We study a fundamental generalization of principal component analysis (PCA) that looks for a small set of common time series, i.e., principal components (PCs), whose filtered versions can explain most of the variances of multiple observed time series. This problem boils down to PCA in the frequency domain, in principle. But, frequency domain processing suffers from aliasing, and brings inconveniences when handling certain time domain properties like nonstationarity, and sparsity. We propose novel time, and $z$-domain costs for such PCA, and study its properties in detail with setting of either finite or diverging filter lengths. We further discuss its implementations, and possible extensions, and present numerical results for empirical performance study. Convolution is used to extract these PCs, thus the name convolutional PCA.

中文翻译：

---

多时间序列的卷积 PCA

我们研究了主成分分析 (PCA) 的基本概括，它寻找一小组常见的时间序列，即主成分 (PC)，其过滤版本可以解释多个观察到的时间序列的大部分方差。原则上，这个问题归结为频域中的 PCA。但是，频域处理会受到混叠的影响，并且在处理某些时域属性（如非平稳性和稀疏性）时会带来不便。我们为这种 PCA 提出了新的时间和 $z$ 域成本，并通过设置有限或发散滤波器长度来详细研究其属性。我们进一步讨论了它的实现和可能的扩展，并提供了用于实证性能研究的数值结果。卷积用于提取这些 PC，因此称为卷积 PCA。