Noise Covariance Estimation in Adaptive Kalman Filtering via sequential Mini-batch Stochastic Gradient Descent Algorithms

Estimation of unknown noise covariances in a Kalman filter is a problem of significant practical interest in a wide array of applications. Although this problem has a long history, reliable algorithms for their estimation were scant, and necessary and sufficient conditions for identifiability of the covariances were in dispute until recently. Necessary and sufficient conditions for covariance estimation and a batch estimation algorithm. This paper presents stochastic gradient descent (SGD) algorithms for noise covariance estimation in adaptive Kalman filters that are an order of magnitude faster than the batch method for similar or better root mean square error (RMSE) and are applicable to non-stationary systems where the noise covariances can occasionally jump up or down by an unknown magnitude. The computational efficiency of the new algorithm stems from adaptive thresholds for convergence, recursive fading memory estimation of the sample cross-correlations of the innovations, and accelerated SGD algorithms. The comparative evaluation of the proposed method on a number of test cases demonstrates its computational efficiency and accuracy.