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The well-established practice of time series analysis involves estimatingdeterministic, non-stationary trend and seasonality components followed bylearning the residual stochastic, stationary components. Recently, it has beenshown that one can learn the deterministic non-stationary components accuratelyusing multivariate Singular Spectrum Analysis (mSSA) in the absence of acorrelated stationary component; meanwhile, in the absence of deterministicnon-stationary components, the Autoregressive (AR) stationary component canalso be learnt readily, e.g. via Ordinary Least Squares (OLS). However, atheoretical underpinning of multi-stage learning algorithms involving bothdeterministic and stationary components has been absent in the literaturedespite its pervasiveness. We resolve this open question by establishingdesirable theoretical guarantees for a natural two-stage algorithm, where mSSAis first applied to estimate the non-stationary components despite the presenceof a correlated stationary AR component, which is subsequently learned from theresidual time series. We provide a finite-sample forecasting consistency boundfor the proposed algorithm, SAMoSSA, which is data-driven and thus requiresminimal parameter tuning. To establish theoretical guarantees, we overcomethree hurdles: (i) we characterize the spectra of Page matrices of stable ARprocesses, thus extending the analysis of mSSA; (ii) we extend the analysis ofAR process identification in the presence of arbitrary bounded perturbations;(iii) we characterize the out-of-sample or forecasting error, as opposed tosolely considering model identification. Through representative empiricalstudies, we validate the superior performance of SAMoSSA compared to existingbaselines. Notably, SAMoSSA's ability to account for AR noise structure yieldsimprovements ranging from 5% to 37% across various benchmark datasets.

SAMoSSA: Multivariate Singular Spectrum Analysis with Stochastic Autoregressive Noise