On Multivariate Singular Spectrum Analysis: Tensor and Matrix Variants

We introduce and analyze two extensions of singular spectrum analysis (SSA) to the multivariate setting: a new variant of the well-known matrix-based method (mSSA) and a novel tensor-based approach (tSSA). Under a spatio-temporal factor model, we establish prediction-error guarantees for mSSA for both imputation and out-of-sample forecasting. By exploiting both spatial and temporal structure, mSSA achieves better rates than univariate SSA and standard matrix estimation methods. The out-of-sample forecasting result of mSSA could be of independent interest for online learning under a spatio-temporal factor model. For tSSA, we characterize its imputation mean squared error and showcase its better sample complexity, compared with mSSA, for certain regimes of N and T. We establish that our spatio-temporal model admits a broad range of time series dynamics including harmonics, polynomials, differentiable periodic functions, and Hölder continuous functions. This is further illustrated via the Hankel calculus, which establishes that the set of time series the model represents is closed under component-wise addition and multiplication. Empirically, on benchmark data sets, mSSA performs competitively with state-of-the-art neural-network time series methods (e.g., DeepAR, long short-term memory) and significantly outperforms classical methods such as vector autoregression (VAR). Consistent with our theory, tSSA achieves improved imputation performance over mSSA in certain regimes of N and T. Finally, we introduce and analyze an additional variant of SSA to estimate the time-varying variance of a time series. To our knowledge, this is the first result providing provable finite-sample performance guarantees for estimating the time-varying variance of a time series.

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2023.0060.