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

Since its first descriptions by Pearson (1901) and by Hotelling (1933), principal component analysis (PCA) has been one of the main multivariate analysis techniques for dimension reduction and feature extraction. PCA has become an essential tool for not just independent and identically distributed (iid) multivariate data, but also for serially correlated multivariate time series data in both the time and frequency domains. In the frequency domain, PCA as a sequential method for finding directions of maximum variability appeared in the work of Brillinger (1964) and Goodman (1967). Brillinger (1969) formulated the principal component series through an optimal linear filtering that transmit a p-dimensional signal through a d-dimensional channel and recovers it with minimum loss of information. A foundational discussion of theory and applications of PCA in frequency domain can be found in Brillinger (2001); recent applications of this framework include uncovering non-coherent block structures (Sundararajan, 2021), time-frequency analysis (Ombao et al., 2005) and change point detection (Jiao et al., 2021). PCA for the frequency domain analysis of high-dimensional multivariate time series faces several challenges. The first challenge, which is not unique to frequency domain PCA and is a challenge for PCA in general, is high-dimensionality. When the dimension is fixed, sample eigenvectors, and consequently sample estimates of the principal components, are consistent and asymptotically normally distributed (Anderson, 1958). However, in highdimensional regimes, where the dimension of the random variable grows, sample PCs fail to be consistent.