General Consistency Results of PCA in High Dimension

• Activity Number: 15
• Type: Topic Contributed
• Date/Time: Sunday, July 29, 2012 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Sungkyu Jung*+ and J. Steve Marron and Jason Fine
• Companies: University of Pittsburgh and The University of North Carolina at Chapel Hill and The University of North Carolina at Chapel Hill

Abstract

Principal component Analysis is a widely used method for dimensionality reduction and visualization of multidimensional data. It becomes common in modern data analytic situation that the dimension $d$ of the observation is much larger than the sample size $n$. This leads to a new domain in asymptotic studies of the estimated principal component analysis, that is, in terms of the limit of $d$. A unified framework for assessing the consistency of principal component estimates in a wide range of asymptotic settings is provided. In particular, our result works for any ratio of dimension and sample size, $d/n \to c$, $c \in [0,\infty]$. We apply this framework to two different statistical situations. When applied to a factor model, we obtain a unified view on the sufficient condition for the consistency of principal component analysis. We propose to use time-varying principal components to model multivariate longitudinal data with an irregular grid. A sufficient condition for the consistency of the estimates is obtained by the proposed tool. Simulation results and real data analysis are included.