  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.
