Besse, Philippe PCA stability and choice of dimensionality. (English) Zbl 0743.62046 Stat. Probab. Lett. 13, No. 5, 405-410 (1992). Summary: A criterion of stability for PCA scatterplots is defined based on a classical distance between projectors. It is constructed as a risk function and can be estimated by bootstrap or jackknife methods. Furthermore, perturbation theory is used to write down a Taylor expansion of the jackknife estimate for reasons of computational cost and in order to obtain an analytic expression for the approximation. The comparative study of these three estimates on real data shows that the last one is easy to compute, sufficiently accurate and helpful in choosing dimensionality in PCA. Cited in 18 Documents MSC: 62H25 Factor analysis and principal components; correspondence analysis 62A09 Graphical methods in statistics Keywords:principal components analysis; PCA scatterplots; classical distance between projectors; risk function; bootstrap; perturbation theory; Taylor expansion; jackknife estimate; dimensionality Software:R PDFBibTeX XMLCite \textit{P. Besse}, Stat. Probab. Lett. 13, No. 5, 405--410 (1992; Zbl 0743.62046) Full Text: DOI References: [1] Becker, R. A.; Chambers, J. M.; Wilks, A. R., The New S Language, a Programming Environment for Data Analysis and Graphics (1988), Wadsworth & Brooks/Cole: Wadsworth & Brooks/Cole Pacific Grove · Zbl 0642.68003 [2] Beran, R.; Srivastava, M. S., Bootstrap tests and confidence regions for functions of a covariance matrix, Ann. Statist., 13, 95-115 (1985) · Zbl 0607.62048 [3] Chatelin, F., Valeurs Propres de Matrices (1988), Masson: Masson Paris · Zbl 0691.65018 [4] Critchley, F., Influence in principal components analysis, Biometrika, 72, 627-636 (1985) · Zbl 0608.62068 [5] Daudin, J. J.; Duby, C.; Trécourt, P., Stability of principal components analysis studied by the bootstrap method, Statistics, 19, 241-258 (1988) · Zbl 0643.62043 [6] Daudin, J. J.; Duby, C.; Trécourt, P., P.C.A. stability studied by the bootstrap and the infinitesimal jackknife method, Statistics, 20, 255-270 (1989) · Zbl 0671.62038 [7] Dauxois, J.; Pousse, A.; Romain, Y., Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference, J. Multivariate Anal., 12, 136-154 (1982) · Zbl 0539.62064 [8] Efron, B., The Jackknife, the Bootstrap and other Resampling Plans (1982), SIAM: SIAM Philadelphia, PA · Zbl 0496.62036 [9] Gauss, The Gauss System Version 2.0 (1988), Aptech Systems: Aptech Systems Kent [10] Jolliffe, I. T., Principal Component Analysis (1986), Springer: Springer New York · Zbl 1011.62064 [11] Kato, T., Perturbation Theory for Linear Operators (1966), Springer: Springer New York · Zbl 0148.12601 [12] McDonald, G. C.; Ayers, J. A., Some applications of the “Chernoff Faces’: A technique for graphically representing multivariate data, (Wang, P. C.C., Graphical Representation of Multivariate Data (1978), Academic Press: Academic Press New York) [13] Winsberg, S., Two techniques: Monotone spline transformations for dimension reduction in PCA and easy to generate metrics for PCA of sampled functions, (van Rijckevorsel, J. L.A.; de Leeuw, J., Components and Correspondance Analysis (1988), Wiley: Wiley London) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.