SAMoSSA: Multivariate Singular Spectrum Analysis with Stochastic Autoregressive Noise

The well-established practice of time series analysis involves estimating deterministic, non-stationary trend and seasonality components followed by learning the residual stochastic, stationary components. Recently, it has been shown that one can learn the deterministic non-stationary components accurately using multivariate Singular Spectrum Analysis (mSSA) in the absence of a correlated stationary component; meanwhile, in the absence of deterministic non-stationary components, the Autoregressive (AR) stationary component can also be learnt readily, e.g. via Ordinary Least Squares (OLS). However, a theoretical underpinning of multi-stage learning algorithms involving both deterministic and stationary components has been absent in the literature despite its pervasiveness. We resolve this open question by establishing desirable theoretical guarantees for a natural two-stage algorithm, where mSSA is first applied to estimate the non-stationary components despite the presence of a correlated stationary AR component, which is subsequently learned from the residual time series. We provide a finite-sample forecasting consistency bound for the proposed algorithm, SAMoSSA, which is data-driven and thus requires minimal parameter tuning. To establish theoretical guarantees, we overcome three hurdles: (i) we characterize the spectra of Page matrices of stable AR processes, thus extending the analysis of mSSA; (ii) we extend the analysis of AR process identification in the presence of arbitrary bounded perturbations; (iii) we characterize the out-of-sample or forecasting error, as opposed to solely considering model identification. Through representative empirical studies, we validate the superior performance of SAMoSSA compared to existing baselines. Notably, SAMoSSA's ability to account for AR noise structure yields improvements ranging from 5% to 37% across various benchmark datasets. Model Deterministic Non-Stationary Component. We adopt the spatio-temporal factor model of [3, 6] described next. The spatio-temporal factor model holds when two assumptions are satisfied: the first assumption concerns the spatial structure, i.e., the structure across the N time series f 1 , . . . , f N ; the second assumption pertains to the "temporal" structure. Before we describe the assumptions, we first define a key time series representation: the Page matrix. Definition 2.1 (Page Matrix). Given a time series f : Z + → R, and an initial time index t 0 > 0, the Page matrix representation over the T entries f (t 0 ), . . . , f (t 0 + T -1) with parameter 1 ≤ L ≤ T is given by the matrix Z(f, L, T, t 0 ) ∈ R L×⌊T /L⌋ with Z(f, L, T, t 0 4 Otherwise, one can apply this algorithm to the two ranges 1, . . . , L × ⌊T /L⌋ and (T mod L) + 1, . . . , T . 5 In the forecasting algorithm, the estimates [ f1(L(m -1) + 1), . . . , f1(L × m -1)] are obtained by applying HSVT on a sub-matrix of Zy 1 which consists of its first L -1 rows. This is done to establish the theoretical results as it helps us avoid dependencies in the noise between y1(Lm) and Fm for m ∈ [T /L]. 6 Note that we assume knowledge of the true parameter p1 (and pn in the multivariate case). 7 Note the subtle difference between x1(t) and x1(t) -precisely, for t > T , x1(t) is an estimate of x1(t) before observing y1(t) (i.e., a forecast), whereas x1(t) is an estimate of x1(t) after observing y1(t).