Online Aleatory Uncertainty Quantification for Probabilistic Time Series Models

Abstract

Engineering problems rely on probabilistic models for decision making tasks. One such approach are the state-space models used for time series forecasting which involve unknown parameters for not only modelling physical phenomena but also for quantifying the model's epistemic and aleatory uncertainties. In practice, estimating some of these parameters can be order of magnitude more computationally demanding than others that prevents existing models from being scaled up for large-scale implementation [1].

For instance, estimating the mean and the epistemic uncertainty for the hidden state variables is computationally cheap as an analytical formulation exist for performing the Bayesian inference. However, the aleatory uncertainty is quantified by variance parameters in the process and observation error covariance matrices, which need to be known accurately for an exact state estimation. In practice, obtaining optimal estimates for these unknown parameters is typically the most computationally demanding task in the state estimation procedure. Even though the observation error matrix can be, in many situations, considered to be known from the measuring instrument specifications, it remains a challenge to develop a computationally efficient method which is able to perform closed-form online estimation of the process error covariance matrix for multiple time series [2].

This article presents an analytical Bayesian inference method called the approximate Gaussian variance inference (AGVI) in the context of time series modeling which will enable performing closed-form online estimation of the multivariate process error's variance parameters. Two applied examples are included where the first case study compares the performance of AGVI with the existing adaptive Kalman filtering (AKF) [3] approaches and the second case study show its application on real datasets from a concrete gravity dam. The proposed method can exceed the performance of existing approaches in terms of its predictive capacity while being up to orders of magnitude faster.

References

[1] S. Sarkka and A. Nummenmaa, ﾓRecursive noise adaptive Kalman filtering by variational Bayesian approximations,ﾔ IEEE Transactions on Automatic control, vol. 54, no. 3, pp. 596ﾖ600, 2009.

[2] Y. Huang, Y. Zhang, Z. Wu, N. Li, and J. Chambers, ﾓA novel adaptive Kalman filter with inaccurate process and measurement noise covariance matrices,ﾔ IEEE Transactions on Automatic Control, vol. 63, no. 2, pp. 594ﾖ601, 2017.

[3] R. Mehra, ﾓOn the identification of variances and adaptive Kalman filtering,ﾔ IEEE Transactions on automatic control, vol. 15, no. 2, pp. 175ﾖ184, 1970.