An efficient particle-based method for maximum likelihood estimation in nonlinear state-space models

Abstract

Data assimilation methods aim at estimating the state of a system by combining observations with a physical model. When sequential data assimilation is considered, the joint distribution of the latent state and the observations is described mathematically using a state-space model, and filtering or smoothing algorithms are used to approximate the conditional distribution of the state given the observations. The most popular algorithms in the data assimilation community are based on the Ensemble Kalman Filter and Smoother (EnKF/EnKS) and its extensions. In this paper we investigate an alternative approach where a Conditional Particle Filter (CPF) is combined with Backward Simulation (BS). This allows to explore efficiently the latent space and simulate quickly relevant trajectories of the state conditionally to the observations. We also tackle the difficult problem of parameter estimation. Indeed, the models generally involve statistical parameters in the physical models and/or in the stochastic models for the errors. These parameters strongly impact the results of the data assimilation algorithm and there is a need for an efficient method to estimate them. Expectation-Maximization (EM) is the most classical algorithm in the statistical literature to estimate the parameters in models with latent variables. It consists in updating sequentially the parameters by maximizing a likelihood function where the state is approximated using a smoothing algorithm. In this paper, we propose an original Stochastic Expectation-Maximization (SEM) algorithm combined to the CPF-BS smoother to estimate the statistical parameters. We show on several toy models that this algorithm provides, with reasonable computational cost, accurate estimations of the statistical parameters and the state in highly nonlinear state-space models, where the application of EM algorithms using EnKS is limited. We also provide a Python source code of the algorithm.