Robust nonlinear principal components

Abstract

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 xi (i=1,…,n) the method finds a function h:R→Rp and a set {t1,…,tn}⊂R that minimize a joint M-scale of the residuals xi−h(ti), 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.