Abstract
In this paper we shall discuss an extension to Gaussian process (GP) regression models, where the measurements are modeled as linear functionals of the underlying GP and the estimation objective is a general linear operator of the process. We shall show how this framework can be used for modeling physical processes involved in measurement of the GP and for encoding physical prior information into regression models in form of stochastic partial differential equations (SPDE). We shall also illustrate the practical applicability of the theory in a simulated application.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
O’Hagan, A.: Curve fitting and optimal design for prediction (with discussion). Journal of the Royal Statistical Society B 40(1), 1–42 (1978)
Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)
Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover (1993)
Carlson, A.B.: Communication Systems: An Introduction to Signals and Noise in Electrical Communication, 3rd edn. McGraw-Hill, New York (1986)
Hayes, M.H.: Statistical Digital Signal Processing and Modeling. John Wiley & Sons, Chichester (1996)
Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Prentice-Hall, Englewood Cliffs (2002)
Guenther, R.B., Lee, J.W.: Partial Differential Equations of Mathematical Physics and Integral Equations. Dover, New York (1988)
Griffiths, D.J.: Introduction to Electrodynamics, 3rd edn. Pearson, London (2008)
Holden, H., Øksendal, B., Ubøe, J., Zhang, T. (eds.): Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach. Birkhäuser, Basel (1996)
Whittle, P.: On stationary processes in the plane. Biometrica 41(3/4), 434–449 (1954)
Matérn, B.: Spatial variation. Technical report, Meddelanden från Statens Skogforskningsinstitut, Band 49 - Nr 5 (1960)
Christakos, G.: Random Field Models in Earth Sciences. Academic Press, London (1992)
Cressie, N.A.C.: Statistics for Spatial Data. Wiley, Chichester (1993)
Gelfand, A.E., Diggle, P.J., Fuentes, M., Guttorp, P.: Handbook of Spatial Statistics. Chapman & Hall/CRC (2010)
Jain, A.K.: Partial differential equations and finite-difference methods in image processing, part 1: Image representation. Journal of Optimization Theory and Applications 23(1) (1977)
Jain, A.K., Jain, J.R.: Partial differential equations and finite difference methods in image processing – part II: Image restoration. IEEE Transactions on Automatic Control 23(5) (1978)
Curtain, R.: A survey of infinite-dimensional filtering. SIAM Review 17(3), 395–411 (1975)
Ray, W.H., Lainiotis, D.G.: Distributed Parameter Systems. Marcel Dekker, New York (1978)
Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, Philadelphia (2004)
Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Applied mathematical Sciences, vol. 160. Springer, Heidelberg (2005)
Hiltunen, P., Särkkä, S., Nissila, I., Lajunen, A., Lampinen, J.: State space regularization in the nonstationary inverse problem for diffuse optical tomography. Inverse Problems 27(2) (2011)
Alvarez, M., Lawrence, N.D.: Sparse convolved Gaussian processes for multi-output regression. In: Koller, D., Schuurmans, D., Bengio, Y., Bottou, L. (eds.) Advances in Neural Information Processing Systems, vol. 21, pp. 57–64. The MIT Press, Cambridge (2009)
Alvarez, M., Luengo, D., Titsias, M.K., Lawrence, N.D.: Efficient multioutput Gaussian processes through variational inducing kernels. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the 13th International Workshop on Artificial Intelligence and Statistics, pp. 25–32 (2010)
Alvarez, M., Lawrence, N.D.: Latent force models. In: van Dyk, D., Welling, M. (eds.) Proceedings of the 12th International Workshop on Artificial Intelligence and Statistics, pp. 9–16 (2009)
Vapnik, V.: Statistical Learning Theory. Wiley, Chichester (1998)
Graepel, T.: Solving noisy linear operator equations by Gaussian processes: Application to ordinary and partial differential equations. In: Proceedings of 20th International Conference on Machine Learning (2003)
Solak, E., Murray-Smith, R., Leithead, W.E., Leith, D., Rasmussen, C.E.: Derivative observations in Gaussian process models of dynamic systems. In: Advances in Neural Information Processing Systems, vol. 15, pp. 1033–1040. MIT Press, Cambridge (2003)
O’Hagan, A.: Bayes-Hermite quadrature. Journal of Statistical Planning and Inference 29, 245–260 (1991)
Papoulis, A.: Probability, Random Variables, and Stochastic Processes. McGraw-Hill, New York (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Särkkä, S. (2011). Linear Operators and Stochastic Partial Differential Equations in Gaussian Process Regression. In: Honkela, T., Duch, W., Girolami, M., Kaski, S. (eds) Artificial Neural Networks and Machine Learning – ICANN 2011. ICANN 2011. Lecture Notes in Computer Science, vol 6792. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21738-8_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-21738-8_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21737-1
Online ISBN: 978-3-642-21738-8
eBook Packages: Computer ScienceComputer Science (R0)