Nonlinear constrained principal component analysis in the quality control framework

Many problems in industrial quality control involve n measurements on p process variables X-n,X-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 Y-n,Y-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 Y-n,Y-q and X-n,X-p variables gives us information on the Y-n,Y-q without expensive measurements and off-line analysis. Finally, there are few uncorrelated latent variables which contain the information about the Y-n,Y-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.