Spatiotemporal Analysis Using Riemannian Composition of Diffusion Operators
This work addresses spatiotemporal analysis for researchers dealing with sensor-based multivariate time-series, but it appears incremental as it builds on existing operator-based and wavelet techniques.
The paper tackles the problem of analyzing multivariate time-series with geometric structure by proposing a Riemannian multi-resolution analysis (RMRA) method that combines manifold learning, Riemannian geometry, and spectral analysis to extract dynamic modes, demonstrating it on simulations and real data.
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.