Nonlinear Constrained Principal Component Analysis for the Multivariate Process Control

Many problems in industrial quality control involve n measurements on p process variables Xn;p. Generally, we need to know how the quality characteristics of a product behavior as process variables change. Nevertheless, there may be two problems: the linear hypothesis is not always respected and q quality variables Yn;q are not measured frequently because of high costs. B-spline transformation remove nonlinear hypothesis while principal component analysis with linear constraints (CPCA) onto subspace spanned by column X matrix. Linking Yn;q and Xn;p variables gives us information on the Yn;q without expensive measurements and off-line analysis. Finally, there are few uncorrelated latent variables which contain the information about the Yn;q and may be monitored by multivariate control charts. The purpose of this paper is to show how the conjoint employment of different statistical methods, such as B-splines, Constrained PCA and multivariate control charts allow a better control on product or service quality by monitoring directly the process variables. The proposed approach is illustrated by the discussion of a real problem in an industrial process.