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Koopman Ensembles for Probabilistic Time Series Forecasting

Anthony Frion, Lucas Drumetz, Guillaume Tochon, Mauro Dalla Mura, Albdeldjalil Aïssa El Bey

TL;DR

This work addresses uncertainty in data-driven Koopman forecasting by training ensembles of Koopman autoencoders with a variance-promoting loss to induce diversity among ensemble members. The authors analytically and empirically show that independent training yields overconfident ensembles, while a calibrated diversity term improves spread and accuracy, with a CRPS-inspired variant providing an alternative balance. On Sentinel-2 time series, the approach improves uncertainty calibration for both area extrapolation and cross-area transfer, offering a practical method for probabilistic dynamical forecasting in remote sensing. The results highlight the value of targeted diversity-promoting objectives for reliable uncertainty quantification in nonlinear dynamical models.

Abstract

In the context of an increasing popularity of data-driven models to represent dynamical systems, many machine learning-based implementations of the Koopman operator have recently been proposed. However, the vast majority of those works are limited to deterministic predictions, while the knowledge of uncertainty is critical in fields like meteorology and climatology. In this work, we investigate the training of ensembles of models to produce stochastic outputs. We show through experiments on real remote sensing image time series that ensembles of independently trained models are highly overconfident and that using a training criterion that explicitly encourages the members to produce predictions with high inter-model variances greatly improves the uncertainty quantification of the ensembles.

Koopman Ensembles for Probabilistic Time Series Forecasting

TL;DR

This work addresses uncertainty in data-driven Koopman forecasting by training ensembles of Koopman autoencoders with a variance-promoting loss to induce diversity among ensemble members. The authors analytically and empirically show that independent training yields overconfident ensembles, while a calibrated diversity term improves spread and accuracy, with a CRPS-inspired variant providing an alternative balance. On Sentinel-2 time series, the approach improves uncertainty calibration for both area extrapolation and cross-area transfer, offering a practical method for probabilistic dynamical forecasting in remote sensing. The results highlight the value of targeted diversity-promoting objectives for reliable uncertainty quantification in nonlinear dynamical models.

Abstract

In the context of an increasing popularity of data-driven models to represent dynamical systems, many machine learning-based implementations of the Koopman operator have recently been proposed. However, the vast majority of those works are limited to deterministic predictions, while the knowledge of uncertainty is critical in fields like meteorology and climatology. In this work, we investigate the training of ensembles of models to produce stochastic outputs. We show through experiments on real remote sensing image time series that ensembles of independently trained models are highly overconfident and that using a training criterion that explicitly encourages the members to produce predictions with high inter-model variances greatly improves the uncertainty quantification of the ensembles.
Paper Structure (10 sections, 15 equations, 3 figures)

This paper contains 10 sections, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Data-driven implementations of the Koopman operator
  4. Uncertainty quantification for neural networks
  5. Proposed methods
  6. Introduction of a variance-promoting loss term
  7. Analysis of the choice of $\lambda$
  8. Using a loss term inspired by the CRPS
  9. Experiments
  10. Conclusion

Figures (3)

  • Figure 1: Forecasting from time 0 by two ensembles for the reflectance of the B7 band (in near infrared) for a Fontainebleau pixel. Here, both ensembles are biased, but the ensemble trained with a variance-promoting loss term ($\lambda=0.5$) yields a higher inter-member variance, and hence a better uncertainty estimate, than the ensemble of independently trained models ($\lambda=0$).
  • Figure 2: Spread-skill plots for two different datasets. Top: spread-skill plot of extrapolation on the training Fontainebleau area. Bottom: spread-skill plot of predictions from time 0 on the test Orléans area.
  • Figure 3: CRPS of ensembles of Koopman autoencoders according to the weight $\lambda$ of their variance-promoting loss term during training. Top: extrapolation on training Fontainebleau area. Bottom: transfer to test Orléans area. The represented values of $\lambda$ are $0, 0.1, 0.5, 0.9, 0.99, 1$.