Selection of the number of principal components based on the fault reconstruction approach applied to a new combined index

The main difficulties in using Principal Components Analysis (PCA) approach is the selection of the optimum number of Principal Components (PCs). A well-defined Variance of Reconstruction Error (VRE) criterion is proposed in order to find the optimum PCA-model giving a best reconstruction of the correlated variables. Given that this unique existing VRE criterion depends implicitly on the Squared Prediction Error (SPE) index, it determines the number of redundancies in data without considering the uncorrelated variables. As a result, this criterion behaves well for the PCA modelling task if all variables are correlated. In this paper, we propose a new combined index that depends on two parameters. By minimizing its bidimensional VRE criterion along these parameters, we can determine firstly the optimum number of PCs and secondly the number of redundancies in data. We show also that this new criterion gives effective results with usual values of confidence level.