关于多元奇异谱分析：张量与矩阵变体
On Multivariate Singular Spectrum Analysis: Tensor and Matrix Variants
AI总结：
本研究提出多元奇异谱分析的矩阵（mSSA）与张量（tSSA）两种变体，证明二者在时空因子模型下的误差缩放与样本复杂度优势，实验验证其性能对标神经网络方法、优于经典方法，还提出首个带有限样本保证的时变方差估计SSA变体。
摘要：
我们提出并分析了奇异谱分析（SSA）向多元场景拓展的两种方法：一种是知名的基于矩阵的方法（mSSA，矩阵多元奇异谱分析）的新变体，另一种是全新的基于张量的方法（tSSA，张量多元奇异谱分析）。\n对于mSSA，在包含N条时间序列、每条序列含T个观测值的时空因子模型下，我们证明其在缺失值插补和样本外预测任务中的预测均方误差的有效缩放规律为$1 / \sqrt{\min(N, T )T}$。这一表现优于两类方法：（i）作为mSSA单变量特例的SSA的$1 /\sqrt{T}$误差缩放；（ii）忽略时间结构的矩阵估计方法的$1/\min(N, T)$误差缩放。mSSA的样本外预测结果对时空因子模型下的在线学习研究也具有独立价值。\n对于tSSA，我们刻画了其插补均方误差特性，并展示在特定的N和T取值区间内，它的样本复杂度优于mSSA。我们证明所提出的时空因子模型可适配广泛的时间序列动态模式，包括谐波序列、多项式序列、可微周期函数以及赫尔德连续函数。我们通过“汉克尔演算”进一步说明这一点，该演算证明该模型所能表示的时间序列集合在分量-wise 加法和乘法运算下是封闭的。\n实验方面，在基准数据集上，mSSA的表现与当前最优的神经网络时间序列方法（如DeepAR、LSTM）相当，且显著优于向量自回归（VAR）等经典方法。与我们的理论结论一致，在特定的N和T取值区间内，tSSA的插补性能优于mSSA。最后，我们提出并分析了SSA的另一种变体，用于估计时间序列的时变方差。据我们所知，这是首个为该任务提供可证明有限样本性能保证的研究成果。
英文摘要：
We introduce and analyze two extensions of Singular Spectrum Analysis (SSA) to the multivariate setting: a new variant of the well-known matrix-based method (mSSA), and a novel tensor-based approach (tSSA). For mSSA, under a spatio-temporal factor model with $N$ time series and $T$ observations per series, we establish that prediction mean-squared-error for both imputation and out-of-sample forecasting effectively scales as $1 / \sqrt{\min(N, T )T}$. This improves over: (i) $1 /\sqrt{T}$ error scaling of SSA, the univariate restriction of mSSA; (ii) $1/\min(N, T)$ error scaling for matrix estimation methods that ignore temporal structure. The out-of-sample forecasting result of mSSA could be of independent interest for online learning under a spatio-temporal factor model. For tSSA, we characterize its imputation mean-squared-error and showcase its better sample complexity, compared to mSSA, for certain regimes of $N$ and $T$. We establish that our spatio-temporal model admits a broad range of time series dynamics including harmonics, polynomials, differentiable periodic functions, and Holder continuous functions. This is further illustrated via the {\em Hankel Calculus}, which establishes that the set of time series the model represents is closed under component-wise addition and multiplication. Empirically, on benchmark datasets, 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). Consistent with our theory, tSSA achieves improved imputation performance over mSSA in certain regimes of $N$ and $T$. Finally, we introduce and analyze an additional variant of SSA to estimate the time-varying variance of a time series. To our knowledge, this is the first result providing provable finite-sample performance guarantees for this task.
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