## 4 Computational Experiments  
In this section, we present comparisons of 2D \(\ell^{1}\) MCDA with standard PCA, [PERSON]'s method and [PERSON]'s method. Of all the robust PCAs that have been developed, only [PERSON]'s method [8, 9], uses \(\ell^{1}\) as its main basis. For this reason, comparison of our \(\ell^{1}\) MCDA, which is \"fully \(\ell^{1}\)\", with [PERSON] and [PERSON]'s method is important. For further contextual awareness, comparison with another widely used robust PCA, for example, [PERSON] and [PERSON]'s projection-pursuit method [4] is equally important. Eight types of distributions were used for the computational experiments:  
* Bivariate Gaussian without and with additional artificial outliers
* Bivariate Student \(t\) with 1 degree of freedom without and with additional artificial outliers
* Three superimposed bivariate Gaussians without and with additional artificial outliers
* Three superimposed bivariate Student \(t\) with 1 degree of freedom without and with additional artificial outliers  
Bivariate Student \(t\) distributions with 1 degree of freedom are heavy-tailed distributions with particularly heavy tails. These distributions were chosen for the computational experiments because they represent a significant computational challenge for standard PCA, the robust PCAs and \(\ell^{1}\) MCDA. All computational results were generated by MATLAB R2009b on a sequential 2.50 GHz computer with 1 GB memory. The quadratic functions in Steps 3 and 5 of the local \(\ell^{1}\) fitting algorithm described in Subsection 3.2 were calculated by the MATLAB \(linprog\) module. The \(q\) and \(q^{*}\) of Steps 2 and 5 of the algorithm for calculating the directions and values of the local maxima of the median radius were chosen to be 5 and 12, respectively. We generated samples from distributions with median \(\ell^{1}\) radius \(\rho\) in the \(\ell^{1}\) direction \(\alpha\) for the following \(\rho\) and \(\alpha\). For the one-main-direction situation, we used samples from bivariate distributions with \(\{\rho,\alpha\}\) = \(\{2.1291,0.2929\}\). Distributions with three main directions were generated by superimposing three distributions with \(\{\rho,\alpha\}\) = \(\{2.3340,-0.4308\}\), \(\{2.1291,0.2929\}\) and \(\{2.1291,0.7071\}\). Samples from the bivariate Student \(t\) distributions were generated by the Matlab \(mvtrnd\) module using the correlation matrix \(A=\left[ \begin{array}{cc}1&0.9\\ 0.9&1\end{array} \right]\). These samples were rotated (in \(\ell^{2}\)) to the \(\ell^{1}\) directions mentioned above. In the figures described below in which we illustrate these distributions, we indicate the points of these samples by red dots. Directions and magnitudes of the maxima of the median radius are indicated by bars extending out from the origin. In the computational experiments, we used data sets consisting of 50, 150 and 500 points. In Figures 1-4, we present examples of the data sets with 500 points (red dots) for one Gaussian distribution, one Student \(t\) distribution, three superimposed Gaussian distributions and three superimposed Student \(t\) distributions, respectively. In order to exhibit the major components clearly, we show in these figures data points only in \([-10,10]\times[-10,10]\). For Student \(t\) distributions, there are still many points outside \([-10,10]\times[-10,10]\). We also generated data sets consisting of data from the distributions described above along with \(5\%\), \(10\%\) and \(20\%\) patterned artificial outliers that represent clutter. For the one-distribution situation, artificial outliers were set up using a uniform statistical distribution on the \(\ell^{1}\) diamond with radius 1000 in the \(\ell^{1}\)-direction window \([-0.9,-0.5]\). For the three-superimposed-distribution situation, artificial outliers were set up using uniform statistical distributions on the \(\ell^{1}\) diamond with radius 1000 in the three different \(\ell^{1}\)-direction windows \([-0.9,-0.8]\), \([-0.3,-0.2]\) and \([0.4,0.5]\). In Figs.