# Projection Pursuit Multivariate Sampling of Parameter Uncertainty
## 2 Sampling Techniques
### Projection Pursuit Multivariate Transform  
Considering a two-dimensional sampling space \((\mathbb{R}^{d}\) where \(d=2)\), a pattern (or a structure), such as clusters, outliers, and skewness, can be instantaneously discovered by simply observing the scatterplot. However, it is not possible to detect the aforementioned patterns when the sampling space is greater than three \((\mathbb{R}^{d}\) where \(d>3)\). Projection pursuit, which was first proposed by [22] and first implemented by [23], can be used to detect these structures in the datasets defined in a high-dimensional sampling space. It makes use of a projection index \(I(\mathbf{u},\mathbf{v})\) that measures the degree of 'interestingness' of the data projected onto the plane spanned by the orthogonal vectors \((\mathbf{u},\mathbf{v})\). The plane that maximizes the projection index is determined by numerical optimization. The dataset is  
Figure 3: A sampling design with \(n=5\) realizations generated by (**a**) maximin LHS and (**b**) LHSMDU in a two-dimensional sampling space where the random variables are independent. generally transformed to normal scores and sphered in advance, that is, the transformed data has a mean of zero, a variance of one, and an identity correlation matrix. The projection index then measures the discrepancy between the distribution of the projected data and the standard Gaussian distribution \(N(0,1)\). For example, if the data is multivariate Gaussian, all of the projections are expected to be Gaussian and no 'interestingness' will be found. Also, as proved by [29], most projections of the multivariate data appear to be approximately Gaussian under appropriate conditions. If one considers that \(I(\mathbf{u},\mathbf{v})=0\) is the perfectly Gaussian case, any projected data that has a non-Gaussian distribution will increase the value of \(I(\mathbf{u},\mathbf{v})\), which indicates the 'interestingness' (or non-Gaussianity). One can also consider many other projection indices that measure the deviation from the standard Gaussian distribution [30; 31; 32]. We use the Fortran program called PPMT.EXE, which was proposed by [24] and is publicly available through the link [[http://www.ccgalberta.com/resources/select-software/](http://www.ccgalberta.com/resources/select-software/)]([http://www.ccgalberta.com/resources/select-software/](http://www.ccgalberta.com/resources/select-software/)), accessed on 20 September 202, in order to demonstrate the generation of a sampling design by the PPMT technique. Let us consider again that the sampling space is two-dimensional, that is, \(k=2\) represents the number of random variables, and \(n=5\) is the number of realizations. The steps of the PPMT procedure for generating the required number of realizations \((k\cdot n)\) are summarized as follows:  
1. Generate \((k\cdot n)\) random numbers from a uniform distribution \(U(0,1)\) using MCS and establish these random numbers in a \((5\times 2)\) matrix \(\mathbf{P}\). 2. Transform the elements of matrix \(\mathbf{P}\) to the standard Gaussian values, that is, \(\mathbf{Y}=G^{-1}[\mathrm{F}(\mathbf{P})]\), where \(G^{-1}[\mathrm{F}(\cdot)]\) is the normal score transform. 3. Compute the variance-covariance matrix of \(\mathbf{Y}\), that is, \(\mathbf{\Sigma}=(1/n)[\mathbf{Y}\mathbf{Y}^{T}]\). 4. Diagonalize \(\mathbf{\Sigma}\), that is, \(\mathbf{\Sigma}=\mathbf{Q}_{1}\wedge_{1}\mathbf{Q}_{1}^{T}\), where \(\mathbf{Q}_{1}\) denotes an orthogonal matrix of the eigenvectors and \(\mathbf{\Lambda}_{1}\) denotes the diagonal matrix of the eigenvalues. 5. Sphere the elements of matrix \(\mathbf{Y}\); that is, \(\mathbf{Y}^{\prime}=\mathbf{S}^{-1/2}\mathbf{Y}\), where \(\mathbf{S}^{-1/2}=\mathbf{Q}_{1}\wedge_{1}^{-1/2}\mathbf{Q}_{1}^{T}\). 6. Project \(\mathbf{Y}^{\prime}\) onto \(k\)-dimensional unit length vector \(\mathbf{\theta}\), that is, \(\mathbf{p}=\mathbf{\theta}\mathbf{Y}^{\prime}\). 7. Determine \(\mathbf{\theta}\) maximizing the projection index \(I(\mathbf{\theta})\) that measures the univariate non-Gaussianity. 8.