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 $T$.
翻译:我们引入并分析多变量单谱分析的变式。 这是一个用来估算和预测多变量时间序列的流行时间序列方法。 在一种时序要素模型下,我们引入了一种多变量单数时间序列分析。 在基于美元的时间序列和每时间序列观测$T美元的情况下,我们为估算和标外预测有效规模设定了预测平均半偏差值值值值值值值值值值为1美元/\sqrt=min(N,T)T}美元。这是一个进步:(一) 1美元/\sqrt{T}美元,用于估算和预测一个多变量时间序列。在SSA的深度时间序列中,将 mSS的最小值限值值限制为1美元/\min(N,T)美元用于矩阵估算方法,但不会在数据中利用时间结构。我们引入的Spotio-时序模型包括任何限定值和产品:调和多数值、不同周期函数,以及Solder 变量的功能。我们的Sad- sambreal laveal laveal-s roal roal-stal-motional-motional-motional rodustrate ma-de romodustry rodustry-motional-motional-motional-motional-moduction romotional-motional-mocal-mocal-motional-mod-s-motion-motion-motion-motion-motion-motion-motion-motion-motion-motional-motion-motion-motion-motion-mocs-motion-mocs-motion-motion-motion-motion-motion-mocs-mocal-mocal-mod-mocal-mod-mod-mods-mods-mods-mocal-mocs-mods-modal-mocal-mod-mod-mod-mod-mod-mod-mod-mod-mod-mods-mod-mod-mod-mod-mod-mod-mod-mod-mod-mo