On the Number of Principal Components

Abstract

This talk presents a confidence interval for the number of important principal components in principal component analysis. An important principal component is defined as a principal component whose variance is close to the variance of the largest principal component. More specifically, a principal component with variance is called important if is sufficiently close to 1 where is the variance of the first principal component. A distance measure for closeness will be defined under the framework of ranking and selection theory. A confidence interval for the number of important principal components will be proposed using a stepwise selection procedure. The proposed interval, which is asymptotic in nature, includes the true important components with a specified confidence. Numerical examples are given to illustrate our procedure. Current studies on confident interval that is based on plug-in estimate of the proportion of the total variance and the applications of principal components analysis to functional magnetic resonance imaging will be briefly introduced.