UBC Theses and Dissertations
Robust principal component analysis via projection pursuit Patak, Zdenek
In principal component analysis (PCA), the principal components (PC) are linear combinations of the variables that minimize some objective function. In the classical setup the objective function is the variance of the PC's. The variance of the PC's can be easily upset by outlying observations; hence, Chen and Li (1985) proposed a robust alternative for the PC's obtained by replacing the variance with an M-estimate of scale. This approach cannot achieve a high breakdown point (BP) and efficiency at the same time. To obtain both high BP and efficiency, we propose to use MM- and τ-estimates in place of the M-estimate. Although outliers may cause bias in both the direction and the size of the PC's, Chen and Li looked at the scale bias only, whereas we consider both. All proposed robust methods are based on the minimization of a non-convex objective function; hence, a good initial starting point is required. With this in mind, we propose an orthogonal version of the least median of squares (Rousseeuw and Leroy, 1987) and a new method that is orthogonal equivariant, robust and easy to compute. Extensive Monte Carlo study shows promising results for the proposed method. Orthogonal regression and detection of multivariate outliers are discussed as possible applications of PCA.
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