Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression

Simo Sarkka, Jouni Hartikainen
Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, PMLR 22:993-1001, 2012.

Abstract

We show how spatio-temporal Gaussian process (GP) regression problems (or the equivalent Kriging problems) can be formulated as infinite-dimensional Kalman filtering and Rauch-Tung-Striebel (RTS) smoothing problems, and present a procedure for converting spatio-temporal covariance functions into infinite-dimensional stochastic differential equations (SDEs). The resulting infinite-dimensional SDEs belong to the class of stochastic pseudo-differential equations and can be numerically treated using the methods developed for deterministic counterparts of the equations. The scaling of the computational cost in the proposed approach is linear in the number of time steps as opposed to the cubic scaling of the direct GP regression solution. We also show how separable covariance functions lead to a finite-dimensional Kalman filtering and RTS smoothing problem, present analytical and numerical examples, and discuss numerical methods for computing the solutions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v22-sarkka12, title = {Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression}, author = {Sarkka, Simo and Hartikainen, Jouni}, booktitle = {Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics}, pages = {993--1001}, year = {2012}, editor = {Lawrence, Neil D. and Girolami, Mark}, volume = {22}, series = {Proceedings of Machine Learning Research}, address = {La Palma, Canary Islands}, month = {21--23 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v22/sarkka12/sarkka12.pdf}, url = {https://proceedings.mlr.press/v22/sarkka12.html}, abstract = {We show how spatio-temporal Gaussian process (GP) regression problems (or the equivalent Kriging problems) can be formulated as infinite-dimensional Kalman filtering and Rauch-Tung-Striebel (RTS) smoothing problems, and present a procedure for converting spatio-temporal covariance functions into infinite-dimensional stochastic differential equations (SDEs). The resulting infinite-dimensional SDEs belong to the class of stochastic pseudo-differential equations and can be numerically treated using the methods developed for deterministic counterparts of the equations. The scaling of the computational cost in the proposed approach is linear in the number of time steps as opposed to the cubic scaling of the direct GP regression solution. We also show how separable covariance functions lead to a finite-dimensional Kalman filtering and RTS smoothing problem, present analytical and numerical examples, and discuss numerical methods for computing the solutions.} }
Endnote
%0 Conference Paper %T Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression %A Simo Sarkka %A Jouni Hartikainen %B Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2012 %E Neil D. Lawrence %E Mark Girolami %F pmlr-v22-sarkka12 %I PMLR %P 993--1001 %U https://proceedings.mlr.press/v22/sarkka12.html %V 22 %X We show how spatio-temporal Gaussian process (GP) regression problems (or the equivalent Kriging problems) can be formulated as infinite-dimensional Kalman filtering and Rauch-Tung-Striebel (RTS) smoothing problems, and present a procedure for converting spatio-temporal covariance functions into infinite-dimensional stochastic differential equations (SDEs). The resulting infinite-dimensional SDEs belong to the class of stochastic pseudo-differential equations and can be numerically treated using the methods developed for deterministic counterparts of the equations. The scaling of the computational cost in the proposed approach is linear in the number of time steps as opposed to the cubic scaling of the direct GP regression solution. We also show how separable covariance functions lead to a finite-dimensional Kalman filtering and RTS smoothing problem, present analytical and numerical examples, and discuss numerical methods for computing the solutions.
RIS
TY - CPAPER TI - Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression AU - Simo Sarkka AU - Jouni Hartikainen BT - Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics DA - 2012/03/21 ED - Neil D. Lawrence ED - Mark Girolami ID - pmlr-v22-sarkka12 PB - PMLR DP - Proceedings of Machine Learning Research VL - 22 SP - 993 EP - 1001 L1 - http://proceedings.mlr.press/v22/sarkka12/sarkka12.pdf UR - https://proceedings.mlr.press/v22/sarkka12.html AB - We show how spatio-temporal Gaussian process (GP) regression problems (or the equivalent Kriging problems) can be formulated as infinite-dimensional Kalman filtering and Rauch-Tung-Striebel (RTS) smoothing problems, and present a procedure for converting spatio-temporal covariance functions into infinite-dimensional stochastic differential equations (SDEs). The resulting infinite-dimensional SDEs belong to the class of stochastic pseudo-differential equations and can be numerically treated using the methods developed for deterministic counterparts of the equations. The scaling of the computational cost in the proposed approach is linear in the number of time steps as opposed to the cubic scaling of the direct GP regression solution. We also show how separable covariance functions lead to a finite-dimensional Kalman filtering and RTS smoothing problem, present analytical and numerical examples, and discuss numerical methods for computing the solutions. ER -
APA
Sarkka, S. & Hartikainen, J.. (2012). Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression. Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 22:993-1001 Available from https://proceedings.mlr.press/v22/sarkka12.html.

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