On Multivariate Singular Spectrum Analysis

We analyze a variant of multivariate singular spectrum analysis (mSSA), a widely used method used to impute and forecast a multivariate time series. Its restriction to a single time series, known as singular spectrum analysis (SSA), has been analyzed recently. Despite its popularity, theoretical understanding of mSSA is absent. Towards this we introduce a spatio-temporal factor model to analyze mSSA. We establish the in-sample prediction error for both imputation and forecasting scales as $1/\sqrt{NT}$, for $N$ time series with $T$ observations per time series. In contrast, for SSA the error scales as $1/\sqrt{T}$ and for popular matrix factorization based time series methods, the error scales as ${1}/{\min(N, T)}$ -- we note these previous results are established only for imputation. Further, we utilize an online learning framework to analyze the one-step-ahead prediction error of mSSA and establish it has a regret of ${1}/{(\sqrt{N}T^{0.04})}$ with respect to in-sample forecasting error. Empirically, we find mSSA outperforms neural network based methods, LSTM and DeepAR, two of the most widely used and empirically effective methods, though they come with no theoretical guarantees. To establish our results, we make three technical contributions. First, we show that the stacked Page Matrix representation has an approximate low-rank structure for a large class of time series models -- in doing so, we introduce a `calculus' for approximate low-rank models. In particular, we establish that such models are closed under linear combinations as well as multiplications. Second, to establish our regret bounds, we extend the theory of online convex optimization to when the constraints are time-varying, a variant not addressed by the current literature. Third, we extend the prediction error analysis of Principle Component Regression to when the covariate matrix is approximately low-rank.