# Projection Pursuit Multivariate Sampling of Parameter Uncertainty
## 1 Introduction  
Mathematical models are frequently used in many disciplines (i.e., natural sciences, social sciences, engineering) in order to realistically estimate the physical processes in question. To construct such models (or outputs), in most cases, one must use a number of input variables. For example, to calculate the original oil in place (OOIP) for a reservoir, five input variables including the thickness of the deposit, deposit area, net oil to gross volume, net porosity, and water saturation should be considered [1]. However, due to the physical and financial constraints related to the sampling scheme, there are generally a limited number of measurements (or observations) of the input variables available for modeling. Therefore, it is imperative that the effect of uncertainty associated with the input variables of the model output be taken into account [2]. There are many sampling techniques that are used to assess the uncertainty associated with the parameters of the models. For example, Monte Carlo simulation (MCS), which relies on a repeated random sampling and statistical analysis, is generally used for this purpose [3, 4]. In MCS, the population is assumed to be independent and identically distributed, and the realizations of a sample are randomly chosen from the population with an equal probability. A pseudo-random number generator, which satisfies a series of statistical tests for randomness [5, 6], is used to generate a sequence of independent numbers (or random variates) from a uniform distribution \(U(0,1)\)[7]. A major drawback of MCS is that the realizations that are chosen completely at random tend to form clusters, which leaves gaps that are not investigated in the sampling space. If one takes a large sampleof realizations, the accuracy of the predicted model output increases and the sampling errors become negligible. However, this is only achievable provided that the run time of MCS is reasonably short. If the MCS run is computationally expensive, then the sampling techniques with better sampling efficiency such as Latin hypercube sampling (LHS) (or stratified random sampling) [8; 9] and its variants should be employed. The original and most simple form of LHS is referred to as classic Latin hypercube sampling (CLHS). In CLHS, the population is divided into a number of non-overlapping strata and MCS is used to generate a realization from each stratum. Because the population is stratified, the heterogeneity in each stratum becomes less, which results in realizations that are more uniform and representative. The stratification is maximized when the number of strata \(n\) is equal to the sample size \((n)\), i.e., \([0,1/n],[1/n,2/n],\ldots,[(n-1)/n,1]\)[10]. In CLHS, as mentioned earlier, the population is marginally stratified, that is, it accounts for only the univariate uniformity of the realizations and does not enforce any multivariate uniformity. To improve the space-filling properties of CLHS designs, many studies in the literature make use of mainly two performance criteria: (1) minimizing the pairwise correlation between the realizations, and (2) maximizing the minimum distance between the realizations [11; 12; 13; 14; 15]. Considering the correlated (or dependent) random variables, several studies [17; 18; 19; 20] propose various methodologies to generate a sample whose correlation matrix is approximately equal to the given (or target) correlation matrix, that is, the joint distribution is reproduced. The two important variants of LHS that can be used to generate realizations that have improved space-filling characteristics are (1) maximin Latin hypercube sampling (maximin LHS) [21] and (2) Latin hypercube sampling with multidimensional uniformity (LHSMDU) [1]. In the former approach, the aim is to generate a sample that maximizes the minimum Euclidean distance between any pair of realizations, which is achieved by generating a large number (i.e., thousands) of sampling designs and choosing the one that has a maximized distance between any pair of realizations.