This paper develops efficient ensemble Kalman filter (EnKF) implementations based on shrinkage covariance estimation. The forecast ensemble members at each step are used to estimate the background error covariance matrix via the Rao-Blackwell Ledoit and Wolf estimator, which has been developed specifically developed to approximate high-dimensional covariance matrices using a small number of samples. Additional samples are taken from the normal distribution described by the background ensemble mean and the estimated background covariance matrix in order to increase the size of the ensemble and reduce the sampling error of the filter. This increase in the size of the ensemble is obtained without running the forward model. After the assimilation step, the additional samples are discarded and only the initial members are propagated. Two implementations are considered. In the EnKF Full-Space (EnKF-FS) approach the assimilation process is performed in the model space, while the EnKF Reduce-Space (EnKF-RS) formulation performs the analysis in the subspace spanned by the ensemble members. Numerical experiments carried out with a quasi-geostrophic model show that the proposed implementations outperform current methods such as the traditional EnKF formulation, square root filters, and inflation-free EnKF implementations. The proposed implementations provide good results with small ensemble sizes ($\sim 10$) and small percentages of observed components from the vector state. These results are similar (and in some cases better) to traditional methods using large ensemble sizes ($\sim 80$) and large percentages of observed components. The computational times of the new implementations remain reasonably low.
Estimation of the covariance matrix of asset returns from high frequency data is complicated by asynchronous returns, market mi- crostructure noise and jumps. One technique for addressing both asynchronous returns and market microstructure is the Kalman-EM (KEM) algorithm. However the KEM approach assumes log-normal prices and does not address jumps in the return process which can corrupt estimation of the covariance matrix. In this paper we extend the KEM algorithm to price models that include jumps. We propose two sparse Kalman filtering approaches to this problem. In the first approach we d…
This paper develops an efficient implementation of the ensemble Kalman filter based on a modified Cholesky decomposition for inverse covariance matrix estimation. This implementation is named EnKF-MC. Background errors corresponding to distant model components with respect to some radius of influence are assumed to be conditionally independent. This allows to obtain sparse estimators of the inverse background error covariance matrix. The computational effort of the proposed method is discussed and different formulations based on various matrix identities are provided. Furthermore, an asymptoti…
Ensemble filters implement sequential Bayesian estimation by representing the probability distribution by an ensemble mean and covariance. Unbiased square root ensemble filters use deterministic algorithms to produce an analysis (posterior) ensemble with prescribed mean and covariance, consistent with the Kalman update. This includes several filters used in practice, such as the Ensemble Transform Kalman Filter (ETKF), the Ensemble Adjustment Kalman Filter (EAKF), and a filter by Whitaker and Hamill. We show that at every time index, as the number of ensemble members increases to infinity, the…
Various methods have been proposed for the nonlinear filtering problem, including the extended Kalman filter (EKF), iterated extended Kalman filter (IEKF), unscented Kalman filter (UKF) and iterated unscented Kalman filter (IUKF). In this paper two new nonlinear Kalman filters are proposed and investigated, namely the observation-centered extended Kalman filter (OCEKF) and observation-centered unscented Kalman filter (OCUKF). Although the UKF and EKF are common default choices for nonlinear filtering, there are situations where they are bad choices. Examples are given where the EKF and UKF per…
Contemporary data assimilation often involves millions of prediction variables. The classical Kalman filter is no longer computationally feasible in such a high dimensional context. This problem can often be resolved by exploiting the underlying multiscale structure, applying the full Kalman filtering procedures only to the large scale vari- ables, and estimating the small scale variables with proper statistical strategies, including multiplicative inflation, representation model error in the observations, and crude localization. The resulting two-scale reduced filters can have close to optima…
This paper uses a probabilistic approach to analyze the converge of an ensemble Kalman filter solution to an exact Kalman filter solution in the simplest possible setting, the scalar case, as it allows us to build upon a rich literature of scalar probability distributions and non-elementary functions. To this end we introduce the bare-bones Scalar Pedagogical Ensemble Kalman Filter (SPEnKF). We show that in the asymptotic case of ensemble size, the expected value of both the analysis mean and variance estimate of the SPEnKF converges to that of the true Kalman filter, and that the variances of…
Kalman filter is a key tool for time-series forecasting and analysis. We show that the dependence of a prediction of Kalman filter on the past is decaying exponentially, whenever the process noise is non-degenerate. Therefore, Kalman filter may be approximated by regression on a few recent observations. Surprisingly, we also show that having some process noise is essential for the exponential decay. With no process noise, it may happen that the forecast depends on all of the past uniformly, which makes forecasting more difficult. Based on this insight, we devise an on-line algorithm for improp…
