Abstract

Data taking value on a Riemannian manifold and observed over a complex spatial domain are becoming more frequent in applications, for example, in environmental sciences and in geoscience. The analysis of these data needs to rely on local models to account for the nonstationarity of the generating random process, the nonlinearity of the manifold, and the complex topology of the domain. In this article, we propose to use a random domain decomposition approach to estimate an ensemble of local models and then to aggregate the predictions of the local models through Fréchet averaging. The algorithm is introduced in complete generality and is valid for data belonging to any smooth Riemannian manifold but it is then described in details for the case of the manifold of positive definite matrices, the hypersphere and the Cholesky manifold. The predictive performances of the method are explored via simulation studies for covariance matrices and correlation matrices, where the Cholesky manifold geometry is used. Finally, the method is illustrated on an environmental dataset observed over the Chesapeake Bay (USA). Supplementary materials for this article are available online.

摘要

在黎曼流形上获得价值并在复杂空间域上观察到的数据在应用中变得越来越频繁，例如在环境科学和地球科学中。这些数据的分析需要依靠局部模型来解释生成随机过程的非平稳性、流形的非线性以及域的复杂拓扑结构。在本文中，我们建议使用随机域分解方法来估计局部模型的集合，然后通过 Fréchet 平均来聚合局部模型的预测。该算法是完全通用的，适用于属于任何光滑黎曼流形的数据，但随后详细描述了正定矩阵流形、超球面和 Cholesky 流形的情况。该方法的预测性能是通过协方差矩阵和相关矩阵的模拟研究来探索的，其中使用了 Cholesky 流形几何。最后，在切萨皮克湾（美国）上观察到的环境数据集上说明了该方法。本文的补充材料可在线获取。