A generalized variance of reconstruction error criterion for determining the optimum number of Principal Components

One of the main difficulties in using Principal Component Analysis (PCA) is the selection of the number of Principal Components (PCs) that constitute the optimal PCA-model. A well-defined Variance of Reconstruction Error (VRE) is proposed to determine this number. It finds the model which corresponds to the best reconstruction of variables by minimization of the reconstructed Squared Prediction Error (SPE) index. This classical VRE criterion behaves well when all variables are correlated because it determines the number of redundancies in data without taking into account the uncorrelated variables. However to represent the system in an optimal way, the number of PCs constituting the PCA-model must be equal to the sum of the number of redundancies and the number of the independent variables in data. It is well known that the reconstruction task depends on the used detection index. Consequently, the best reconstruction is not unique and can be obtained differently. In this paper, we generalize the VRE criterion to any quadratic detection index. We show that the optimum number of PCs can be chosen by minimizing the VRE of the combined index.