Advertisement

Nonlinear Constrained Principal Component Analysis in the Quality Control Framework

  • Michele Gallo
  • Luigi D’Ambra
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

Abstract

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

Keywords

Principal Component Analysis Process Variable Quality Characteristic Multivariate Linear Model Robust Principal Component Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ABECASSIS, J., DURAND, J.F., MEOT, J.M., and VASSEUR, P. (1992): Etude non linéaire de l’influence des conditions de séchage sur la qualité des pâtes alimentaires par l’analysis en composantes principales par rapport à des variables instrumentales. Agro-Industrie et Methodes Statistiques, 3èmes Journees Europeennes.Google Scholar
  2. D’AMBRA, L., and LAURO, N. (1982): Analisi in componenti principali in rapporto ad un sottospazio di riferimento. Italian Journal of Applied Statistics, 1.Google Scholar
  3. DAUDIN, J.J., DUBY, C., and TRECOURT, P. (1988): Stability of stability of principal com-ponent analysis studied by bootstrap method. Statistics , 19, 2.CrossRefMathSciNetGoogle Scholar
  4. DE BOOR, C. (1978): A practical guide to splines. Springer, N.Y.zbMATHGoogle Scholar
  5. DURAND, J.F. (1993): Generalized principal component analysis with respect to instrumental variables via univariate spline trasformations. Computational Statistics Data Analysis, 16,423-440.zbMATHCrossRefMathSciNetGoogle Scholar
  6. EUBANK, R.L. (1988): Smoothing splines and non parametric regression. Markel Dekker and Bosel, N.Y.Google Scholar
  7. GIFI, A. (1990): Nonlinear Multivariate Analysis. Wiley, Chichester, England.zbMATHGoogle Scholar
  8. GROVE, D.M., WOODS, D.C., and LEWIS, S.M. (2004): Multifactor B-Spline Mixed Mod-els in Designed Experiments for the Engine Mapping Problem. Journal of Quality Tech-nology, 36, 4, 380-391.Google Scholar
  9. IZERNMAN, A.J. (1975): Reduced-rank regression for the multivariate linear model. Journal of Multivariate Analysis, 5.Google Scholar
  10. HUBERT, M., ROUSSEEUW, P.J., and BRANDEN, K.V. (2005): ROBPCA: A New Ap-proach to Robust Principal Component Analysis. Technometrics, 47, 1.CrossRefMathSciNetGoogle Scholar
  11. MACGREGOR, J.F. and KOURTI, T. (1995): Statistical Process Control of Multivariate Pro-cesses. Control Engineering Practice, 3, 3, 403-414.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Michele Gallo
    • 1
  • Luigi D’Ambra
    • 2
  1. 1.Department of Social ScienceUniversity of Naples — L’OrientaleNaplesItaly
  2. 2.Department of Mathematics and StatisticsUniversity of Naples — Federico IINaplesItaly

Personalised recommendations