Hyperparameter inference in nonlinear Gaussian state-space model.

TANDEO, PIERRE; LUCINI, MARÍA MAGDALENA

Valparaiso

Congreso; XII Latin American Congress of Probability and Mathematical Statistics; 2012

The state-space model is an hidden Markov model where the state is in a continuous space. The most classical state-space formulation is linear with additive Gaussian errors, also known as linear dynamic system in the statistics community. The optimal solution of this sytem is given by the classical Kalman Filter (KF). In this presentation, we focus on the nonlinear Gaussian state-space model, where the state and/or observation operators are nonlinear functions. In this case, the KF is not applicable and we try to mimic it. One possibility is to use linear approximations of the operators such as the Extended Kalman Filter (EKF). Another way is to use particle filters (based on Monte Carlo simulations). Among them, we find the Ensemble Kalman Filter (EnKF) and the Sequential Importance Resampling (SIR). In this presentation, we focus on the EnKF that avoid linearizations of operators and that requires small number of particules compared to the SIR filter (cf. \cite{evensen2009data}, p.260). In addition to the state inference, we present the estimation of the hyperparameters of the system. They correspond to the errors of the initial state, the state evolution and the observation measurement. They are all supposed to be Gaussianly distributed with zero mean. The tunning of the 3 covariance matrices controls the quality of the state esimates and constitutes an important framework in the data assimilation community. Here we work on the Maximum Likelihood Estimates (MLE) of those covariances. We use the classical Expectation Maximization (EM) algorithm well suited for models with incomplete data such as state-space formulations. More precisely, we use the Monte Carlo Ensemble Kalman Smoother (EnKS) to approximate the expected likelihood function on the E-step. We present results of the hyperparameter estimation on classical nonlinear models (cf. \cite{kitagawa1996monte} and examples therein) and the chaotic Lorenz model. The results indicate that the method is well suited for high nonlinear problems with low state dimensions. Finally, we talk about future work on hyperparameter inference in data assimilation where the dimension of the state is higher.