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. 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 demonstrate that if the true filtering distribution is close to Gaussian in the joint space of state and data, in the weighted total variation metric, then the true-filter is well approximated by the ensemble Kalman filter, in the same metric. Finally, we 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.
翻译:集合卡尔曼滤波器在应用中被广泛使用,因为对于高维过滤问题,具有不会出现重量聚集等鲁棒性,例如粒子滤波器不能共享。然而,在高斯设置之外没有量化其作为真实过滤分布的准确度的理论。 为了解决这个问题,我们提供了除了高斯设置之外的均值场集合卡尔曼滤波器准确性的第一次分析。 我们为此滤波器重写了更新方程,也重写了真实的过滤分布方程。 我们用概率测度上的映射来定义这些方程,然后引入加权总变差度量来计算两个滤波器之间的距离。 我们证明了定义两个滤波器的映射在这种度量下的各种稳定性估计。 我们使用这些稳定估计证明,如果在状态和数据的联合空间中,真实的过滤分布在加权总变差距离度量下接近高斯分布,则在相同度量下,真实过滤器就可以很好地近似于集合卡尔曼滤波器。 最后,我们将这些结果推广到高斯投影滤波器,并将其视为无扰动卡尔曼滤波器的一种均值场描述。