# Projection Pursuit Multivariate Sampling of Parameter Uncertainty
## 4 Conclusions  
The main objective of this paper was to introduce PPMT as an efficient and easily applicable tool for the assessment of parameter uncertainty of the models defined in the multidimensional sampling spaces. The study considered four other sampling techniques for comparison including MCS, which is a general technique for random sampling from a given distribution, CLHS, which is a stratified random sampling technique, and two variants, the (maximin LHS and LHSMDU) of the LHS technique. Two synthetic case studies where various sample sizes (ranging from \(n=10\) to \(n=10\),000) were used considering two- and five-dimensional sampling spaces were conducted in order to assess the sampling performance of PPMT.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{7}{c}{**MCS Equivalent Number of Realizations**} \\ \hline
**Reals \#** & **CLHS** & \multicolumn{2}{c}{**Maximin LHS**} & \multicolumn{2}{c}{**LHSMDU**} & \multicolumn{2}{c}{**PPMT**} \\ \hline
10 & 20 & **24** & 22 & **25** & 26 & **30** & 30 & **32** \\
100 & 195 & **200** & 199 & **204** & 202 & **208** & 281 & **312** \\
1000 & 1631 & **1657** & 1690 & **1703** & 1699 & **1715** &  
In the first case study, the sampling space was considered to be two-dimensional and \(n=20\) realizations of two random variables were generated 500 times. The visual and numerical results shown in Figures 6, 7, and 12 clearly indicated that in comparison to the other sampling techniques, PPMT appeared to be the best technique that enforces the multidimensional uniformity among the realizations defined in a two-dimensional sampling space. When the two random variables were correlated according to the target correlation matrix, PPMT again outperformed the other sampling techniques by yielding the fewest discrepancy values.  
As for the second case study where the sampling space was considered to be five-dimensional, the results of the simulation study presented in Figures 8-11 clearly indicated that PPMT yielded the fewest sampling errors in comparison to all other sampling techniques in question and generated realizations whose empirical CDFs were rather close to the underlying CDF of the variable of interest (OOIP). The other important outcome of the simulation study, which is worth mentioning, is that the CPU capacity (or run time) required for the PPMT procedure was a lot less than those required for maximin LHS and LHSMDU. Therefore, PPMT can be considered an easily applicable technique that can be used for random sampling in Monte Carlo analysis. The Python code demonstrating the implementation of the PPMT as a sampling technique in the case of correlated random variables is available through [[https://github.com/Oktay-Erten/ppmt_parameter_uncertainty](https://github.com/Oktay-Erten/ppmt_parameter_uncertainty)]([https://github.com/Oktay-Erten/ppmt_parameter_uncertainty](https://github.com/Oktay-Erten/ppmt_parameter_uncertainty)), accessed on 20 September 2022.  
Conceptualization, C.V.D.; methodology, C.V.D. and O.E.; software, O.E. and F.P.L.P.; writing--original draft preparation, O.E.; writing--review and editing, O.E. and F.P.L.P.; supervision, C.V.D. All authors have read and agreed to the published version of the manuscript. This research received no external funding.  
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The authors thank the industrial sponsors of the Centre for Computational Geostatistics (CCG) for providing the resources to prepare this manuscript.  
The authors declare no conflicts of interest.  
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