Multivariate time-series have become abundant in recent years, as many data-acquisition systems record information through multiple sensors simultaneously. In this paper, we assume the variables pertain to some geometry and present an operator-based approach for spatiotemporal analysis. Our approach combines three components that are often considered separately: (i) manifold learning for building operators representing the geometry of the variables, (ii) Riemannian geometry of symmetric positive-definite matrices for multiscale composition of operators corresponding to different time samples, and (iii) spectral analysis of the composite operators for extracting different dynamic modes. We propose a method that is analogous to the classical wavelet analysis, which we term Riemannian multi-resolution analysis (RMRA). We provide some theoretical results on the spectral analysis of the composite operators, and we demonstrate the proposed method on simulations and on real data.

近年来，随着许多数据采集系统通过多个传感器同时记录信息，多元时间序列变得越来越丰富。在本文中，我们假设变量属于某些几何形状，并提出一种基于算子的时空分析方法。我们的方法结合了通常单独考虑的三个组成部分：（i）用于构建表示变量几何形状的算子的流形学习，（ii）对称正定矩阵的黎曼几何，用于对应于不同时间样本的算子的多尺度组合，以及（ iii)光谱分析用于提取不同动态模式的复合算子。我们提出了一种类似于经典小波分析的方法，我们将其称为黎曼多分辨率分析（RMRA）。我们提供了一些关于复合算子谱分析的理论结果，并在模拟和实际数据上演示了所提出的方法。