Model error covariances estimation in the mapping particle filter using an online expectation-maximization algorithm
The estimation of hyperparameters, such as model error and observational covariances, of hidden Markov models in sequential Monte Carlo techniques poses a big challenge. The posterior density associated to hyperparameters collapses when using the standard augmented state techniques. The expectation-maximization algorithm is a suitable method that permits to maximize the complete observation likelihood using observations distributed in time. However, the direct application of the expectation-maximization algorithm is prohibitive even for low dimensional spaces. This is because the complete likelihood calculation requires the application of a particle smoother for the whole sequence in each iteration of the algorithm. Based on the pioneer work of Neal and Hinton (1998), we develop an incremental expectation-maximization algorithm which changes the density for only the current state of the system. Under this approximation the algorithm only requires a sequential Monte Carlo filter and avoids the use of a smoother. The implementation combines the online expectation-maximization algorithm with the recently developed mapping particle filter to estimate model error covariances in the hidden Markov model. The evaluation is conducted in Lorenz-63 and Lorenz-96 experiments. An excellent convergence of the hyperparameters is obtained after 10-50 cycles. The only tuning parameter is the learning rate of the hyperparameters which is evaluated in twin experiments. A highly efficient algorithm is obtained since its application does not require a backward recursion except between the previous and the actual state variables.
Prof. Manuel Pulido University of Reading