Explicit Link Between Periodic Covariance Functions and State Space Models

Arno Solin, Simo Särkkä
Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, PMLR 33:904-912, 2014.

Abstract

This paper shows how periodic covariance functions in Gaussian process regression can be reformulated as state space models, which can be solved with classical Kalman filtering theory. This reduces the problematic cubic complexity of Gaussian process regression in the number of time steps into linear time complexity. The representation is based on expanding periodic covariance functions into a series of stochastic resonators. The explicit representation of the canonical periodic covariance function is written out and the expansion is shown to uniformly converge to the exact covariance function with a known convergence rate. The framework is generalized to quasi-periodic covariance functions by introducing damping terms in the system and applied to two sets of real data. The approach could be easily extended to non-stationary and spatio-temporal variants.

Cite this Paper

BibTeX
@InProceedings{pmlr-v33-solin14, title = {{Explicit Link Between Periodic Covariance Functions and State Space Models}}, author = {Solin, Arno and Särkkä, Simo}, booktitle = {Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics}, pages = {904--912}, year = {2014}, editor = {Kaski, Samuel and Corander, Jukka}, volume = {33}, series = {Proceedings of Machine Learning Research}, address = {Reykjavik, Iceland}, month = {22--25 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v33/solin14.pdf}, url = {https://proceedings.mlr.press/v33/solin14.html}, abstract = {This paper shows how periodic covariance functions in Gaussian process regression can be reformulated as state space models, which can be solved with classical Kalman filtering theory. This reduces the problematic cubic complexity of Gaussian process regression in the number of time steps into linear time complexity. The representation is based on expanding periodic covariance functions into a series of stochastic resonators. The explicit representation of the canonical periodic covariance function is written out and the expansion is shown to uniformly converge to the exact covariance function with a known convergence rate. The framework is generalized to quasi-periodic covariance functions by introducing damping terms in the system and applied to two sets of real data. The approach could be easily extended to non-stationary and spatio-temporal variants.} }
Endnote
%0 Conference Paper %T Explicit Link Between Periodic Covariance Functions and State Space Models %A Arno Solin %A Simo Särkkä %B Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2014 %E Samuel Kaski %E Jukka Corander %F pmlr-v33-solin14 %I PMLR %P 904--912 %U https://proceedings.mlr.press/v33/solin14.html %V 33 %X This paper shows how periodic covariance functions in Gaussian process regression can be reformulated as state space models, which can be solved with classical Kalman filtering theory. This reduces the problematic cubic complexity of Gaussian process regression in the number of time steps into linear time complexity. The representation is based on expanding periodic covariance functions into a series of stochastic resonators. The explicit representation of the canonical periodic covariance function is written out and the expansion is shown to uniformly converge to the exact covariance function with a known convergence rate. The framework is generalized to quasi-periodic covariance functions by introducing damping terms in the system and applied to two sets of real data. The approach could be easily extended to non-stationary and spatio-temporal variants.
RIS
TY - CPAPER TI - Explicit Link Between Periodic Covariance Functions and State Space Models AU - Arno Solin AU - Simo Särkkä BT - Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics DA - 2014/04/02 ED - Samuel Kaski ED - Jukka Corander ID - pmlr-v33-solin14 PB - PMLR DP - Proceedings of Machine Learning Research VL - 33 SP - 904 EP - 912 L1 - http://proceedings.mlr.press/v33/solin14.pdf UR - https://proceedings.mlr.press/v33/solin14.html AB - This paper shows how periodic covariance functions in Gaussian process regression can be reformulated as state space models, which can be solved with classical Kalman filtering theory. This reduces the problematic cubic complexity of Gaussian process regression in the number of time steps into linear time complexity. The representation is based on expanding periodic covariance functions into a series of stochastic resonators. The explicit representation of the canonical periodic covariance function is written out and the expansion is shown to uniformly converge to the exact covariance function with a known convergence rate. The framework is generalized to quasi-periodic covariance functions by introducing damping terms in the system and applied to two sets of real data. The approach could be easily extended to non-stationary and spatio-temporal variants. ER -
APA
Solin, A. & Särkkä, S.. (2014). Explicit Link Between Periodic Covariance Functions and State Space Models. Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 33:904-912 Available from https://proceedings.mlr.press/v33/solin14.html.

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