Abstract. The ensemble Kalman filter has become a popular data assimilation technique in the geosciences.However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman-Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman-Bucy filter is consistent with the classic Kalman-Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system.

This paper is concerned with the filtering problem in continuous-time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter which provides an exact solution for the linear Gaussian problem, (ii) the ensemble Kalman-Bucy filter (EnKBF) which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems, and (iii) the feedback particle filter (FPF) which represents an extension of the EnKBF and furthermore provides for an consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of non-uniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.

Atmospheric circulation is often clustered in so‐called circulation regimes, which are persistent and recurrent patterns. For the Euro‐Atlantic sector in winter, most studies identify four regimes: the Atlantic Ridge, Scandinavian Blocking and the two phases of the North Atlantic Oscillation. These results are obtained by applying k‐means clustering to the first several empirical orthogonal functions (EOFs) of geopotential height data. Studying the observed circulation in reanalysis data, it is found that when the full field data are used for the k‐means cluster analysis instead of the EOFs, the optimal number of clusters is no longer four but six. The two extra regimes that are found are the opposites of the Atlantic Ridge and Scandinavian Blocking, meaning they have a low‐pressure area roughly where the original regimes have a high‐pressure area. This introduces an appealing symmetry in the clustering result. Incorporating a weak persistence constraint in the clustering procedure is found to lead to a longer duration of regimes, extending beyond the synoptic time‐scale, without changing their occurrence rates. This is in contrast to the commonly used application of a time‐filter to the data before the clustering is executed, which, while increasing the persistence, changes the occurrence rates of the regimes. We conclude that applying a persistence constraint within the clustering procedure is a better way of stabilizing the clustering results than low‐pass filtering the data.

An essential component of therapeutic drug/biomarker monitoring (TDM) is to combine patient data with prior knowledge for model‐based predictions of therapy outcomes. Current Bayesian forecasting tools typically rely only on the most probable model parameters (maximum a posteriori (MAP) estimate). This MAP‐based approach, however, does neither necessarily predict the most probable outcome nor does it quantify the risks of treatment inefficacy or toxicity. Bayesian data assimilation (DA) methods overcome these limitations by providing a comprehensive uncertainty quantification. We compare DA methods with MAP‐based approaches and show how probabilistic statements about key markers related to chemotherapy‐induced neutropenia can be leveraged for more informative decision support in individualized chemotherapy. Sequential Bayesian DA proved to be most computationally efficient for handling interoccasion variability and integrating TDM data. For new digital monitoring devices enabling more frequent data collection, these features will be of critical importance to improve patient care decisions in various therapeutic areas.