Summary
An alternating least squares algorithm based on maximal correlation between variables is proposed to introduce linear and nonlinear interaction terms in PC A. Such algorithm fits a model in which principal components are data driven dimensionality reduction functions. The detection of meaningful interaction is a way to specify such general unknown functions. As an example the highly nonlinear structure of a circle is recovered by the first component of a nonlinear interactive PCA of the circle coordinates.
The author was partially supported by a grant of the Italian Ministry of University, Scientific and Technological Research (M.U.R.S.T. 40%, n.940326)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bekker, P. and De Leeuw, J. (1988). Relations between variants of nonlinear Principal Component Analysis. In: J. van Rijckevorsel and J. De Leeuw (Eds), Component and Correspondence Analysis, 1–31. New York: Wiley
De Boor, C. (1978). A practical Guide to Splines. New York: Springer Verlag
De Leeuw, J. (1982). Nonlinear principal component analysis. In H. Caussinus et al.(eds), COMPSTAT 82, 77–86. Proceedings in Computational Statistics, part I. Wien: Physika-Verlag.
De Leeuw, J. and van Rijckevorsel, J.L.A. (1980). HOMALS and PRINCALS. In Diday E. et al. (eds) Data Analysis and Informatics, 231–241. Amsterdam: North Holland.
Gebelein, H. (1941). Das statistische Problem der Korrelation als Varations-und Eigenwertproblem und sein Zusammenhang mit der Ausgleichungsrechnung. Z. Angew. Math. Mech. 21, 364–379.
Gifi, A. (1990). Nonlinear Multivariate Analysis. New York: Wiley
Gnanadesikan, R. and Wilk, M.B. (1969). Data analytic methods in multivariate statistical analysis. Multivariate Analysis, II (P.R. Krishnaiah, editor), Academic Press, New York.
Guttman, L. (1959). Metricizing rank-ordered or unordered data for a linear factor analysis. Sankhya, A, 21, 257–68.
Hastie, T. and Stuetzle, W. (1989). Principal Curves. J. of the American Statistical Association, 84, 502–516.
Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. J. Educ. Psychol., 24, 417–441, 498–520.
Jackson, J.E. (1991). A User’s Guide to Principal Components. New York: Wiley.
Koyak, R. (1985). Optimal Transformations for Multivariate Linear Reduction Analysis. Unpublished PhD thesis. University of California, Berkeley, California: Dept. of Statistics.
Koyak, R. (1987). On measuring internal dependence in a set of random variables. The Annals of Statistics, 15, 1215–1228.
Kruskal, J.B. and Shepard, R.N. (1974). A nonmetric variety of linear factor analysis. Psychometrika, 39, 123–57.
LeBlanc, M. and Tibshirani, R. (1994). Adaptive Principal Surfaces. J. of the American Statistical Association, 89, 53–64.
Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. Phil. Mag. (6), 2, 559–572.
Rao, C.R. (1964). The use and interpretation of principal component analysis in applied research. Sankhya, A, 26, 329–358.
Renyi, A. (1959). On measures of dependence. Acta. Math. Acad. Sci. Hungar., 10, 441–451.
van Rijckevorsel, J.L.A. and De Leeuw, J. (eds) (1988). Component and Correspondence Analysis. New York: Wiley.
van Rijckevorsel, J.L.A. and De Leeuw, J. (1992). Some results about the importance of knot selection in non linear multivariate analysis. The Italian Journal of Applied Statistics, vol. 4 (4), 429–451.
van Rijckevorsel, J.L.A. and Tessitore, G. (1993). An algorithm for multivariate adaptive component and correspondence analysis. Bulletin of the International Statistical Institute, 49th session. Contributed papers, 2, 513–14, Firenze, 1993.
Schumaker, L. (1981). Spline Functions: Basic Theory. New York: Wiley.
Thurstone, L.L. (1947). Multiple factor analysis. Chicago:University of Chicago press.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Physica-Verlag Heidelberg
About this paper
Cite this paper
Costanzo, G.D., van Rijckevorsel, J.L.A. (1996). Interaction in Nonlinear Principal Components Analysis. In: Härdle, W., Schimek, M.G. (eds) Statistical Theory and Computational Aspects of Smoothing. Contributions to Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-642-48425-4_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-48425-4_17
Publisher Name: Physica-Verlag HD
Print ISBN: 978-3-7908-0930-5
Online ISBN: 978-3-642-48425-4
eBook Packages: Springer Book Archive