Robust Estimation for Linear State-Space Models with Stochastic Covariates

The Kalman Filter (KF) is used to estimate and predict state-space models. It frequently encounters uncertain covariates, such as those derived from noisy measurements of physical signals (e.g., sensor readings, weather data, etc.). Our aim is to account for their uncertainties and develop robust KF approaches when incorporating the uncertainties of the covariates into the model; that is, by treating them as stochastic variables with known variances. To enhance the robustness of the KF viewed as a Bayesian method that estimates the posterior distribution of the system state conditioned on past observations, we propose two methods. The linear Bayes estimator yields an explicit, unbiased forecast with minimal variance. The optimal risk estimator is nonlinear and remains robust even when the covariates are highly noisy. This second method is approximated using a sequential Monte Carlo technique.