粒子卡尔曼滤波：一种非线性贝叶斯滤波器框架
Particle Kalman Filtering: A Nonlinear Bayesian Framework for Ensemble Kalman Filters
摘要：
本文研究了一种基于状态概率分布函数的高斯混合表示的最优非线性贝叶斯滤波器的逼近方案。得到的滤波器类似于粒子滤波器，但与粒子滤波器的不同之处在于，粒子滤波器中的标准权重类型校正由具有高斯混合中的相关协方差矩阵的卡尔曼类型校正来补充。我们表明，该滤波器是卡尔曼滤波器和粒子滤波器之间的算法，因此被称为粒子卡尔曼滤波器（PKF）。在PKF中，非线性滤波问题的解决方案被表示为并行操作的“卡尔曼滤波器集合”的加权平均。然而，运行卡尔曼滤波器的集合对于现实的大气和海洋数据同化问题在计算上是禁止的。因此，我们考虑通过集合卡尔曼滤波器（EnKF）的“集合”来构建PKF，并将其称为粒子EnKF（PEnKF）。我们表明，不同类型的EnKF可以被认为是特殊情况下的PEnKF。与粒子滤波器中的情况类似，我们还在PEnKF中引入了重采样步骤，以降低权重崩溃的风险并提高滤波器的性能。给出了强非线性Lorenz-96模型的数值实验结果并进行了讨论。
英文摘要：
This paper investigates an approximation scheme of the optimal nonlinear Bayesian filter based on the Gaussian mixture representation of the state probability distribution function. The resulting filter is similar to the particle filter, but is different from it in that, the standard weight-type correction in the particle filter is complemented by the Kalman-type correction with the associated covariance matrices in the Gaussian mixture. We show that this filter is an algorithm in between the Kalman filter and the particle filter, and therefore is referred to as the particle Kalman filter (PKF). In the PKF, the solution of a nonlinear filtering problem is expressed as the weighted average of an "ensemble of Kalman filters" operating in parallel. Running an ensemble of Kalman filters is, however, computationally prohibitive for realistic atmospheric and oceanic data assimilation problems. For this reason, we consider the construction of the PKF through an "ensemble" of ensemble Kalman filters (EnKFs) instead, and call the implementation the particle EnKF (PEnKF). We show that different types of the EnKFs can be considered as special cases of the PEnKF. Similar to the situation in the particle filter, we also introduce a re-sampling step to the PEnKF in order to reduce the risk of weights collapse and improve the performance of the filter. Numerical experiments with the strongly nonlinear Lorenz-96 model are presented and discussed.
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