## 3 Computational Experiments  
In this section, we present comparisons of our \(\ell_{1}\) MCDA with [PERSON] and [PERSON]'s robust PCA [5] and [PERSON] _et al._'s \(L_{1}\) PCA [17]. Since both [PERSON] and [PERSON]'s and [PERSON] _et al._'s PCA methods are also widely used for the data with outliers, comparisons of our \(\ell_{1}\) MCDA with their methods are imperative. The following eight types of distributions were used for the computational experiments:  
* \(n\)-Dimensional (\(n\geq 3\)) Gaussian without and with additional artificial outliers;
* \(n\)-Dimensional (\(n\geq 3\)) Student \(t\) (degree of freedom = 1) without and with additional artificial outliers;
* Four superimposed \(n\)-dimensional (\(n\geq 3\)) Gaussians without and with additional artificial outliers;
* Four superimposed \(n\)-dimensional (\(n\geq 3\)) Student \(t\) (degree of freedom = 1) without and with additional artificial outliers. All computational results were generated using MATLAB codes in [18] and MATLAB R2012a [19] on a 2.50 GHz PC with 4 GB memory. Samples from Gaussian and Student \(t\) distributions were generated using the MATLAB mvnrnd and mvtrnd modules, respectively, with the covariance/correlation matrix  
\[\sigma(n)= \begin{bmatrix}1&b&\cdots&b\\ b&1&\cdots&b\\ \vdots&\vdots&\ddots&\vdots\\ b&b&\cdots&1\end{bmatrix} \]  
where \(0<b<1\) and \(n\) is the size of the covariance/correlation matrix. In our experiment, the computational time for each sample was restricted to 3600 s.  
For the one-major-direction situation, the ratio of the length of the longest major direction to that of other major directions (which are equal for the covariance/correlation matrix \(\sigma(n)\)) is set to be a constant \(C\) for all \(n\). To accomplish this, we set the ratio of the maximum eigenvalue (\(\alpha_{max}\)) to the minimal eigenvalue (\(\alpha_{min}\)) to be \(C^{2}\), _i.e._, \(\frac{\alpha_{max}}{\alpha_{min}}=\frac{1+(n-1)b}{1-b}=C^{2}\). Hence, \(b=\frac{C^{2}-1}{n-1+C^{2}}\) in the covariance/correlation matrix \(\sigma(n)\). In our experiment, we chose \(C=10\), which requires that \(b=\frac{99}{n+99}\). Therefore, for the one-major-direction situation, the major direction is \((\frac{1}{\sqrt{n}},\ldots,\frac{1}{\sqrt{n}})^{T}\) and the ratio of the length of the  
The artificial outliers are generated from a uniform distribution on a simplex. The \(n\) vertices of the simplex are randomly chosen from the intersection of the ball \(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=1500^{2}\) with the hyperplane \(a_{1}x_{1}+a_{2}x_{2}+\ldots+a_{n}x_{n}=1000\). For each distribution in the data, the angle between the normal of the hyperplane \((a_{1},a_{2},\ldots,a_{n})^{T}\) and the major direction of the distribution is 45\({}^{\circ}\). In Figures 5-8, we present examples with 10% artificial outliers (depicted by blue dots) for one Gaussian distribution, one Student \(t\) distribution, four superimposed Gaussian distributions and four superimposed Student \(t\) distributions (points from distributions depicted by red dots), respectively. Figure 4: Sample from four overlaid Student \(t\) distributions. Figure 3: Sample from four overlaid Gaussian distributions. Figure 5: Sample from one Gaussian distribution with 10% artificial outliers. Figure 6: Sample from one Student \(t\) distribution with 10% artificial outliers. Figure 7: Sample from four Gaussian distributions with 10% artificial outliers. 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. The neighbors required in Steps 4 and 6 of the proposed \(\ell_{1}\) MCDA were calculated by the \(k\)-nearest-neighbors (kNN) method [20]. Other methods for identifying neighbors may also be used.