On Multivariate Singular Spectrum Analysis and Its Variants

Abstract

We introduce and analyze a simpler and practical variant of multivariate singular spectrum analysis (mSSA), a known time series method to impute and forecast multivariate time series. Towards this, we introduce a spatio-temporal factor model to analyze mSSA. We establish that given $N$ time series and $T$ observations per time series, the in-sample prediction error for both imputation and forecasting under mSSA scales as $1/\sqrt{\min(N, T) T}$. This is an improvement over: (i) the $1/\sqrt{T}$ error scaling one gets for SSA, which is the restriction of mSSA to a univariate time series; (ii) the ${1}/{\min(N, T)}$ error scaling one gets for Temporal Regularized Matrix Factorized (TRMF), a matrix factorization based method for time series prediction. That is, mSSA exploits both the `temporal' and `spatial' structure in a multivariate time series. Our experimental results using various benchmark datasets confirm the characteristics of the spatio-temporal factor model as well as our theoretical findings -- the variant of mSSA we introduce empirically performs as well or better compared to popular neural network based time series methods, LSTM and DeepAR. We discuss various extensions of mSSA we introduce: (i) a variant of mSSA to estimate the time-varying variance of a time series; (ii) a tensor variant of mSSA we call tSSA to further exploit the `temporal' and `spatial' structure in a multivariate time series. The spatio-temporal model considered in our work includes the usual components used to model dynamics in time series analysis such as trends (low order polynomials), seasonality (finite sum of harmonics) and linear time-invariant systems. An important representation result of this work, which might be of interest more broadly, is the `calculus' for such models that we introduce: specifically, instances of the spatio-temporal factor model are closed under addition and multiplication.