# Projection Pursuit Multivariate Sampling of Parameter Uncertainty
## 3 Case Studies
### Quality Assessments of Sampling Designs  
The quality of each design generated by the sampling techniques is visually inspected and numerically assessed according to the magnitude of the values of each criterion given in Equations (5) and (7). Considering the first case study where the sampling space is two-dimensional, one can see in Figures 6 and 7 that given the realizations generated by MCS, a large area of the sampling space is not investigated. This is mainly because MCS generates realizations completely at random; therefore, it is expected that a cluster of realizations is formed in the sampling space. The sampling designs generated by CLHS appear to be better than the ones generated by MCS. This is due to the fact that CLHS enforces the univariate uniformity of the realizations, that is, the realizations are drawn from each stratum, which guarantees that there is only one realization from each row and each column of the sampling space. However, no multidimensional uniformity is considered in CLHS. As for maximin LHS, LHSMDU, and PPMT through which one can take into account the multidimensional uniformity, the sampling designs shown in Figures 6 and 7 clearly indicate significant improvements in terms of the space-filling properties of the realizations. In addition to the visual inspection of the designs, we also numerically assess the quality of each sampling design using \(WL2\) statistics (Equation (5)). The box plots indicating the distribution of the aforementioned statistic considering the cases where the random variables are uncorrelated and correlated are presented in Figure 12.  
It can clearly be seen in Figure 12a,b that MCS yields the largest discrepancy values according to the \(WL2\) statistic, and PPMT, on the other hand, appears to outperform CLHS, maximin LHS, and LHSMDU. This is mainly due to the fact that the sampling designs generated by PPMT have no restrictions due to the Latin hypercube design. The median value of the discrepancy generated by MCS is approximately twice as much as the those generated by CLHS and its variants. The medians of the discrepancy values generated  
Figure 12: The box plots of the \(WL2\) statistics: (**a**) uncorrelated case, and (**b**) correlated case. by maximin LHS and LHSMDU are rather close and slightly lower than that generated by CLHS. Considering the second case study, one can see in Figures 8 and 10 that the empirical CDFs of the samples of varying sizes generated by PPMT appear to be closer to the underlying OOIP CDF than those generated by the other sampling techniques. It is again clear in the same figures that the most noticeable discrepancy between the empirical CDFs and the underlying OOIP CDF is in the MCS case. Similarly, Figures 9 and 11 clearly show that the fewest errors indicating the discrepancy between each empirical OOIP CDF and the underlying OOIP CDF are generated by PPMT. Considering the case where the random variables are correlated, the magnitudes of the error values appear to be greater for all of the sampling techniques than the uncorrelated case; however, PPMT outperforms the other sampling techniques in the correlated case as well. In addition to the aforementioned comparisons, one should also know how many realizations should be generated by MCS to ensure a specified statistical accuracy. This can be calculated using the error values generated by each sampling technique in the power law function (Equation (8)) that is fitted to the MCS error values versus the number of realizations. Table 4 contains the approximate number of realizations that should be generated by MCS to ensure the same statistical accuracy achieved by the other sampling techniques. The approximate number of realizations that should be generated by MCS is calculated using the coefficients (Tables 2 and 3) of the power law functions fitted to the MCS case considering the cases where the random variables are deemed to be uncorrelated and correlated.