# Projection Pursuit Multivariate Sampling of Parameter Uncertainty
## 3 Case Studies  
### Synthetic Bivariate Case  
In the first case study, we consider that the sampling space is two-dimensional (\(k=2\)) and that 20 realizations (\(n=20\)) of each random variable are generated using MCS, CLHS, maximin LHS, LHSMDU, and PPMT, considering the cases where the random variables are uncorrelated and correlated. It is noted that the values of the realizations that are generated are only in the range of \([0,1]\), that is, \([0,1]\) bounds are interpreted as the probability. To stabilize the distance measures between all of the realizations, 500 sets of the sample of \(n=20\) are generated using each sampling technique. To assess the quality of the sampling designs yielded by each technique, we use the Wraparound L2 (WL2) statistics [34] that measure the discrepancy between the number of design realizations per subvolume in comparison to the same number of uniformly distributed realizations across the sampling space.  
\[WL2=-\left(\frac{4}{3}\right)^{p}+\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n} \prod_{k=1}^{p}\left(\frac{3}{2}-\left|z_{i}^{k}-z_{j}^{k}\right|\left(1-\left| z_{i}^{k}-z_{j}^{k}\right|\right)\right), \tag{5}\]  
where \(z_{i}^{k}\) and \(z_{j}^{k}\) are the elements of the vectors \(\mathbf{z}^{1},\ldots,\mathbf{z}^{n}\); \(i,j=1,\ldots,n\) denotes the number of realizations, and \(k=1,\ldots,p\) denotes the number of random variables.  
We first consider the case where the two random variables are uncorrelated. The realizations are straightforwardly generated by MCS and CLHS. Considering maximin LHS and LHSMDU, the additional parameters required by the procedures include the number of iterations in maximin LHS and \(m\) value in LHSMDU, as explained in Section 2.2. We select the number of iterations as 1000 in the maximin LHS procedure and consider the \(m\) value to be equal to 5 in the LHSMDU procedure. As for PPMT, the entire procedure, as explained in Section 2.3, consists of generating realizations by MCS, mapping these realizations according to a standard Gaussian CDF, using these realizations to generate PPMT scores, and back-transforming the PPMT scores to the uniform distribution. Figure 6 shows four sets of sample with \(n=20\) realizations generated by MCS, CLHS, maximin LHS, LHSMDU, and PPMT.  
The contours shown in Figure 6 are the probability contours of the multivariate Gaussian distribution. Because the random variables are independent, their covariance matrix is an identity matrix. Therefore, the probability contours, as shown in Figure 6, represent a circle shape.  
We now consider the case where the random variables are positively correlated according to the following covariance matrix \(\mathbf{C}\):  
\[\mathbf{C}=\begin{bmatrix}1&0.85\\ 0.85&1\end{bmatrix}\]  
As can be seen from the elements of the matrix \(\mathbf{C}\), the variances of both random variables are one and the strength of the linear relation between the random variables, as determined by the covariance (or correlation coefficient), is 0.85. The target correlation is imposed to the realizations of the independent random variables through the Cholesky decomposition (Equation (1)). Figure 7 shows the four sets of samples with \(n=20\) correlated realizations generated by each sampling technique.  
The contours shown in Figure 7 represent the probability contours of the multivariate Gaussian distribution. Considering the target correlation matrix \(\mathbf{C}\), the probability contours exhibit an elliptical shape.