## 4 Computational Experiments
5, 6, 7 and 8, we present examples of 500-point data sets with 10% artificial outliers (depicted by blue dots) for one Gaussian distribution, one Student \(t\) distribution,  
Figure 1: Sample from one Gaussian distribution without artificial outliers. Figure 2: Sample from three superimposed Gaussian distributions without artificial outliers. three superimposed Gaussian distributions and three superimposed Student \(t\) distributions (points from distributions depicted by red dots), respectively. For each type of data, we carried out 100 computational experiments, each time with a new sample from the statistical distribution(s), including the uniform distributions that generated the outliers. To measure the accuracy of the results, we calculated the average and standard deviation (over 100 computational experiments) of the error of each main \(\ell^{1}\) direction and the average and standard deviation of the error of the median radius in that direction vs. the theoretical values of the direction of maximum spread and the median radius in that direction of the continuum distribution. In Tables 1-5, we present computational results for the sets of 500 points. Computational results for 50 and 150 points were analogous to those for 500 points. Figure 4: Sample from three superimposed Gaussian distributions with 10% artificial outliers. Figure 3: Sample from one Gaussian distribution with 10% artificial outliers. \begin{table}
\begin{tabular}{l c c c c} \hline  & **Standard PCA** & **Croux+Ruiz-Gazen** & **Ke+Kanade** & \(\ell^{1}\)**MCDA** \\ \hline av. error & 0.00005 & \(-0.0034\) & \(-0.0063\) & \(-0.0081\) \\ \hline std. dev. & 0.0051 & 0.0563 & 0.0084 & 0.0105 \\ \hline av. error & \(-0.0053\) & 0.0219 & — & 0.0349 \\ \hline std. dev. & 0.0901 & 0.1733 & — & 0.1683 \\ \hline \end{tabular}
\end{table} 
Table 1: Average error (av. error) and standard deviation of the error (std. dev.) of \(\ell^{1}\) direction \(\theta\) for one Gaussian distribution. \begin{table}
\begin{tabular}{l c c c} \hline  & **Standard PCA** & **Croux+Ruiz-Gazen** & **Ke+Kanade** & \(\ell^{1}\)**MCDA** \\ \hline av. error & 1.0046 & \(-0.0168\) & 0.9148 & \(-0.0105\) \\ \hline std. dev. & 0.0184 & 0.0837 & 0.0243 & 0.0114 \\ \hline av. error & 261.9384 & 0.3185 & — & 0.1138 \\ \hline std. dev. & 3.4384 & 0.1482 & — & 0.2028 \\ \hline \end{tabular}
\end{table} 
Table 2: Average error (av. error) and standard deviation of the error (std. dev.) of \(\ell^{1}\) direction \(\theta\) for one Gaussian distribution with 10% artificial outliers. \begin{table}
\begin{tabular}{l c c c c} \hline  & **Standard PCA** & **Croux+Ruiz-Gazen** & **Ke+Kanade** & \(\ell^{1}\)**MCDA** \\ \hline av. error & 0.0353 & 0.0614 & 0.0474 & 0.0163 \\ \hline std. dev. & 0.1983 & 0.0743 & 0.1738 & 0.01717 \\ \hline av. error & 147.2380 & \(-0.7139\) & — & 0.1438 \\ \hline std. dev. & 258.1932 & 0.1692 & — & 0.2302 \\ \hline \end{tabular}
\end{table} 
Table 3: Average error (av. error) and standard deviation of the error (std. dev.) of \(\ell^{1}\) direction \(\theta\) for one Student \(t\) distribution. _Remark 4_ Since \(\ell^{1}\) MCDA is an \(\ell^{1}\) method, one may ask why the accuracy of the results is measured using averages and standard deviations rather than, for example, \(\ell^{1}\) measures such as medians and other quantiles. The statistical distributions of the directions and median radii that are calculated by \(\ell^{1}\) MCDA are zero-tailed and (apparently) light-tailed, respectively, which indicates that averages and standard deviations are more appropriate measures than quantiles. _Remark 5_ No results for [PERSON] and [PERSON]'s method are provided in Tables 1-4 because [PERSON] and [PERSON]'s method does not yield radius information. The results in Table 1 indicate that, as theoretically predicted, standard PCA performs better than any of the other three methods for data from one single Gaussian distribution.