Defining dynamics via Gaussian processes



Two classic flavours together, Gaussian Processes and dynamical_systems, where the dynamics are modelled by a Gaussian process.

Here we use Gaussian processes to define the dynamics, in particular to learn nonparametric transition, observation or state densities. This is what [Turner, Deisenroth, and Rasmussen (2010);Frigola, Chen, and Rasmussen (2014);Frigola et al. (2013);EleftheriadisIdentification2017] do.

This is distinct from calculating a Gaussian process posterior via a state filter, which is another way you can combine the concepts of dynamics and Gaussian process.

c.f. Variational filters and particle filters.

Possible the same, recurrent Gaussian Processes? 🏗 (Mattos et al. 2016, 2017; Föll et al. 2017).

References

Candela, J.Q., A. Girard, J. Larsen, and C.E. Rasmussen. 2003. “Propagation of Uncertainty in Bayesian Kernel Models - Application to Multiple-Step Ahead Forecasting.” In 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP ’03)., 2:II-701-4. Hong Kong, China: IEEE. https://doi.org/10.1109/ICASSP.2003.1202463.
Eleftheriadis, Stefanos, Tom Nicholson, Marc Deisenroth, and James Hensman. 2017. “Identification of Gaussian Process State Space Models.” In Advances in Neural Information Processing Systems 30, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, 5309–19. Curran Associates, Inc. http://papers.nips.cc/paper/7115-identification-of-gaussian-process-state-space-models.pdf.
Föll, Roman, Bernard Haasdonk, Markus Hanselmann, and Holger Ulmer. 2017. “Deep Recurrent Gaussian Process with Variational Sparse Spectrum Approximation.” arXiv:1711.00799 [stat], November. http://arxiv.org/abs/1711.00799.
Frigola, Roger, Yutian Chen, and Carl Edward Rasmussen. 2014. “Variational Gaussian Process State-Space Models.” In Advances in Neural Information Processing Systems 27, edited by Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, 3680–88. Curran Associates, Inc. http://papers.nips.cc/paper/5375-variational-gaussian-process-state-space-models.pdf.
Frigola, Roger, Fredrik Lindsten, Thomas B Schön, and Carl Edward Rasmussen. 2013. “Bayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC.” In Advances in Neural Information Processing Systems 26, edited by C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger, 3156–64. Curran Associates, Inc. http://papers.nips.cc/paper/5085-bayesian-inference-and-learning-in-gaussian-process-state-space-models-with-particle-mcmc.pdf.
Huber, Marco F. 2014. “Recursive Gaussian Process: On-Line Regression and Learning.” Pattern Recognition Letters 45 (August): 85–91. https://doi.org/10.1016/j.patrec.2014.03.004.
Mattos, César Lincoln C., Zhenwen Dai, Andreas Damianou, Guilherme A. Barreto, and Neil D. Lawrence. 2017. “Deep Recurrent Gaussian Processes for Outlier-Robust System Identification.” Journal of Process Control, DYCOPS-CAB 2016, 60 (December): 82–94. https://doi.org/10.1016/j.jprocont.2017.06.010.
Mattos, César Lincoln C., Zhenwen Dai, Andreas Damianou, Jeremy Forth, Guilherme A. Barreto, and Neil D. Lawrence. 2016. “Recurrent Gaussian Processes.” In Proceedings of ICLR. http://arxiv.org/abs/1511.06644.
Mohammadi, Hossein, Peter Challenor, and Marc Goodfellow. 2021. “Emulating Computationally Expensive Dynamical Simulators Using Gaussian Processes.” arXiv:2104.14987 [stat], April. http://arxiv.org/abs/2104.14987.
Nickisch, Hannes, Arno Solin, and Alexander Grigorevskiy. 2018. “State Space Gaussian Processes with Non-Gaussian Likelihood.” In International Conference on Machine Learning, 3789–98. http://proceedings.mlr.press/v80/nickisch18a.html.
Turner, Ryan, Marc Deisenroth, and Carl Rasmussen. 2010. “State-Space Inference and Learning with Gaussian Processes.” In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, 868–75. http://proceedings.mlr.press/v9/turner10a.html.
Wilkinson, William J., Paul E. Chang, Michael Riis Andersen, and Arno Solin. 2020. “State Space Expectation Propagation: Efficient Inference Schemes for Temporal Gaussian Processes.” In ICML. http://arxiv.org/abs/2007.05994.

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