The Ensemble Kalman Filter and Related Filters

Abstract

Temporal phenomena are abundant in nature and in man-created activity. Traditionally, mathematicians have modeled these phenomena by differential equations while statisticians have relied on empirically based time series models. It is a natural challenge to combine these two modeling approaches, and hidden Markov models have proven efficient in doing so. The inherent local characteristics of differential equations justifies the Markov assumption while the empirical data is linked to the variables of interest through likelihood functions. Evaluation of the hidden Markov model can be done by Bayesian inversion. R. E. Kalmans’ celebrated paper (Kalman 1960) was based on this line of thought. Under very specific assumptions about linearity and Gaussianity exact analytical solutions can be determined for the Bayesian inversion. Whenever deviations from these assumptions occur however, one has to rely on approximations. This gives room for a large variety of approaches including linearizations and simulation based inference. The current paper focuses on simulation based inference of hidden Markov models and on ensemble Kalman filters in particular. The ensemble Kalman filter was introduced by Evensen in the papers Evensen (1994) and Burgers et al. (1998). The