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International Workshop on Advanced Analysis and Learning on Temporal Data

AALTD 2019: Advanced Analytics and Learning on Temporal Data pp 183–198Cite as

Learning Stochastic Dynamical Systems via Bridge Sampling

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Part of the Lecture Notes in Computer Science book series (LNAI,volume 11986)

Abstract

We develop algorithms to automate discovery of stochastic dynamical system models from noisy, vector-valued time series. By discovery, we mean learning both a nonlinear drift vector field and a diagonal diffusion matrix for an Itô stochastic differential equation in \(\mathbb {R}^d\). We parameterize the vector field using tensor products of Hermite polynomials, enabling the model to capture highly nonlinear and/or coupled dynamics. We solve the resulting estimation problem using expectation maximization (EM). This involves two steps. We augment the data via diffusion bridge sampling, with the goal of producing time series observed at a higher frequency than the original data. With this augmented data, the resulting expected log likelihood maximization problem reduces to a least squares problem. We provide an open-source implementation of this algorithm. Through experiments on systems with dimensions one through eight, we show that this EM approach enables accurate estimation for multiple time series with possibly irregular observation times. We study how the EM method performs as a function of the amount of data augmentation, as well as the volume and noisiness of the data.

Keywords

  • Stochastic differential equations
  • Nonparametric estimation
  • Diffusion bridges
  • Expectation maximization

H. S. Bhat was partially supported by NSF award DMS-1723272. Both authors acknowledge use of the MERCED computational cluster, funded by NSF award ACI-1429783.

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References

  1. Archambeau, C., Opper, M., Shen, Y., Cornford, D., Shawe-Taylor, J.S.: Variational inference for diffusion processes. In: Advances in Neural Information Processing Systems, pp. 17–24 (2008)

    Google Scholar 

  2. Batz, P., Ruttor, A., Opper, M.: Variational estimation of the drift for stochastic differential equations from the empirical density. J. Stat. Mech: Theory Exp. 2016(8), 083404 (2016)

    CrossRef  MathSciNet  Google Scholar 

  3. Batz, P., Ruttor, A., Opper, M.: Approximate Bayes learning of stochastic differential equations. Phys. Rev. E 98, 022109 (2018)

    CrossRef  MathSciNet  Google Scholar 

  4. Bhat, H.S., Madushani, R.W.M.A.: Nonparametric adjoint-based inference for stochastic differential equations. In: 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA), pp. 798–807 (2016)

    Google Scholar 

  5. Bhattacharya, R.N., Waymire, E.C.: Stochastic Processes with Applications. SIAM (2009)

    Google Scholar 

  6. Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Nat. Acad. Sci. 113(15), 3932–3937 (2016)

    CrossRef  MathSciNet  Google Scholar 

  7. Chen, S., Shojaie, A., Witten, D.M.: Network reconstruction from high-dimensional ordinary differential equations. J. Am. Stat. Assoc. 112(520), 1697–1707 (2017)

    CrossRef  MathSciNet  Google Scholar 

  8. Ghahramani, Z., Roweis, S.T.: Learning nonlinear dynamical systems using an EM algorithm. In: Advances in Neural Information Processing Systems (NIPS), pp. 431–437 (1999)

    Google Scholar 

  9. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-662-12616-5

    CrossRef  MATH  Google Scholar 

  10. Mangan, N.M., Brunton, S.L., Proctor, J.L., Kutz, J.N.: Inferring biological networks by sparse identification of nonlinear dynamics. IEEE Trans. Mol. Biol. Multi-Scale Commun. 2(1), 52–63 (2016)

    CrossRef  Google Scholar 

  11. Mangan, N.M., Kutz, J.N., Brunton, S.L., Proctor, J.L.: Model selection for dynamical systems via sparse regression and information criteria. Proc. R. Soc. A 473(2204), 20170009 (2017)

    CrossRef  MathSciNet  Google Scholar 

  12. van der Meulen, F., Schauer, M., van Waaij, J.: Adaptive nonparametric drift estimation for diffusion processes using Faber-Schauder expansions. Statistical Inference for Stochastic Processes, pp. 1–26 (2017)

    Google Scholar 

  13. van der Meulen, F., Schauer, M., van Zanten, H.: Reversible jump MCMC for nonparametric drift estimation for diffusion processes. Comput. Stat. Data Anal. 71, 615–632 (2014)

    CrossRef  MathSciNet  Google Scholar 

  14. Müller, H.G., Yao, F., et al.: Empirical dynamics for longitudinal data. Ann. Stat. 38(6), 3458–3486 (2010)

    CrossRef  MathSciNet  Google Scholar 

  15. Nicolau, J.: Nonparametric estimation of second-order stochastic differential equations. Econ. Theory 23(05), 880 (2007)

    CrossRef  MathSciNet  Google Scholar 

  16. Papaspiliopoulos, O., Roberts, G.O.: Importance sampling techniques for estimation of diffusion models. Stat. Methods Stoch. Differ. Equ. 124, 311–340 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Papaspiliopoulos, O., Roberts, G.O., Stramer, O.: Data augmentation for diffusions. J. Comput. Graph. Stat. 22(3), 665–688 (2013)

    CrossRef  MathSciNet  Google Scholar 

  18. Quade, M., Abel, M., Kutz, J.N., Brunton, S.L.: Sparse identification of nonlinear dynamics for rapid model recovery. Chaos 28(6), 063116 (2018)

    CrossRef  MathSciNet  Google Scholar 

  19. Raissi, M., Karniadakis, G.E.: Hidden physics models: machine learning of nonlinear partial differential equations. J. Comput. Phys. 357, 125–141 (2018)

    CrossRef  MathSciNet  Google Scholar 

  20. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Machine learning of linear differential equations using Gaussian processes. J. Comput. Phys. 348, 683–693 (2017)

    CrossRef  MathSciNet  Google Scholar 

  21. Rawat, S.: Learning governing equations for stochastic dynamical systems. Ph.D. thesis, University of California, Merced (2018). Advisor: H.S. Bhat

    Google Scholar 

  22. Raziperchikolaei, R., Bhat, H.: A block coordinate descent proximal method for simultaneous filtering and parameter estimation. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning. Proceedings of Machine Learning Research, Long Beach, California, USA, 09–15 June 2019, vol. 97, pp. 5380–5388. PMLR (2019)

    Google Scholar 

  23. Roberts, G.O., Stramer, O.: On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm. Biometrika 88(3), 603–621 (2001)

    CrossRef  MathSciNet  Google Scholar 

  24. Rudy, S.H., Brunton, S.L., Proctor, J.L., Kutz, J.N.: Data-driven discovery of partial differential equations. Sci. Adv. 3(4), e1602614 (2017)

    CrossRef  Google Scholar 

  25. Ruttor, A., Batz, P., Opper, M.: Approximate Gaussian process inference for the drift function in stochastic differential equations. In: Advances in Neural Information Processing Systems, pp. 2040–2048 (2013)

    Google Scholar 

  26. Schaeffer, H., Caflisch, R., Hauck, C.D., Osher, S.: Sparse dynamics for partial differential equations. Proc. Nat. Acad. Sci. 110(17), 6634–6639 (2013)

    CrossRef  MathSciNet  Google Scholar 

  27. Schaeffer, H.: Learning partial differential equations via data discovery and sparse optimization. Proc. R. Soc. A: Math., Phys. Eng. Sci. 473(2197), 20160446 (2017)

    CrossRef  MathSciNet  Google Scholar 

  28. Schaeffer, H., Tran, G., Ward, R.: Extracting sparse high-dimensional dynamics from limited data. SIAM J. Appl. Math. 78(6), 3279–3295 (2018)

    CrossRef  MathSciNet  Google Scholar 

  29. Schauer, M., van der Meulen, F., van Zanten, H.: Guided proposals for simulating multi-dimensional diffusion bridges. Bernoulli 23(4A), 2917–2950 (2017)

    CrossRef  MathSciNet  Google Scholar 

  30. Schön, T.B., Svensson, A., Murray, L., Lindsten, F.: Probabilistic learning of nonlinear dynamical systems using sequential Monte Carlo. Mech. Syst. Signal Process. 104, 866–883 (2018)

    CrossRef  Google Scholar 

  31. Stuart, A.M.: Inverse problems: a Bayesian perspective. Acta Numerica 19, 451–559 (2010)

    CrossRef  MathSciNet  Google Scholar 

  32. Tran, G., Ward, R.: Exact recovery of chaotic systems from highly corrupted data. Multiscale Model. Simul. 15(3), 1108–1129 (2017)

    CrossRef  MathSciNet  Google Scholar 

  33. Verzelen, N., Tao, W., Müller, H.G.: others: Inferring stochastic dynamics from functional data. Biometrika 99(3), 533–550 (2012)

    CrossRef  MathSciNet  Google Scholar 

  34. Vrettas, M.D., Opper, M., Cornford, D.: Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations. Phys. Rev. E 91(1), 012148 (2015)

    CrossRef  MathSciNet  Google Scholar 

  35. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Universitext, 6th edn. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-642-14394-6

    CrossRef  MATH  Google Scholar 

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Authors and Affiliations

  1. Applied Mathematics Unit, University of California, Merced, CA, 95343, USA

    Harish S. Bhat & Shagun Rawat

Authors
  1. Harish S. Bhat
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  2. Shagun Rawat
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Correspondence to Harish S. Bhat .

Editor information

Editors and Affiliations

  1. Orange Labs, Lannion, France

    Vincent Lemaire

  2. Inria, University of Rennes, Rennes, France

    Simon Malinowski

  3. University of East Anglia, Norwich, UK

    Prof. Anthony Bagnall

  4. Orange Labs, Châtillon, France

    Alexis Bondu

  5. Irisa, Agrocampus Ouest, Rennes, France

    Dr. Thomas Guyet

  6. University of Rennes 2, Rennes, France

    Romain Tavenard

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Bhat, H.S., Rawat, S. (2020). Learning Stochastic Dynamical Systems via Bridge Sampling. In: Lemaire, V., Malinowski, S., Bagnall, A., Bondu, A., Guyet, T., Tavenard, R. (eds) Advanced Analytics and Learning on Temporal Data. AALTD 2019. Lecture Notes in Computer Science(), vol 11986. Springer, Cham. https://doi.org/10.1007/978-3-030-39098-3_14

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  • DOI: https://doi.org/10.1007/978-3-030-39098-3_14

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