Ensemble Kalman Methods for High-Dimensional Hierarchical Dynamic Space-Time Models

Abstract

We propose a new class of filtering and smoothing methods for inference in highdimensional, nonlinear, non-Gaussian, spatio-temporal state-space models. The main idea is to combine the ensemble Kalman filter and smoother, developed in the geophysics literature, with state-space algorithms from the statistics literature. Our algorithms address a variety of estimation scenarios, including on-line and off-line state and parameter estimation. We take a Bayesian perspective, for which the goal is to generate samples from the joint posterior distribution of states and parameters. The key benefit of our approach is the use of ensemble Kalman methods for dimension reduction, which allows inference for high-dimensional state vectors. We compare our methods to existing ones, including ensemble Kalman filters, particle filters, and particle MCMC. Using a real data example of cloud motion and data simulated under a number of nonlinear and non-Gaussian scenarios, we show that our approaches outperform these existing methods.

Deep integro-difference equation models for spatio-temporal forecasting

“…This component of variation, as well as measurement error, can be determined offline, as we did in Section 4.4. Alternatively, the ensemble Kalman filter may be placed within a parameter estimation framework (e.g., Katzfuss et al, 2019), wherein the variance components are estimated. Even in a framework where parameter estimation is needed, use of the CNN-IDE may still be beneficial since it precludes estimation of the dynamical parameters, which are sometimes difficult to estimate in both Bayesian and maximum-likelihood settings.…”

“…Parameter estimation with the EnKF can generally be done in an iterative framework (e.g., Gibson and Ninness, 2005;Zammit Mangion et al, 2011;Katzfuss et al, 2019); since such algorithms are well-established, we omit details. An advantage of the CNN-IDE, however, is that parameters corresponding to the CNN can be reasonably estimated offline using complete data generated from a numerical physical model or an analyses (this is the strategy we adopt in Section 4).…”

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“…EnKF is a sequential Monte Carlo algorithm, which updates the state values following a similar principle as KF but with the ensemble of samples [16]. Supposing the ensemble {̂− 1 } =1: is a sample from (̂− 1 , Σ −1 ) , i.e., the filtering distribution of at time − 1 , and ̂− 1 is the estimation of at time − 1, the excursion of EnKF starts with the state forecast:…”

“…Here Σ −1 is the estimation of the covariance matrix of the forecast distribution ̃− 1 . An efficient EnKF algorithm requires a small number of ensembles compared to the state dimension N. However, estimating of the state variables with small ensembles is challenging, because the ensemble may collapse and result in the similar degeneration problem as in PF [16]. A few techniques have been proposed in the literature to alleviate this issue, and we adopted the tapering method [23] in this study.…”

“…In stage one, multiple sets of candidate parameters are generated, and for each set of parameters the ensemble KF (EnKF) is applied to track the state values. The reason for choosing EnKF is that it has the capability to capture the evolution of high dimensional states with a small group of ensembles [15,16]. Then the estimated state values will be used as the initial inputs to generate model predictions in stage two.…”