The ensemble Gaussian mixture filter (EnGMF) is a powerful filter for highly non-Gaussian and non-linear models that has practical utility in the case of a small number of samples, and theoretical convergence to full Bayesian inference in the ensemble limit. We aim to increase the utility of the EnGMF by introducing an ensemble-local notion of covariance into the kernel density estimation (KDE) step for the prior distribution. We prove that in the Gaussian case, our new ensemble-localized KDE technique is exactly the same as more traditional KDE techniques. We also show an example of a non-Gaussian distribution that can fail to be approximated by canonical KDE methods, but can be approximated exactly by our new KDE technique. We showcase our new KDE technique on a simple bivariate problem, showing that it has nice qualitative and quantitative properties, and significantly improves the estimate of the prior and posterior distributions for all ensemble sizes tested. We additionally show the utility of the proposed methodology for sequential filtering for the Lorenz '63 equations, achieving a significant reduction in error, and less conservative behavior in the uncertainty estimate with respect to traditional techniques.

翻译：集成高斯混合滤波（EnGMF）是一种适用于强非高斯和非线性模型的有效滤波方法，在小样本情况下具有实用价值，并且在集成极限下理论上收敛于完全贝叶斯推断。我们旨在通过在先验分布的核密度估计（KDE）步骤中引入集成局部化协方差概念，提升EnGMF的实用性。我们证明，在高斯情形下，我们提出的新集成局部化KDE技术与传统KDE方法完全相同。我们还展示了一个非高斯分布的例子，该分布无法被经典KDE方法近似，但可以通过我们的新KDE技术精确近似。我们在一个简单的二元问题上展示了新KDE技术的性能，表明其具有优良的定性和定量特性，并在所有测试的集成规模下显著改进了先验和后验分布的估计。此外，我们展示了所提方法在Lorenz '63方程序贯滤波中的实用性，实现了误差的显著降低，并在不确定性估计方面表现出比传统技术更不保守的行为。