Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
166 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Estimation of Dynamic Gaussian Processes (2311.17871v1)

Published 29 Nov 2023 in eess.SY and cs.SY

Abstract: Gaussian processes provide a compact representation for modeling and estimating an unknown function, that can be updated as new measurements of the function are obtained. This paper extends this powerful framework to the case where the unknown function dynamically changes over time. Specifically, we assume that the function evolves according to an integro-difference equation and that the measurements are obtained locally in a spatial sense. In this setting, we will provide the expressions for the conditional mean and covariance of the process given the measurements, which results in a generalized estimation framework, for which we coined the term Dynamic Gaussian Process (DGP) estimation. This new framework generalizes both Gaussian process regression and Kalman filtering. For a broad class of kernels, described by a set of basis functions, fast implementations are provided. We illustrate the results on a numerical example, demonstrating that the method can accurately estimate an evolving continuous function, even in the presence of noisy measurements and disturbances.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (17)
  1. G. Atluri, A. Karpatne, and V. Kumar, “Spatio-Temporal Data Mining,” ACM Computing Surveys, vol. 51, no. 4, pp. 1–41, jul 2019.
  2. R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Journal of Basic Engineering, vol. 82, no. 1, pp. 35–45, mar 1960.
  3. M. P. Deisenroth and H. Ohlsson, “A general perspective on Gaussian filtering and smoothing: Explaining current and deriving new algorithms,” in Proceedings of the 2011 American Control Conference.   IEEE, jun 2011, pp. 1807–1812.
  4. S. Särkkä, J. Hartikainen, L. Svensson, and F. Sandblom, “On the relation between Gaussian process quadratures and sigma-point methods,” Journal of Advances in Information Fusion, vol. 11, no. 1, pp. 31–46, apr 2016.
  5. S. Reece and S. Roberts, “An introduction to Gaussian processes for the Kalman filter expert,” in 2010 13th International Conference on Information Fusion.   IEEE, jul 2010, pp. 1–9.
  6. J. Hartikainen and S. Särkkä, “Kalman filtering and smoothing solutions to temporal Gaussian process regression models,” Proceedings of the 2010 IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2010, pp. 379–384, 2010.
  7. M. Todescato, A. Carron, R. Carli, G. Pillonetto, and L. Schenato, “Efficient spatio-temporal Gaussian regression via Kalman filtering,” Automatica, vol. 118, p. 109032, aug 2020.
  8. J. Quiñonero, Q. Quiñonero-Candela, C. E. Rasmussen, and C. M. De, “A Unifying View of Sparse Approximate Gaussian Process Regression,” The Journal of Machine Learning Research, vol. 6, pp. 1939–1959, dec 2005.
  9. H. Liu, Y.-S. Ong, X. Shen, and J. Cai, “When Gaussian Process Meets Big Data: A Review of Scalable GPs,” IEEE Transactions on Neural Networks and Learning Systems, vol. 31, no. 11, pp. 4405–4423, nov 2020.
  10. M. F. Huber, “Recursive Gaussian process: On-line regression and learning,” Pattern Recognition Letters, vol. 45, no. 1, pp. 85–91, aug 2014.
  11. N. Cressie and G. Johannesson, “Fixed rank kriging for very large spatial data sets,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 70, no. 1, pp. 209–226, jan 2008.
  12. J. Hensman, N. Durrande, and A. Solin, “Variational Fourier Features for Gaussian Processes,” Journal of Machine Learning Research, vol. 18, pp. 1–52, 2018.
  13. T. Lee, “Adaptive Learning Kalman Filter with Gaussian Process,” in 2020 American Control Conference (ACC), vol. 2020-July.   IEEE, jul 2020, pp. 4442–4447.
  14. J. Ko and D. Fox, “GP-BayesFilters: Bayesian filtering using Gaussian process prediction and observation models,” in 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems.   IEEE, sep 2008, pp. 3471–3476.
  15. S. Sarkka, A. Solin, and J. Hartikainen, “Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering,” IEEE Signal Processing Magazine, vol. 30, no. 4, pp. 51–61, jul 2013.
  16. C. K. Wikle and N. Cressie, “A Dimension-Reduced Approach to Space-Time Kalman Filtering,” Biometrika, vol. 86, no. 4, pp. 815–829, 1999.
  17. K. V. Mardia, C. Goodall, E. J. Redfern, and F. J. Alonso, “The Kriged Kalman filter,” Test, vol. 7, no. 2, pp. 217–282, dec 1998.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now