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Multiplicative Latent Force Models

  • Daniel J. TaitEmail author
  • Bruce J. Worton
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 296)

Abstract

Bayesian modelling of dynamic systems must achieve a compromise between providing a complete mechanistic specification of the process while retaining the flexibility to handle those situations in which data is sparse relative to model complexity, or a full specification is hard to motivate. Latent force models achieve this dual aim by specifying a parsimonious linear evolution equation with an additive latent Gaussian process (GP) forcing term. In this work we extend the latent force framework to allow for multiplicative interactions between the GP and the latent states leading to more control over the geometry of the trajectories. Unfortunately inference is no longer straightforward and so we introduce an approximation based on the method of successive approximations and examine its performance using a simulation study.

Keywords

Gaussian processes Latent force models 

Notes

Acknowledgements

The authors would like to thank Dr. Durante and two anonymous referees for their valuable comments and suggestions on an earlier version of this paper, which substantially improved the content and clarity of the article. Daniel J. Tait is supported by an EPSRC studentship.

References

  1. 1.
    Alvarez, M., Luengo, D., Lawrence, N.D.: Latent force models. In: van Dyk, D., Welling, M. (eds.) Proc. Mach. Learn. Res. 5, 9–16 (2009)Google Scholar
  2. 2.
    Dowson, D.C., Landau, B.V.: The Fréchet distance between multivariate normal distributions. J. Multivar. Anal. 12, 450–455 (1982)CrossRefGoogle Scholar
  3. 3.
    Gao, P., Honkela, A., Rattray, M., Lawrence, N.D.: Gaussian process modelling of latent chemical species: applications to inferring transcription factor activities. Bioinform. 24, i70–i75 (2008)CrossRefGoogle Scholar
  4. 4.
    Iserles, A., Norsett, S.P.: On the solution of linear differential equations in Lie groups. Philos. Trans. Roy. Soc. A, Math., Phys. and Eng. Sci. 357, 983–1019 (1999)Google Scholar
  5. 5.
    Lawrence, N.D., Sanguinetti, G., Rattray, M.: Modelling transcriptional regulation using Gaussian processes. In: Schölkopf, B., Platt, J.C., Hofmann, T. (eds.) Adv. Neural Inf. Process. Syst. 19, 785–792 (2007)Google Scholar
  6. 6.
    Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)zbMATHGoogle Scholar
  7. 7.
    Risken, H.: The Fokker-Planck Equation: Methods of Solution and Applications. Springer, Berlin (1989)CrossRefGoogle Scholar
  8. 8.
    Strand, J.L.: Random ordinary differential equations. J. Differ. Equ. 7, 538–553 (1970)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Titsias M.K., Lawrence, N.D., Rattray, M.: Efficient sampling for Gaussian process inference using control variables. In: Koller, D., Schuurmans, D., Bengio, Y., Bottou, L. (eds.) Adv. Neural Inf. Process. Syst. 21, 1681–1688 (2009)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsUniversity of EdinburghEdinburghScotland

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