ROBPCA

Hubert, M., Rousseeuw, P. J., Vanden Branden, K. Technometrics 47, 2005, 64-79. ROBPCA A new approach to robust principal components. 

Another approach to robust PCA has been proposed by Hubert et al. [52] and is called ROBPCA. This method combines ideas of both projection pursuit and robust covariance estimation. The projection pursuit part is used for the initial dimension reduction. Some ideas based on the MCD estimator are then applied to this lowerdimensional data space. Simulations have shown that this combined approach yields more accurate estimates than the raw projection pursuit algorithm RAPCA. The complete description of the ROBPCA method is quite involved, so here we will only outline the main stages of the algorithm. 

Note that this criterion cannot be used with ROBPCA, as the method does not yield all of the j) eigenvalues (as then it would become impossible to compute the MCD estimator in the final stage of the algorithm). But we can apply it on the eigenvalues of the covariance matrix of y(, that was constructed in the second stage of the algorithm. 

FIGURE 6.9 Outlier map of the glass data set based on three principal components computed with (a) CPCA and (b) ROBPCA [52],... 

Note that, as for the MCD estimator, the robustness of the RPCR algorithm depends on the value of h, which is chosen in the ROBPCA algorithm and in the LTS and MCD regression. Although it is not really necessary, it is recommended that the same value be used in both steps. 

Next, we can construct outlier maps as in Sections 6.5.5 and 6.4.2.3. ROBPCA yields the PC A outlier map displayed in Figure 6.12a. We see that there are no PCA leverage points, but there are some orthogonal outliers, the largest being 23, 7, and 20. The result of the regression step is shown in Figure 6.12b. It exposes the robust distances of the residuals (or the standardized residuals if q = 1) vs. the score... 

A robust method, RSIMPLS, has been developed by Hubert and Vanden Branden [65], It starts by applying ROBPCA on the joint x- and y-variables to replace Sjy and S, by robust estimates, and then proceeds analogously to the SIMPLS algorithm. More precisely, to obtain robust scores, ROBPCA is first applied on the joint x- and y-variables Ln m = (Xnp, Y,v/) with m = p + q. Assume that we select k0 components. This yields a robust estimate of the center of Z, jj,z = and, following... 

The third group of robust approaches to PCA combines the PP idea with the construction of robust covariance. An example of such a hybrid method is robust principal component analysis (ROBPCA) proposed by Hubert et al. [27]. 

The four algorithmic steps of the ROBPCA method can be presented as follows ... 

ROBPCA a version of robust principal component analysis... 

Hubert M, Rousseeuw PJ, Vanden Branden K. ROBPCA a new approach to robust principal component analysis. Technometrics 2005 47 64—79. 

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