Localized Sparse Principal Component Analysis of Multivariate Time Series in the Frequency Domain

Principal component analysis has been a main tool in multivariate analysis for estimating a low dimensional linear subspace that explains most of the variability in the data. However, in high-dimensional regimes, naive estimates of the principal loadings are not consistent and difficult to interpret. In the context of time series, principal component analysis of spectral density matrices can provide valuable, parsimonious information about the behavior of the underlying process, particularly if the principal components are interpretable in that they are sparse in coordinates and localized in frequency bands. In this paper, we introduce a formulation and consistent estimation procedure for interpretable principal component analysis for high-dimensional time series in the frequency domain. An efficient frequency-sequential algorithm is developed to compute sparse-localized estimates of the low-dimensional principal subspaces of the signal process. The method is motivated by and used to understand neurological mechanisms from high-density resting-state EEG in a study of first episode psychosis.主成分分析（Principal Component Analysis）长期以来都是多元分析领域的主流工具，用于估计可解释数据中绝大多数变异的低维线性子空间。然而在高维场景下，对主载荷的朴素估计并不具备一致性，且难以实现有效解释。在时间序列分析的语境中，谱密度矩阵的主成分分析能够为潜在时序过程的行为提供极具价值且简洁凝练的信息，尤其当主成分具备可解释性时——即它们在坐标维度上呈现稀疏性，且在频段上具有局域化特性。本文针对频域下的高维时间序列，提出了可解释主成分分析的建模框架与一致性估计流程。本文开发了一种高效的频域序贯算法，用于计算信号过程低维主子空间的稀疏局域化估计结果。本方法基于一项针对首发精神病患者的研究提出，并被用于通过高密度静息态脑电图（electroencephalogram, EEG）解析神经机制。