SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 2, Page 681-707, January 2021.
Model error covariances play a central role in the performance of data assimilation methods applied to nonlinear state-space models. However, these covariances are largely unknown in most of the applications. A misspecification of the model error covariance has a strong impact on the computation of the posterior probability density function, leading to unreliable estimations and even to a total failure of the assimilation procedure. In this work, we propose the combination of the expectation maximization (EM) algorithm with an efficient particle filter to estimate the model error covariance using a batch of observations. Based on the EM algorithm principles, the proposed method encompasses two stages: the expectation stage, in which a particle filter is used with the present updated value of the model error covariance as given to find the probability density function that maximizes the likelihood, followed by a maximization stage, in which the expectation under the probability density function found in the expectation step is maximized as a function of the elements of the model error covariance. This novel algorithm here presented combines the EM algorithm with a fixed point algorithm and does not require a particle smoother to approximate the posterior densities. We demonstrate that the new method accurately and efficiently solves the linear model problem. Furthermore, for the chaotic nonlinear Lorenz-96 model the method is stable even for observation error covariance 10 times larger than the estimated model error covariance matrix and also is successful in moderately large dimensional situations where the dimension of the estimated matrix is 40 x 40.

中文翻译：

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使用期望最大化算法和粒子流滤波器的模型误差估计

SIAM / ASA不确定性量化杂志，第9卷，第2期，第681-707页，2021年1月。
模型误差协方差在应用于非线性状态空间模型的数据同化方法的性能中起着核心作用。但是，在大多数应用中这些协方差在很大程度上是未知的。模型误差协方差的错误指定对后验概率密度函数的计算有很大影响，从而导致估计不可靠，甚至导致同化过程完全失败。在这项工作中，我们提出了期望最大化（EM）算法与高效粒子滤波器的组合，以使用一批观测值来估计模型误差协方差。基于EM算法原理，该方法包括两个阶段：期望阶段，其中使用带有给定模型误差协方差的当前更新值的粒子滤波器，以找到使似然性最大化的概率密度函数，然后是最大化阶段，在该阶段中，在期望步骤中找到的概率密度函数下的期望根据模型误差协方差的元素最大化。这里提出的这种新颖算法将EM算法与不动点算法结合在一起，不需要粒子平滑器即可近似后验密度。我们证明了该新方法可以准确有效地解决线性模型问题。此外，