On Multivariate Singular Spectrum Analysis and its Variants
We introduce and analyze a simpler, practically useful variant of
multivariate singular spectrum analysis (mSSA), a known time series method to
impute (or de-noise) and forecast a multivariate time series. Towards this, we
introduce a spatio-temporal factor model to analyze mSSA. This model 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. 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 of SSA, which is the
restriction of mSSA to univariate time series; (ii) the ${1}/{\min(N, T)}$
error scaling 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 and our theoretical
findings -- our variant of mSSA empirically performs as well or better compared
to popular neural network based time series methods, LSTM and DeepAR.