SAMoSSA: Multivariate Singular Spectrum Analysis with Stochastic Autoregressive Noise

TL;DR

SAMoSSA introduces a theoretically grounded two-stage approach combining multivariate Singular Spectrum Analysis and autoregressive modeling to improve time series forecasting, especially in the presence of AR noise.

Contribution

This paper provides the first theoretical guarantees for a two-stage method combining mSSA and AR modeling, with finite-sample forecasting bounds.

Findings

SAMoSSA outperforms existing methods by 5-37% on benchmark datasets.

Theoretical analysis extends mSSA to AR processes with bounded perturbations.

Empirical results validate the effectiveness of SAMoSSA in real-world scenarios.

Abstract

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.