Sparse Principal Component Analysis for High Dimensional Multivariate Time Series

We study sparse principal component analysis (sparse PCA) for high dimensional multivariate vector autoregressive (VAR) time series. By treating the transition matrix as a nuisance parameter, we show that sparse PCA can be directly applied on analyzing multivariate time series as if the data are i.i.d. generated. Under a double asymptotic framework in which both the length of the sample period T and dimensionality d of the time series can increase (with possibly d≫T), we provide explicit rates of convergence of the angle between the estimated and population leading eigenvectors of the time series covariance matrix. Our results suggest that the spectral norm of the transition matrix plays a pivotal role in determining the final rates of convergence. Implications of such a general result is further illustrated using concrete examples. The results of this paper have impacts on different applications, including financial time series, biomedical imaging, and social media, etc.