoogle Scholar
  49. Roberts, G.O., Rosenthal, J.S.: Optimal scaling for various Metropolis–Hastings algorithms. Stat. Sci. 16(4), 351–367 (2001) CrossRefMATHMathSciNetGoogle Scholar
  50. Roberts, G.O., Rosenthal, J.S.: Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. Applied probability 44(2), 458–475 (2007) CrossRefMATHMathSciNetGoogle Scholar
  51. Roberts, G.O., Rosenthal, J.S.: Examples of adaptive MCMC. J. Comput. Graph. Stat. 18(2), 349–367 (2009) CrossRefMathSciNetGoogle Scholar
  52. Ross, J.V., Pagendam, D.E., Pollett, P.K.: On parameter estimation in population models II: multi-dimensional processes and transient dynamics. Theor. Popul. Biol. 75(2), 123–132 (2009) CrossRefMATHGoogle Scholar
  53. Rößler, A.: Runge-Kutta methods for Ito stochastic differential equations with scalar noise. BIT Numer. Math. 46, 97–110 (2006) CrossRefMATHGoogle Scholar
  54. Särkkä, S.: Recursive Bayesian inference on stochastic differential equations. Doctoral dissertation, Helsinki University of Technology (2006) Google Scholar
  55. Särkkä, S.: On unscented Kalman filtering for state estimation of continuous-time nonlinear systems. IEEE Trans. Autom. Control 52(9), 1631–1641 (2007) CrossRefGoogle Scholar
  56. Särkkä, S.: Continuous-time and continuous-discrete-time unscented Rauch-Tung-Striebel smoothers. Signal Process. 90(1), 225–235 (2010) CrossRefMATHGoogle Scholar
  57. Särkkä, S., Sarmavuori, J.: Gaussian filtering and smoothing for continuous-discrete dynamic systems. Signal Process. 93, 500–510 (2013) CrossRefGoogle Scholar
  58. Särkkä, S., Solin, A.: On continuous-discrete cubature Kalman filtering. In: Proceedings of SYSID 2012, pp. 1210–1215 (2012) Google Scholar
  59. Schweppe, F.C.: Evaluation of likelihood functions for Gaussian signals. IEEE Trans. Inf. Theory 11, 61–70 (1965) CrossRefMATHMathSciNetGoogle Scholar
  60. Singer, H.: Parameter estimation of nonlinear stochastic differential equations: simulated maximum likelihood versus extended Kalman filter and Itô-Taylor expansion. J. Comput. Graph. Stat. 11, 972–995 (2002) CrossRefGoogle Scholar
  61. Singer, H.: Generalized Gauss-Hermite filtering. AStA Adv. Stat. Anal. 92(2), 179–195 (2008a) CrossRefMathSciNetGoogle Scholar
  62. Singer, H.: Nonlinear continuous time modeling approaches in panel research. Stat. Neerl. 62(1), 29–57 (2008b) MATHGoogle Scholar
  63. Singer, H.: Continuous-discrete state-space modeling of panel data with nonlinear filter algorithms. AStA Adv. Stat. Anal. 95(4), 375–413 (2011) CrossRefMATHMathSciNetGoogle Scholar
  64. Snyder, C., Bengtsson, T., Bickel, P., Anderson, J.: Obstacles to high-dimensional particle filtering. Mon. Weather Rev. 136(12), 4629–4640 (2008) CrossRefGoogle Scholar
  65. Sørensen, H.: Parametric inference for diffusion processes observed at discrete points in time: a survey. Int. Stat. Rev. 72(3), 337–354 (2004) CrossRefGoogle Scholar
  66. Socha, L.: Linearization Methods for Stochastic Dynamic Systems. Springer, Berlin (2008) MATHGoogle Scholar
  67. Stramer, O., Bognar, M.: Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach. Bayesian Anal. 6(2), 231–258 (2011) CrossRefMathSciNetGoogle Scholar
  68. Stramer, O., Bognar, M., Schneider, P.: Bayesian inference for discretely sampled Markov processes with closed-form likelihood expansions. J. Financ. Econ. 8(4), 450–480 (2010) Google Scholar
  69. Van Kampen, N.G.: Stochastic processes in physics and chemistry, 3rd edn. Elsevier, Amsterdam (2007) Google Scholar
  70. Vihola, M.: Robust adaptive Metropolis algorithm with coerced acceptance rate. Stat. Comput. 22(5), 997–1008 (2012) CrossRefMATHMathSciNetGoogle Scholar
  71. Wu, Y., Hu, D., Wu, M., Hu, X.: A numerical-integration perspective on Gaussian filters. IEEE Trans. Signal Process. 54(8), 2910–2921 (2006) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Simo Särkkä
    • 1
  • Jouni Hartikainen
    • 1
  • Isambi Sailon Mbalawata
    • 1
  • Heikki Haario
    • 1
  1. 1.AaltoFinland

Personalised recommendations