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Generalized principal component analysis with respect to instrumental variables via univariate spline transformations. (English) Zbl 0937.62604
Summary: A method is proposed for a nonlinear structural analysis of multivariate data, that is termed a generalized principal component analysis with respect to instrumental variables via spline transformations (or spline-PCAIV). This method combines features of multiresponse additive spline regression analysis and principal component analysis. The solution of the corresponding linear problem belongs to the set of the feasible solutions and constitutes the first step of the associated iterative algorithm. Introducing adapted metrics in principal component analysis leads to an interpretation of the method as an optimal canonical analysis. Examples related to distorted pattern recognition, multivariate regression analysis and nonlinear discriminant analysis show how spline-PCAIV works.

62H25 Factor analysis and principal components; correspondence analysis
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[1] Becker, R. A.; Chambers, J. M.; Wilks, A. R.: The new S language. (1988) · Zbl 0642.68003
[2] Bonifas, L.; Escoufier, Y.; Gonzales, P. L.; Sabatier, R.: Choix de variables en analyse en composantes principales. Revue de statistique appliquée, XXXII, 5-15 (1984) · Zbl 0583.62053
[3] Breiman, L.; Friedman, J. H.: Estimating optimal transformations for multiple regression and correlation (with discussion). Journal of the American statistical association 80, 580-618 (1985) · Zbl 0594.62044
[4] Breiman, L.; Ihaca, R.: Nonlinear discriminant analysis via scaling and ACE. Technical report (1989)
[5] Buja, A.; Hastie, T.; Tibshirani, R.: Linear smoothers and additive models. The annals of statistics 17, 453-555 (1989) · Zbl 0689.62029
[6] Van Der Burg, E.; De Leeuw, J.: Nonlinear redundancy analysis. British journal of mathematical and statistical psychology 43, 217-230 (1990)
[7] Durand, J. F.: Additive spline discriminant analysis. Computational statistics, I, 145-150 (1992)
[8] Escoufier, Y.: Principal components analysis with respect to instrumental variables. European courses in advanced statistics, 285-299 (1987)
[9] Escoufier, Y.; Holmes, S.: Data analysis in France. Biometric bulletin 7, No. 3, 27-28 (1990)
[10] Fisher, R. A.: The use of multiple measurements in taxonomic problems. Annals of eugenics 7, 179-188 (1936)
[11] Friedman, J. H.: Multivariate adaptive regression splines (with discussion). The annals of statistics 19, 1-123 (1991)
[12] Friedman, J. H.; Silverman, B. W.: Flexible parsimonious smoothing and additive modeling. Technometrics 31, 3-39 (1989) · Zbl 0672.65119
[13] Hastie, T.; Tibshirani, R.: Generalized additive models. (1990) · Zbl 0747.62061
[14] Magnus, J.; Neudecker, H.: Matrix differential calculus with applications in statistics and econometrics. (1988) · Zbl 0651.15001
[15] Ramsay, J. O.: Monotone regression splines in action. Statistical science 3, 425-461 (1988)
[16] Rao, C. R.: The use and the interpretation of principal component analysis in applied research. Sankhyā A 26, 329-358 (1964) · Zbl 0137.37207
[17] Van Rijckvorsel, J.: Fuzzy coding and B-splines. Component and correspondence analysis, 33-55 (1988)
[18] Robert, P.; Escoufier, Y.: A unifying tool for linear multivariate methods: the RV-coefficient. Applied statistics 25, 257-265 (1976)
[19] Rogers, G. S.: Matrix derivatives. 2 (1980) · Zbl 0463.15005
[20] Sabatier, R.: Méthodes factorielles en analyse des données: approximations et prise en compte des variables concomitantes. Thèse d’état (1987)
[21] Schumaker, L. L.: Spline functions: basic theory. (1981) · Zbl 0449.41004
[22] Zaamoun, S.: Fonctions splines en analyse de données. Thèse de doctorat (1989)
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