Robust nonlinear principal components

• Published in: Statistics and computing Vol. 25; no. 2; pp. 439 - 448
• Main Authors: Maronna, Ricardo A., Méndez, Fernanda, Yohai, Víctor J.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.03.2015
• Subjects: Artificial Intelligence; Computation; Contamination; Estimators; Mathematical analysis; Mathematics and Statistics; Nonlinearity; Probability and Statistics in Computer Science; Regression; Splines; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; Splines; Principal curves; S-estimators
• ISSN: 0960-3174; 1573-1375
• DOI: 10.1007/s11222-013-9442-0

All known approaches to nonlinear principal components are based on minimizing a quadratic loss, which makes them sensitive to data contamination. A predictive approach in which a spline curve is fit minimizing a residual M-scale is proposed for this problem. For a p -dimensional random sample x i ( i =1,…, n ) the method finds a function h : R → R p and a set { t 1 ,…, t n }⊂ R that minimize a joint M-scale of the residuals x i − h ( t i ), where h ranges on the family of splines with a given number of knots. The computation of the curve then becomes the iterative computing of regression S-estimators. The starting values are obtained from a robust linear principal components estimator. A simulation study and the analysis of a real data set indicate that the proposed approach is almost as good as other proposals for row-wise contamination, and is better for element-wise contamination.