Sparse Principal Component Analysis for High Dimensional Vector Autoregressive Models
This work addresses the challenge of dimensionality reduction in time series analysis for fields like econometrics or signal processing, but it is incremental as it extends existing sparse PCA methods to a specific time series context.
The paper tackles the problem of performing sparse principal component analysis (PCA) on high-dimensional vector autoregressive time series, showing that applying sparse PCA directly on such data yields explicit non-asymptotic convergence rates for eigenvector estimation, with the spectral norm of the transition matrix influencing these rates.
We study sparse principal component analysis for high dimensional vector autoregressive time series under a doubly asymptotic framework, which allows the dimension $d$ to scale with the series length $T$. We treat the transition matrix of time series as a nuisance parameter and directly apply sparse principal component analysis on multivariate time series as if the data are independent. We provide explicit non-asymptotic rates of convergence for leading eigenvector estimation and extend this result to principal subspace estimation. Our analysis illustrates that the spectral norm of the transition matrix plays an essential role in determining the final rates. We also characterize sufficient conditions under which sparse principal component analysis attains the optimal parametric rate. Our theoretical results are backed up by thorough numerical studies.