Selection of the Number of Principal Components: The Variance of the Reconstruction Error Criterion with a Comparison to Other Methods

Abstract

One of the main difficulties in using principal component analysis (PCA) is the selection of the number of principal components (PCs). There exist a plethora of methods to calculate the number of PCs, but most of them use monotonically increasing or decreasing indices. Therefore, the decision to choose the number of principal components is very subjective. In this paper, we present a method based on the variance of the reconstruction error to select the number of PCs. This method demonstrates a minimum over the number of PCs. Conditions are given under which this minimum corresponds to the true number of PCs. Ten other methods available in the signal processing and chemometrics literature are overviewed and compared with the proposed method. Three data sets are used to test the different methods for selecting the number of PCs: two of them are real process data and the other one is a batch reactor simulation.