Nonlinear Filtering: Ensemble-Based Approaches

A wide variety of time series models can be represented within the framework of the state space model. The state space model is composed by a system equation that governs the temporal state evolution and by an observation equation that maps the state to the observation. The problem of state estimation is to evaluate conditional distributions, that is, probability density functions of the state given observations. The Markovian property of the system equation gives us recursive formulas for the evaluation of the conditional distributions. If we assume a linear Gaussian state space model, we can skip a direct evaluation of the conditional distributions and it is sufficient to evaluate the mean vectors and the covariance matrices of the distributions. Under this assumption, the recursive formulas are automatically reduced to simpler ones, dealing with the mean vectors and covariance matrices, and they are well known as the Kalman filter and the smoothing algorithms. If we discard the assumption either of linearity or Gaussian, on the other hand, we have to evaluate the conditional distributions themselves. In the ensemble-based filtering approaches, the conditional distributions are expressed by many of their realizations. Here three of ensemble-based nonlinear filters are reviewed: the ensemble Kalman filter (EnKF), the particle filter (PF), and the Monte-Carlo mixture Kalman filter (MCMKF). In the EnKF, each of the realization develops according to the system equation, then is modified by the observation through the approximated Kalman gain matrix. In the PF, while the realizations develop in the same way as in the EnKF, their update step is resampling with weights proportion to the predictive likelihoods. The MCMKF deals with multiple models simultaneously, and the realizations generated from the different models are resampled with weights of the predictive likelihoods. Application of these filters to geophysical phenomena are also presented.