Extended Ensemble Kalman Filters for High-Dimensional Hierarchical State-Space Models

Matthias Katzfuss, Jonathan R. Stroud and Christopher K. Wikle

Texas A&M University, Georgetown University, and University of Missouri

April 2017

The ensemble Kalman filter (EnKF) is a computational technique for approximate inference on the state vector in spatio-temporal state-space models. It has been successfully used in many real-world nonlinear data-assimilation problems with very high dimensions, such as weather forecasting. However, the EnKF is most appropriate for additive Gaussian state-space models with linear observation equation and without unknown parameters. Here, we consider a broader class of hierarchical state-space models, which includes two additional layers: The parameter layer allows handling of unknown variables that cannot be easily included in the state vector, while the transformation layer can be used to model non-Gaussian observations. For Bayesian inference in such hierarchical state-space models, we propose a general class of extended EnKFs, which approximate inference on the state vector in suitable existing Bayesian inference techniques (e.g., Gibbs sampler or particle filter) using the EnKF or the related ensemble Kalman smoother. Extended EnKFs enable approximate, computationally feasible filtering and smoothing in many high-dimensional, nonlinear, and non-Gaussian spatio-temporal models with unknown parameters. We highlight several interesting examples, including assimilation of heavy-tailed and discrete data, and filtering and smoothing inference on model parameters.

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