AGVI

This repository provides illustrative examples for application of the approximate Gaussian variance inference (AGVI) method. AGVI is an online Bayesian inference method that allows inferring the process error variance terms analytically through closed-form equations. The method provides unbiased and statistically consistent estimates for both the mean as well as the uncertainty associated with the variance terms.

Examples

Two Case studies are provided in this repository. Case study $1$ focuses on the online estimation of error variance for a linear time-varying (LTV) model. The study includes statistical consistency tests to showcase the filter's optimality, t-tests to prove that the estimates are unbiased, empirical validation of uncertainty associated with error variance estimates, and analysis of the impact of $\frac{Q}{R}$ ratio on the posterior mean estimate $\mu_{\mathtt{T}|\mathtt{T}}$ of the error variance term. The AGVI method is also compared to two adaptive Kalman filtering (AKF) approaches: the sliding window variational adaptive Kalman filter} (SWVAKF) and the measurement difference method (MDM). These AKF methods fall under different categories, with MDM being a correlation method and SWVAKF being a Bayesian method. Case study $2$ presents a simulated multivariate random walk model with a full process error covariance matrix $\mathbf{Q}$ and a comparison of the AGVI method to the AKF methods.

Contributions

• Bhargob Deka (code development and methodology)
• James-A. Goulet (methodology)

License

This project is licensed under the MIT License.

Acknowledgement

This project was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC), Hydro-Qu'ebec (HQ), Hydro-Qu'ebec's Research Institute (IREQ), Institute For Data Valorization (IVADO). The authors thank Luong Ha Nguyen, Post doc, Department of Civil, Geologic and Mining Engineering, Polytechnique Montr'eal for his help in the project.

References

• Deka, B., Ha Nguyen, L., Amiri, S., & Goulet, J. A. (2022). The Gaussian multiplicative approximation for state‐space models. Structural Control and Health Monitoring, 29(3), e2904.
• Goulet, J. A., Nguyen, L. H., & Amiri, S. (2021). Tractable approximate Gaussian inference for Bayesian neural networks. The Journal of Machine Learning Research, 22(1), 11374-11396.
• Huang, Y., Zhu, F., Jia, G., & Zhang, Y. (2020). A slide window variational adaptive Kalman filter. IEEE Transactions on Circuits and Systems II: Express Briefs, 67(12), 3552-3556.
• Kost, O., Duník, J., & Straka, O. (2022). Measurement Difference Method: A Universal Tool for Noise Identification. IEEE Transactions on Automatic Control, 68(3), 1792-1799.