On Multivariate Singular Spectrum Analysis and its Variants
This work addresses the challenge of accurate multivariate time series analysis for fields like economics or signal processing, offering a method that improves error rates and competes with advanced neural techniques, though it is incremental as it builds on mSSA.
The authors tackled the problem of imputing and forecasting multivariate time series by introducing a variant of multivariate singular spectrum analysis (mSSA) under a spatio-temporal factor model, achieving prediction mean-squared-error scaling as 1/√(min(N,T)T), which improves over existing methods like SSA and matrix estimation, and empirically performs competitively with neural-network methods on benchmark datasets.
We introduce and analyze a variant of multivariate singular spectrum analysis (mSSA), a popular time series method to impute and forecast a multivariate time series. Under a spatio-temporal factor model we introduce, given $N$ time series and $T$ observations per time series, we establish prediction mean-squared-error for both imputation and out-of-sample forecasting effectively scale as $1 / \sqrt{\min(N, T )T}$. This is an improvement over: (i) $1 /\sqrt{T}$ error scaling of SSA, the restriction of mSSA to a univariate time series; (ii) $1/\min(N, T)$ error scaling for matrix estimation methods which do not exploit temporal structure in the data. The spatio-temporal model we introduce includes any finite sum and products of: harmonics, polynomials, differentiable periodic functions, and Holder continuous functions. Our out-of-sample forecasting result could be of independent interest for online learning under a spatio-temporal factor model. Empirically, on benchmark datasets, our variant of mSSA performs competitively with state-of-the-art neural-network time series methods (e.g. DeepAR, LSTM) and significantly outperforms classical methods such as vector autoregression (VAR). Finally, we propose extensions of mSSA: (i) a variant to estimate time-varying variance of a time series; (ii) a tensor variant which has better sample complexity for certain regimes of $N$ and $T$.