On Multivariate Singular Spectrum Analysis: Tensor and Matrix Variants

Tensor and Matrix Methods for Multivariate Time Series, with Provable Guarantees

Real-world time series data—from financial markets to electricity demand—are often multivariate, noisy, and riddled with missing observations. In “On multivariate singular spectrum analysis: Tensor and matrix variants,” Anish Agarwal, Abdullah Alomar, and Devavrat Shah introduce and rigorously analyze two extensions of singular spectrum analysis (SSA) for this challenging setting. Their matrix variant (mSSA) achieves imputation and forecasting error scaling as [Formula: see text] improving over both univariate SSA and standard matrix estimation methods that ignore temporal structure. A novel tensor variant (tSSA) further improves sample complexity in certain regimes, strengthening the link between time series analysis and tensor estimation. These results are made possible by a spatio-temporal factor model that accommodates a rich class of dynamics—harmonics, polynomials, and smooth periodic functions—which the authors extend further by introducing a “Hankel calculus” establishing closure under component-wise addition and multiplication. Empirically, mSSA matches state-of-the-art deep learning methods while significantly outperforming classical approaches such as vector autoregression.