## 7 Results  
Table 1 presents the trimmed means to evaluate the cross-validation schemes when the M\({}^{2}\) statistic was used in the simulation study. It can be seen that, without contamination (i.e., outliers = 0%), the best method is the EK82 method because it minimizes the value of the criterion. This result is to be expected because the standard SVD provides very good results in the absence of outliers. It can also be seen that, of the resistant criteria considered to choose the rank of the matrices, the one that provided the best results was PRESS50. This indicates that using only 50% of the smallest residuals by cross-validation may be sufficient to select the rank. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline \multicolumn{10}{c}{**0.9-Trimmed Means of Procrustes Values**} \\ \hline
**Outliers = 0\%** & \multicolumn{3}{c}{**Methods**} & \multicolumn{3}{c}{**Outliers = 10\%**} & \multicolumn{3}{c}{**Methods**} \\ \hline
**Criterion** & **EK01** & **EK13** & **EK82** & **EK84** & **Criterion** & **EK01** & **EK13** & **EK82** & **EK84** \\ \hline
**PRESS** & 928 & 411 & 405 & 2326 & PRESS & 5,929,179 & 9,858,840 & 10,533,642 & 2726 \\
**PRESS75** & 845 & 367 & 357 & 2284 & PRESS75 & 6,503,877 & 10,519,282 & 10,592,223 & 2764 \\
**PRESS50** & 802 & 357 & **315** & 2273 & PRESS50 & 9,915,026 & 13,550,429 & 13,794,540 & 2771 \\ \(W_{\text{m}}\) & 1773 & 1547 & 1563 & 2364 & \(W_{\text{m}}\) & 16,969,901 & 23,472,838 & 23,671,632 & 2703 \\ \(W_{\text{m}}\)(Max) & 1431 & 1017 & 1000 & 2359 & \(W_{\text{m}}\)(Max) & 16,969,901 & 23,472,838 & 23,671,632 & 2706 \\ \(W_{\text{m}}\)**75** & 1222 & 841 & 826 & 2295 & \(W_{\text{m}}\)75 & 6,890,112 & 10,644,089 & 11,219,853 & 2741 \\ \(W_{\text{m}}\)**50(Max) & 1102 & 647 & 614 & 2292 & \(W_{\text{m}}\)55(Max) & 8,091,972 & 11,470,833 & 11,918,769 & 2748 \\ \(W_{\text{m}}\)**50** & 1177 & 822 & 806 & 2280 & \(W_{\text{m}}\)50 & 10,431,960 & 15,358,453 & 15,708,124 & 2753 \\ \(W_{\text{m}}\)**50(Max) & 1041 & 640 & 619 & 2279 & \(W_{\text{m}}\)80(Max) & 13,724,787 & 18,830,770 & 19,093,611 & 2760 \\ \hline
**Outliers = 5\%** & \multicolumn{3}{c}{**Methods**} & \multicolumn{3}{c}{**Outliers = 20\%**} & \multicolumn{3}{c}{**Methods**} \\ \hline
**Criterion** & **EK01** & **EK13** & **EK82** & **EK84** & **Criterion** & **EK01** & **EK13** & **EK82** & **EK84** \\ \hline
**PRESS** & 1,349,656 & 5,347,643 & 5,517,974 & 2503 & PRESS & 16,219,919 & 20,858,768 & 23,719,760 & 3794 \\
**PRESS75** & 1,593,606 & 5,978,138 & 6,160,294 & 2500 & PRESS75 & 16,585,561 & 20,389,484 & 23,240,805 & 3875 \\
**PRESS50** & 2,085,285 & 7,208,180 & 7,361,273 & **2491** & **PRESS50** & 24,876,820 & 26,596,384 & 26,604,794 & 3896 \\ \(W_{\text{m}}\) & 6,121,399 & 12,296,349 & 12,339,272 & 2513 & \(W_{\text{m}}\) & 32,095,002 & 43,746,109 & 47,429,575 & **3491** \\ \(W_{\text{m}}\)(Max) & 6,145,795 & 1,296,349 & 12,339,272 & 2511 & \(W_{\text{m}}\)(Max) & 32,020,042 & 43,746,109 & 47,429,575 & 3529 \\ \(W_{\text{m}}\)**75** & 1,802,253 & 6,876,657 & 7,043,231 & 2498 & \(W_{\text{m}}\)75 & 17,200,893 & 20,911,456 & 23,301,683 & 3710 \\ \(W_{\text{m}}\)**50(Max) & 2,427,846 & 8,446,853 & 8,567,883 & 2500 & \(W_{\text{m}}\)75(Max) & 18,713,152 & 21,288,046 & 23,488,022 & 3736 \\ \(W_{\text{m}}\)**50** & 2,659,677 & 8,260,863 & 8,433,306 & 2494 & \(W_{\text{m}}\)80 & 23,995,906 & 27,336,626 & 27,389,422 & 3769 \\ \(W_{\text{m}}\)**50(Max) & 4,369,880 & 10,172,078 & 10,266,390 & 2493 & \(W_{\text{m}}\)80(Max) & 28,428,412 & 31,084,469 & 29,862,533 & 3799 \\ \hline \hline \end{tabular} In bold, the method with the lowest statistic value in each percentage of outliers.  
\end{table} 
Table 1: The 0.9-trimmed means of Procrustes values in the simulation study. On the other hand, when there is some degree of contamination in the simulations, according to Procrustes statistics, methods EK01, EK13 and EK84 always present better performances (lower values) than EK82 using any of the criteria considered (resistant or not).