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We propose an approximation to the forward filter backward sampler (FFBS) algorithm for large‐scale spatio‐temporal smoothing. FFBS is commonly used in Bayesian statistics when working with linear Gaussian state‐space models, but it requires inverting covariance matrices which have the size of the latent state vector. The computational burden associated with this operation effectively prohibits its applications in high‐dimensional settings. We propose a scalable spatio‐temporal FFBS approach based on the hierarchical Vecchia approximation of Gaussian processes, which has been previously successfully used in spatial statistics. On simulated and real data, our approach outperformed a low‐rank FFBS approximation. [ABSTRACT FROM AUTHOR]

Spatio-temporal filtering is a common and challenging task in many environmental applications, where the evolution is often nonlinear and the dimension of the spatial state may be very high. We propose a scalable filtering approach based on a hierarchical sparse Cholesky representation of the filtering covariance matrix. At each time point, we compress the sparse Cholesky factor into a dense matrix with a small number of columns. After applying the evolution to each of these columns, we decompress to obtain a hierarchical sparse Cholesky factor of the forecast covariance, which can then be updated based on newly available data. We illustrate the Cholesky evolution via an equivalent representation in terms of spatial basis functions. We also demonstrate the advantage of our method in numerical comparisons, including using a high-dimensional and nonlinear Lorenz model. [ABSTRACT FROM AUTHOR]

The ensemble Kalman filter (EnKF) is a popular technique for data assimilation in high-dimensional nonlinear state-space models. The EnKF represents distributions of interest by an ensemble, which is a form of dimension reduction that enables straightforward forecasting even for complicated and expensive evolution operators. However, the EnKF update step involves estimation of the forecast covariance matrix based on the (often small) ensemble, which requires regularization. Many existing regularization techniques rely on spatial localization, which may ignore long-range dependence. Instead, our proposed approach assumes a sparse Cholesky factor of the inverse covariance matrix, and the nonzero Cholesky entries are further regularized. The resulting method is highly flexible and computationally scalable. In our numerical experiments, our approach was more accurate and less sensitive to misspecification of tuning parameters than tapering-based localization. [ABSTRACT FROM AUTHOR]

We propose a new class of filtering and smoothing methods for inference in high-dimensional, 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 online 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. for this article are available online. [ABSTRACT FROM AUTHOR]

This paper proposes new methodology for sequential state and parameter estimation within the ensemble Kalman filter. The method is fully Bayesian and propagates the joint posterior distribution of states and parameters over time. To implement the method, the authors consider three representations of the marginal posterior distribution of the parameters: a grid-based approach, a Gaussian approximation, and a sequential importance sampling (SIR) approach with kernel resampling. In contrast to existing online parameter estimation algorithms, the new method explicitly accounts for parameter uncertainty and provides a formal way to combine information about the parameters from data at different time periods. The method is illustrated and compared to existing approaches using simulated and real data. [ABSTRACT FROM AUTHOR]

The ensemble Kalman filter (EnKF) is a computational technique for approximate inference in state-space models. In typical applications, the state vectors are large spatial fields that are observed sequentially over time. The EnKF approximates the Kalman filter by representing the distribution of the state with an ensemble of draws from that distribution. The ensemble members are updated based on newly available data by shifting instead of reweighting, which allows the EnKF to avoid the degeneracy problems of reweighting-based algorithms. Taken together, the ensemble representation and shifting-based updates make the EnKF computationally feasible even for extremely high-dimensional state spaces. The EnKF is successfully used in data-assimilation applications with tens of millions of dimensions. While it implicitly assumes a linear Gaussian state-space model, it has also turned out to be remarkably robust to deviations from these assumptions in many applications. Despite its successes, the EnKF is largely unknown in the statistics community. We aim to change that with the present article, and to entice more statisticians to work on this topic. [ABSTRACT FROM PUBLISHER]