论文标题
关于多元奇异频谱分析及其变体
On Multivariate Singular Spectrum Analysis and its Variants
论文作者
论文摘要
我们介绍和分析了多元奇异频谱分析（MSSA）的变体，这是一种流行的时间序列方法，用于启用和预测多元时间序列。在我们引入的时空因素模型下，鉴于$ n $时间序列和每次时间序列的$ t $观测值，我们为插补和样本外预测有效地扩展为$ 1 / \ sqrt {\ sqrt {\ min（n，t）t} $。这是一个改进：（i）$ 1 /\ sqrt {t} $ SSA的错误缩放，将MSSA限制到单变量时间序列； （ii）$ 1/\ min（n，t）$对于不利用数据中时间结构的矩阵估计方法的错误缩放。我们介绍的时空模型包括任何有限总和和产品：谐波，多项式，可区分的周期函数和持有人连续函数。在时空因素模型下，我们的样本外预测结果对于在线学习可能具有独立的兴趣。从经验上讲，在基准数据集上，我们的MSSA变体通过最先进的神经网络时间序列方法（例如DeepAR，LSTM）竞争性能，并且明显优于诸如矢量自动化（VAR）的经典方法。最后，我们提出了MSSA的扩展：（i）估计时间序列的时变差异的变体； （ii）一种张量变种，对于$ n $和$ t $的某些制度具有更好的样本复杂性。
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 $T$.
