Spatial statistics often involves Cholesky decomposition of covariance matrices. To ensure scalability to high dimensions, several recent approximations have assumed a sparse Cholesky factor of the precision matrix. We propose a hierarchical Vecchia approximation, whose conditional-independence assumptions imply sparsity in the Cholesky factors of both the precision and the covariance matrix. This remarkable property is crucial for applications to high-dimensional spatiotemporal filtering. We present a fast and simple algorithm to compute our hierarchical Vecchia approximation, and we provide extensions to nonlinear data assimilation with non-Gaussian data based on the Laplace approximation. In several numerical comparisons, including a filtering analysis of satellite data, our methods strongly outperformed alternative approaches.

空间统计通常涉及协方差矩阵的 Cholesky 分解。为了确保高维度的可扩展性，最近的几个近似值假设了精度矩阵的稀疏 Cholesky 因子。我们提出了一种分层 Vecchia 近似，其条件独立假设意味着精度和协方差矩阵的 Cholesky 因子的稀疏性。这种显着的特性对于高维时空滤波的应用至关重要。我们提出了一种快速简单的算法来计算我们的分层 Vecchia 近似，并且我们提供了基于拉普拉斯近似的非高斯数据的非线性数据同化的扩展。在几个数值比较中，包括对卫星数据的过滤分析，我们的方法大大优于替代方法。