# Projection Pursuit Multivariate Sampling of Parameter Uncertainty
## 2 Sampling Techniques  
### Monte Carlo Simulation  
MCS (or simple random sampling) is a technique through which a sample of the population is constructed using a random sequence of numbers. The deterministic parameters of the population can then be estimated from each sample [25]. The inverse transform method (or inversion sampling) is used to generate a realization through MCS [4]. Let \(Z\) be a random variable whose cumulative distribution function (CDF), which is monotonically non-decreasing, is denoted by \(\{F(z),\ a\leq z\leq b\}\), and the inverse CDF (or a quantile function) of \(Z\) is defined by \(F^{-1}(u)=\inf\{z\in[a,b]:F(z)\geq u\},\ 0<u<1\). Considering that \(U\sim U(0,1)\) is also a random variable that has a standard uniform distribution, then \(z=F^{-1}(u)\), which can also be observed from \(\Pr(Z\leq z)=\Pr(F^{-1}(u)\leq z)=\Pr(u\leq F(z))=F(z)\).  
For example, consider that \(n=5\) is the required number of realizations and \(k=2\) is the number of independent Gaussian random variables, \(X_{1}\perp X_{2}\sim N(0,1)\), that is, the sampling space is two-dimensional and orthogonal. Independent random numbers from a uniform distribution \(U(0,1)\) (i.e., the pairs exhibit a uniform distribution in the unit square, and similarly, the triplets also have a uniform distribution in the unit cube) are generated, that is, \(\mathbf{p}_{i}=[p_{1i}\ p_{2i}\ldots p_{ni}]^{T},\ i=1,\ldots,k=2\) and \(n=5\). These numbers are then established in a matrix \(\mathbf{P}\) where \((p_{ij})\in\mathbb{R}^{5\times 2}\):  
\[\mathbf{P}=\begin{bmatrix}0.06&0.09\\ 0.93&0.51\\ 0.82&0.66\\ 0.99&0.40\\ 0.14&0.76\end{bmatrix}\]  
Each element of the matrix \(\mathbf{P}\) is mapped according to a target CDF, which yields the independent realizations in the Gaussian unit \(\mathbf{x}_{i}=[x_{1i}\ x_{2i}\ldots x_{ni}]^{T},\ i=1,\ldots,k=2\) and \(n=5\). For instance, considering a probability of \(p_{42}=0.40\) associated with the variable \(X_{2}\), the corresponding realization can be obtained as \(x_{42}=F^{-1}(0.40)=-0.24\), where \(F^{-1}(\cdot)\) denotes the inverse of the Gaussian CDF for the variable \(X_{2}\). The MCS sampling design indicating the realizations given in matrix \(\mathbf{P}\) and mapping these probabilities according to the given CDF are presented in Figure 1a,b, respectively.  
Figure 1: (**a**) A sampling design with \(n=5\) realizations generated by MCS in a two-dimensional sampling space where the random variables are independent, (**b**) Mapping the quantiles according to the standard Gaussian distribution using the inverse transform method.  
The matrix \(\mathbf{X}\), \((x_{ij})\in\mathbb{R}^{52}\) whose elements are the realizations of \(X_{1}\) and \(X_{2}\) is then defined by  
\[\mathbf{X}=\begin{bmatrix}-1.54&-1.32\\ 1.44&0.03\\ 0.90&0.41\\ 2.31&-0.24\\ -1.08&0.72\end{bmatrix}\]  
In the case where \(X_{1}\) and \(X_{2}\) are correlated according to a given target correlation matrix \(\mathbf{C}\) where \((c_{ij})\in\mathbb{R}^{2x2}\), the linear dependency between \(X_{1}\) and \(X_{2}\), can be added via Cholesky decompsition [26; 27] of \(\mathbf{C}\), that is,  
\[\mathbf{C}=\mathbf{L}\cdot\mathbf{L}^{T}, \tag{1}\]  
where \(\mathbf{L}\) is the lower triangular matrix and \(\mathbf{L}^{T}\) (where the superscript \(T\) denotes transposition) is the upper triangular matrix. The correlated realizations of \(X_{1}\) and \(X_{2}\) are computed by multiplying the matrix \(\mathbf{L}\) by the matrix \(\mathbf{X}^{T}\) :  
\[\mathbf{X}^{*}=\mathbf{L}\cdot\mathbf{X}^{T}, \tag{2}\]  
The resulting matrix \(\mathbf{X}^{*}\) where \((x_{ij}^{*})\in\mathbb{R}^{2x5}\) contains the realizations of \(X_{1}\) and \(X_{2}\) that have a correlation matrix, which is close to the target correlation matrix \(\mathbf{C}\). The corresponding dependent quantiles (or probabilities) \(0\leq p_{ij}\leq 1\); \(i=1,\dots,n=5\) and \(j=1,\dots,\)\(k=2\) can be drawn from the standard Gaussian CDF as \(p_{ij}=F(x_{ij}^{*})\).