## 3 Computational Experiments
To measure the accuracy of the results, we calculated the average over 100 computational experiments of the absolute error of each major direction and the average of the relative 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 distribution. The theoretical value of major direction for the one-major-direction case is \((1/\sqrt{n},\cdots,1/\sqrt{n})^{T}\) and the theoretical values of major directions for the four-major-directions case are \((1/\sqrt{n},\cdots,1/\sqrt{n})^{T}\), \((1,0,\cdots,0)^{T}\), \((0,1,0,\cdots,0)^{T}\), \((0,0,1,0,\cdots,0)^{T}\), respectively. The theoretical value of median radius for each major direction is calculated as the value of \(r^{*}\) such that \(\int_{0}^{r^{*}}f(r|\boldsymbol{\theta})dr/\int_{0}^{+\infty}f(r|\boldsymbol{ \theta})dr=0.5\) by numerical integration, where \(f(r|\boldsymbol{\theta})\) is the conditional probability density function of radius \(r\) along the major direction \(\boldsymbol{\theta}\). In Tables 1-4, we present computational results for the sets of 8000 points with local neighborhood of size \(I=160\) (2% of total points) for the one-major-direction situation. We compared our method with the methods that appeared in the literature [5,17]. Specifically, the MATLAB version of pcaPP[21] package of the R language for statistical computing [22] was used to implement the robust PCA proposed by [PERSON] and [PERSON] in [5]. [PERSON] _et al._'s \(L_{1}\) PCA [17] was implemented by using MATLAB and IBM ILOG CPLEX 12.5[23]. The computational results for those two PCA methods are also summarized in Tables 1-4. Table 5 shows the average computational time for each method when \(n=10\). For higher dimensions, the average computational time of \(\ell_{1}\) MCDA is shorter than the other two PCA methods. In Tables 6-9, we present the computational results for 32,000 points with local neighborhood of size \(I=320\), \(160\) and \(80\) (_i.e._, 2%, 1% and 0.5% of total points, respectively) for the four-major-directions situation. Figure 8: Sample from four Student \(t\) distributions with 10% artificial outliers. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(n\) & \multicolumn{2}{c}{\(\ell_{1}\) MCDA} & \multicolumn{2}{c}{**[PERSON] + [PERSON]**} & \multicolumn{2}{c}{**[PERSON] _et al._**} \\ \hline  & **av. abs\_err** & **av. rel\_err** & **av. abs\_err** & **av. rel\_err** & **av. abs\_err** & **av. rel\_err** \\  & **of angle** & **ra. of length** & **of angle** & **ra. of length** & **of angle** & **ra. of length** \\ \hline
3 & 0.0112 & 0.0314 & 0.0289 & 0.0681 & 0.0556 & 0.0675 \\
8 & 0.0233 & 0.2498 & 0.0399 & 0.2177 & 0.0118 & 0.1586 \\
10 & 0.0277 & 0.3129 & 0.0308 & 0.2683 & 0.0090 & 0.2256 \\
30 & 0.0315 & 0.5772 & 0.0112 & 0.5308 & 0.0052 & 0.5099 \\
50 & 0.0363 & 0.6539 & 0.0136 & 0.6269 & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The results of \(\ell_{1}\) MCDA and Robust PCAs on one Gaussian distribution. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\) & \multicolumn{2}{c}{\(\ell_{1}\) MCDA} & \multicolumn{2}{c}{**[PERSON] + [PERSON]**} & \multicolumn{2}{c}{**[PERSON] _et al._**} \\ \hline  & **av. abs\_err** & **av. rel\_err** & **av. abs\_err** & **av. rel\_err** & **av. abs\_err** & **av. rel\_err** \\  & **of angle** & **ra. of length** & **of angle** & **ra. of length** & **of angle** & **ra. of length** \\ \hline
3 & 0.0276 & 0.0921 & 0.0346 & 0.1224 & 0.0557 & 0.1310 \\
8 & 0.0423 & 0.2035 & 0.0483 & 0.2113 & 0.0118 & 0.1800 \\
10 & 0.0441 & 0.2717 & 0.0455 & 0.2625 & 0.0090 & 0.2352 \\
30 & 0.0446 & 0.5367 & 0.0113 & 0.5163 & 0.0053 & 0.4869 \\
50 & 0.0448 & 0.6350 & 0.0140 & 0.6100 & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The results of \(\ell_{1}\) MCDA and Robust PCAs on one Student \(t\) distribution.