SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2549-2587, December 2024.
Abstract. The ensemble Kalman filter is widely used in applications because, for high-dimensional filtering problems, it has a robustness that is not shared, for example, by the particle filter; in particular, it does not suffer from weight collapse. However, there is no theory which quantifies its accuracy as an approximation of the true filtering distribution, except in the Gaussian setting. To address this issue, we provide the first analysis of the accuracy of the ensemble Kalman filter beyond the Gaussian setting. We prove two types of results: The first type comprises a stability estimate controlling the error made by the ensemble Kalman filter in terms of the difference between the true filtering distribution and a nearby Gaussian, and the second type uses this stability result to show that, in a neighborhood of Gaussian problems, the ensemble Kalman filter makes a small error in comparison with the true filtering distribution. Our analysis is developed for the mean-field ensemble Kalman filter. We rewrite the update equations for this filter and for the true filtering distribution in terms of maps on probability measures. We introduce a weighted total variation metric to estimate the distance between the two filters, and we prove various stability estimates for the maps defining the evolution of the two filters in this metric. Using these stability estimates, we prove results of the first and second types in the weighted total variation metric. We also provide a generalization of these results to the Gaussian projected filter, which can be viewed as a mean-field description of the unscented Kalman filter.

中文翻译：

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均值场系综卡尔曼滤波器：近高斯设置


SIAM 数值分析杂志，第 62 卷，第 6 期，第 2549-2587 页，2024 年 12 月。

抽象。集成卡尔曼滤波器在应用中被广泛使用，因为对于高维滤波问题，它具有粒子滤波器所不具备的稳健性;特别是，它不会遭受重量崩溃。但是，除了高斯设置之外，没有理论可以将其准确性量化为真实滤波分布的近似值。为了解决这个问题，我们首次分析了超出高斯设置范围的集合卡尔曼滤波的精度。我们证明了两种类型的结果：第一种类型包括一个稳定性估计，控制集成卡尔曼滤波器在真实滤波分布和附近高斯分布之间的差异方面所做的误差，第二种类型使用这个稳定性结果来表明，在高斯问题的邻域中，集成卡尔曼滤波器与真实滤波分布相比误差很小。我们的分析是针对均值场集成卡尔曼滤波开发的。我们重写了此过滤器的更新方程，以及根据概率测度上的映射的真实过滤分布的更新方程。我们引入了一个加权总变分指标来估计两个过滤器之间的距离，并证明了定义该指标中两个过滤器演变的映射的各种稳定性估计值。使用这些稳定性估计值，我们证明了加权总变异度量中第一种和第二种类型的结果。我们还将这些结果推广到高斯投影滤波器，可以将其视为无迹卡尔曼滤波器的平均场描述。