## 2 Methods Revisited
### Design Point and Sensitivity Factors from Simulations  
The obtaining of the design point and sensitivity factors (from the FORM standpoint) based on a previous study [12] is described herein. The basic idea is that simulated data close to the limit state surface (within a prescribed tolerance, \(e\)) can be retrieved, as well as their associated sampled values for each or the random variables given by the LSF (input values combinations considered as a point in the hyperspace), and a multi-linear regression is performed to approximate the linearized limit state surface at the design point. Before the multi-linear regression is performed to fit the hyperplane, the considered points are mapped into a standard normal space. Such hyperplane is an approximation of the LSF, \(g\), and is used to assess the design point and sensitivity factors. The mathematical formulation is given below. A set of \(n\) independent random variables is denoted by \(\mathbf{X}=(x_{1},\,x_{2},\,\ldots,\,x_{n})\). \(\mathbf{X}^{i}\)_defines_ the \(j\)-th randomly generated value of \(\mathbf{X}\). For a given number of crude Monte Carlo simulations, \(n_{sim}\), \(X_{js}\) which satisfy the criterion below (slightly changed from the original formulation in [12]) are selected  
\[\left|0-g\big{(}\mathbf{X}^{k}\big{)}\right|<e_{l} \tag{8}\]  
where \(\mathbf{X}^{k}\) in the LSF is included to emphasize that it is a function of a set of random variables. [PERSON] and [PERSON] [12] used \(e=0.05\) (i.e., 5%) in their study (instead of \(e_{l}\) in Equation (8) and defined below). However, it was noticed that this value could be inadequate depending on the units and magnitude of the considered random variables. Therefore, the distance between zero and the smallest simulated value of \(g\) in absolute terms (\(l_{low}\)) is used to set the tolerance as the fraction given by \(l_{low}\) multiplied by \(e\) (i.e., \(e_{l}=e\times l_{low}\) is used instead of \(e\) in Equation (8)). The selected values of \(\mathbf{X}^{k}\) based on the described criterion are then mapped into a standard normal space [17] using  
\[\mathbf{Z}=[z_{1},z_{2},\ldots,z_{n}]=\left[\Phi^{-1}(F_{1}(x_{1})),\Phi^{-1}(F_{ 2}(x_{2})),\ldots,\Phi^{-1}(F_{n}(x_{n}))\right] \tag{9}\]  
where \(\mathbf{Z}\) is the image of \(\mathbf{X}\) in standard normal space, \(\Phi\) is the normal standard CDF and \(F_{i},I=1,\,2,\,\ldots\,\), \(n\), are the CDFs of the random variables \(x_{i}\). In the standard normal space, a linear function is fitted to the set of selected points using multi-linear regression. Such linear function is given by  
\[\sum_{i=1}^{n}b_{i}z_{i}+c=0 \tag{10}\]  
where \(b_{i}\) and \(c\) are constants to be determined in the multi-linear regression analysis. The resulting linear regression equation is to be used in the same sense as the FORM [17] to estimate the design point and sensitivity factors. The latter are denoted by \(\alpha_{i}\), \(i=1,\,2,\,\ldots\,\,n\), and given by the gradient vector of Equation (10) as indicated below  
\[\alpha_{i}=-b_{i}\big{/}\sqrt{\sum_{i=1}^{n}b_{i}^{2}} \tag{11}\]  
Note that the reliability index can also be estimated as the smallest distance between the linear surface and the origin as  
\[\beta=\sqrt[4]{\sum_{i=1}^{n}b_{i}^{2}} \tag{12}\]As regards the design point (in the normal space) it is given by  
\[z_{id}=\alpha_{i}\beta \tag{13}\]  
The design point in the original space can now be determined with the inverse transformation of \(z_{id}\), as  
\[x_{id}=F_{i}^{-1}(\Phi(z_{id})) \tag{14}\]  
The subscript \(i\) in Equations (11)-(14) is associated to the random variable \(x_{i}\). The inverse transformation in Equation (14) is dependent on the PDF of \(x_{id}\) and, in the case of the Longuet-Higgins distribution [20] used for the coastal engineering example, the inverse transformations of \(z_{id}\) requires of a numerical approach to be determined.