The sample covariance matrix of a random vector is a good estimate of the true covariance matrix if the sample size is much larger than the length of the vector. In high-dimensional problems, this condition is never met. As a result, in high dimensions the Ensemble Kalman Filter's (EnKF) ensemble does not contain enough information to specify the prior covariance matrix accurately. This necessitates the need for regularization of the analysis (observation update) problem. We propose a regularization technique based on a new spatial model. The model is a constrained version of the general Gaussian process convolution model. The constraints include local stationarity and smoothness of local spectra. We regularize EnKF by postulating that its prior covariances obey the spatial model. Placing a hyperprior distribution on the model parameters and using the likelihood of the prior ensemble data allows for an optimized use of both the ensemble and the hyperprior. The respective estimator is shown to be consistent. Its neural Bayes implementation proved to be both accurate and computationally efficient. In simulation experiments, the new technique led to substantially better EnKF performance than several existing techniques.

翻译：若样本量远大于向量长度，则随机向量的样本协方差矩阵是真实协方差矩阵的良好估计量。在高维问题中，此条件永远无法满足。因此，在高维情况下，集合卡尔曼滤波（EnKF）的集合所包含的信息不足以精确确定先验协方差矩阵，这导致分析（观测更新）问题必须进行正则化处理。本文提出一种基于新型空间模型的正则化技术。该模型是广义高斯过程卷积模型的约束版本，其约束条件包括局部平稳性与局部谱平滑性。我们通过假设EnKF的先验协方差服从该空间模型来实现正则化。通过对模型参数设置超先验分布并利用先验集合数据的似然函数，可实现集合信息与超先验信息的最优化利用。研究证明该估计量具有一致性，其神经贝叶斯实现方案兼具计算精度与效率优势。仿真实验表明，相较于多种现有技术，新方法显著提升了EnKF的性能表现。