Robust nonlinear principal components

RICARDO A. MARONNA; FERNANDA MENDEZ; VÍCTOR J. YOHAI

STATISTICS AND COMPUTING

SPRINGER

Lugar: Berlin; Año: 2015 vol. 25 p. 439 - 448

0960-3174

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 $h : R\rightarrow R^p$ and a set ${t1, . . . , tn}\subset R$ 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.