## 4 Computational Experiments
The results in Tables 2-4 indicate that, as expected, standard PCA does not handle data with artificial outliers and/or from heavy-tailed distributions well. The results in Tables 2-4 show, consistent with theoretical and \begin{table}
\begin{tabular}{c c c} \hline \hline
**Distribution** & **av. error** & **std. dev.** \\ \hline  & \(-0.0091\) & 0.0103 \\
3 Gaussians & 0.0124 & 0.0201 \\  & 0.0111 & 0.0103 \\ \hline  & \(-0.0093\) & 0.0221 \\
3 Gaussians with \(10\%\) outliers & \(-0.0084\) & 0.0252 \\  & 0.0124 & 0.0110 \\ \hline  & \(-0.0092\) & 0.0224 \\
3 Student \(t\) & 0.0130 & 0.0202 \\  & \(-0.0110\) & 0.0193 \\ \hline  & 0.0143 & 0.0224 \\
3 Student \(t\) with \(10\%\) outliers & \(-0.0159\) & 0.0252 \\  & 0.0130 & 0.0243 \\ \hline  & 0.0832 & 0.1593 \\
3 Gaussians & 0.1291 & 0.1788 \\  & 0.1402 & 0.1891 \\ \hline  & 0.1382 & 0.1632 \\
3 Gaussians with \(10\%\) outliers & 0.1389 & 0.1738 \\  & 0.1537 & 0.1838 \\ \hline  & 0.1839 & 0.2582 \\
3 Student \(t\) & 0.1783 & 0.2633 \\  & 0.1478 & 0.2537 \\ \hline  & 0.1987 & 0.2638 \\
3 Student \(t\) with \(10\%\) outliers & 0.1733 & 0.2837 \\  & 0.2018 & 0.2738 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average error (av. error) and standard deviation of the error (std. dev.) of \(\ell^{1}\) direction \(\theta\) calculated by \(\ell^{1}\) MCDA for three superimposed distributions without and with 10% artificial outliers. computational evidence available in the previous literature, the advantages of [PERSON] and [PERSON]'s projection-pursuit method and of [PERSON]'s \(\ell^{1}\) factorization method over standard PCA. These results also show in nearly all cases a marked advantage of \(\ell^{1}\) MCDA over [PERSON]'s projection-pursuit and [PERSON]'s \(\ell^{1}\) factorization. For the multiple superimposed distributions considered in Table 5, standard PCA, [PERSON]'s projection-pursuit and [PERSON] and [PERSON]'s \(\ell^{1}\) factorization each provide only one main direction for the superimposed distributions and do not yield any meaningful information about the individual main directions. For this reason, no results for these three methods are presented in Table 5. It is worth noting in Table 5 that the accuracy of \(\ell^{1}\) MCDA for three superimposed distributions without and with artificial outliers is just as good as the accuracy of \(\ell^{1}\) MCDA for single distributions without and with artificial outliers. As the computing times reported in Table 6 indicate, 2D \(\ell^{1}\) MCDA in its current implementation costs 30 to 40 times as much as standard PCA, roughly 3 times as much as [PERSON] and [PERSON]'s factorization and 30% to 40% less than [PERSON] and [PERSON]'s projection-pursuit (all methods sequentially implemented). The wide applicability of \(\ell^{1}\) MCDA in comparison with [PERSON] and [PERSON]'s factorization method justifies an increase of computing time by a factor of 3. Moreover, the computing time of \(\ell^{1}\) MCDA is likely to decrease as the method is further investigated and more efficient implementations are designed. Computational results for 5% artificial outliers were analogous to the results for 10% artificial outliers. Computational results for \(\ell^{1}\) MCDA with 20% artificial outliers in the data were not as accurate as those for 10% outliers. A quantitative description of the robustness of \(\ell^{1}\) MCDA in terms of breakdown point (proportion of outliers beyond which the method produces errors that are arbitrarily large) or influence function (how the result changes when one point is changed) will depend not only on the radii and angular coordinates of the points that are changed but also on clustering patterns of these points in relation to the other points. This task is beyond the scope of this paper but will be an objective of future research.