Nonlinear PCA combining principal curves and RBF-networks for process monitoring

The use of principal component analysis (PCA) for process monitoring applications has attracted much attention recently. PCA is the optimal linear transformation with respect to minimizing the mean square prediction error but it only considers second order statistics. If the data have nonlinear dependencies, an important issue is to develop a technique which takes higher order statistics into account and which can eliminate dependencies not removed by PCA. Recognizing the shortcomings of PCA, a nonlinear extension of PCA is developed. The purpose of this paper is to present a nonlinear generalization of PCA (NLPCA) by combining the principal curves and RBF-Networks. The NLPCA model consists of two RBF networks where the nonlinear transformations of the input variables (that characterize the nonlinear principal component analysis) are modelled as a linear sum of radially symmetric kernel functions by using the first network. The nonlinear principal component, which represents the desired output of the first network, are obtained by the principal curves algorithm. The second network tries to perform the inverse transformation by reproducing the original data. The proposed approach is illustrated by a simulation example.