On Multivariate Singular Spectrum Analysis
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We analyze a variant of multivariate singular spectrum analysis (mSSA), a widely used multivariate time series method, which we find to perform competitively with respect to the state-of-art neural network time series methods (LSTM, DeepAR). Its restriction for single time series, singular spectrum analysis (SSA), has been analyzed recently. Despite its popularity, theoretical understanding of mSSA is absent. Towards this, we introduce a natural spatio-temporal factor model to analyze mSSA. We establish the in-sample prediction error for imputation and forecasting under mSSA scales as 1/√(NT), for N time series with T observations per time series. In contrast, for SSA the error scales as 1/√(T) and for matrix factorization based time series methods, the error scales as 1/min(N, T). We utilize an online learning framework to analyze the one-step-ahead prediction error of mSSA and establish it has a regret of 1/(√(N)T^0.04) with respect to in-sample forecasting error. By applying mSSA on the square of the time series observations, we furnish an algorithm to estimate the time-varying variance of a time series and establish it has in-sample imputation / forecasting error scaling as 1/√(NT). To establish our results, we make three technical contributions. First, we establish that the "stacked" Page Matrix time series representation, the core data structure in mSSA, has an approximate low-rank structure for a large class of time series models used in practice under the spatio-temporal factor model. Second, we extend the theory of online convex optimization to address the variant when the constraints are time-varying. Third, we extend the analysis prediction error analysis of Principle Component Regression beyond recent work to when the covariate matrix is approximately low-rank.
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