Recursive estimation of the stochastic model based on the Kalman filter formulation

Based on the batch expectation–maximization (EM) and recursive least-squares algorithms, we develop a new recursive variance components estimation (Recursive-VCE) algorithm that applies a Kalman filter and validates it by a simulated kinematic precise point positioning (PPP) experiment and a PPP test on real-world data. The Recursive-VCE algorithm processes the observations in an epoch-by-epoch or a group-by-group manner. Once new observations are obtained, it updates the estimates of the variance components in a recursive way or on the fly. Therefore, it does not require significant computing resources to store sufficiently large training datasets. The resulting algorithm is simple and able to be easily adapted to determine time-varying behaviours and is shown to converge faster than the batch EM algorithm because the EM algorithm updates the parameters only once after dealing with all the data. Hence, it is a good complement to other batch VCE methods, and its application in real-time data processing is promising.