Sparse principal component analysis for high dimensional multivariate time series

We study sparse principal component analy- sis (sparse PCA) for high dimensional multi- variate vector autoregressive (VAR) time se- ries. 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 asymp- totic framework in which both the length of the sample period T and dimensionality d of the time series can increase (with possi- bly d ≥ T), we provide explicit rates of con- vergence 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 tran- sition matrix plays a pivotal role in deter- mining the final rates of convergence. Im- plications of such a general result is further illustrated using concrete examples. The re- sults of this paper have impacts on different applications, including financial time series, biomedical imaging, and social media, etc.