Learning the Dynamics of Sparsely Observed Interacting Systems

Linus Bleistein, Adeline Fermanian, Anne-Sophie Jannot, Agathe Guilloux
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:2603-2640, 2023.

Abstract

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-bleistein23a, title = {Learning the Dynamics of Sparsely Observed Interacting Systems}, author = {Bleistein, Linus and Fermanian, Adeline and Jannot, Anne-Sophie and Guilloux, Agathe}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {2603--2640}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/bleistein23a/bleistein23a.pdf}, url = {https://proceedings.mlr.press/v202/bleistein23a.html}, abstract = {We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.} }
Endnote
%0 Conference Paper %T Learning the Dynamics of Sparsely Observed Interacting Systems %A Linus Bleistein %A Adeline Fermanian %A Anne-Sophie Jannot %A Agathe Guilloux %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-bleistein23a %I PMLR %P 2603--2640 %U https://proceedings.mlr.press/v202/bleistein23a.html %V 202 %X We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.
APA
Bleistein, L., Fermanian, A., Jannot, A. & Guilloux, A.. (2023). Learning the Dynamics of Sparsely Observed Interacting Systems. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:2603-2640 Available from https://proceedings.mlr.press/v202/bleistein23a.html.

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