## 2 An Efficient Algorithm for Moment-Independent Importance Analysis
### The Modified 4 PEM  
To improve the efficiency of MII estimation, the key is to obtain the PDF \(f_{Y}(y)\) and \(f_{Y|X_{i}}(y)\) with a low computational cost. PEM is an efficient way to propagate the probability characteristics. Its key point is to obtain the moments of the output \(Y\) according to the several moments of the input variable \(\mathbf{X}\) by \(m\times n\) times propagations of \(m\) points through the input-output system \(Y=g(\mathbf{X})\). Designate the first r central moments of \(X_{i}\) as \(\alpha_{r,i}\). The first three are the mean, the standard deviation and the skew. PEM sampling coefficients \(\zeta_{i,j}\) and the corresponding weights \(w_{i,j}\) can be determined by [15]  
\[\left\{ \begin{array}{l}\sum\limits_{j=1}^{m}w_{i,j}=1/n\ i=1\sim n;\ j=1 \sim m;\\ \sum\limits_{j=1}^{m}w_{i,j}\big{(}\zeta_{i,j}\big{)}^{r}=\lambda_{i,r}\ r=1 \sim 2m-1\end{array} \right. \tag{5}\]  
where \(\lambda_{i,r}\) is the ratio of \(\alpha_{r,i}\) to the \(r\)-th power of \(\alpha_{2,i}\), i.e., \(\lambda_{i,r}=\frac{\alpha_{r,i}}{\alpha_{2,i}^{2}}\) and \(\lambda_{i,1}\)= 0, \(\lambda_{i,2}\)= 1 for \(r\)= 1, \(r\)= 2, respectively. With the solutions of \(\zeta_{i,j}\) and \(w_{i,j}\) from Equation (5), the \(q\)-th order raw moment of \(Y\) can be evaluated by [16]  
\[E[Y^{q}]\approx\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}w_{i,j}\big{(}g\big{(} \mu_{1}\mu_{2}\cdots\mathbf{\lambda}_{i,j}\cdots\mu_{n}\big{)}\big{)}^{q} \tag{6}\]  
in which \(Y_{i,j}=g\big{(}\mu_{1}\mu_{2}\cdots\mathbf{\lambda}_{i,j}\cdots\mu_{n}\big{)}\) means the realization of \(Y\) when \(X_{i}\) is taking its location denoted by \(\lambda_{i,j}=\mu_{i}+\zeta_{i,j}\alpha_{2,i}\) and the other variables of \(\mathbf{X}\) fixing on their mean values \(\mu\). When m = 2, the PEM is specialized as 2 PEM denoted by the solution of Equation (5) [14; 15]  
\[\left\{ \begin{array}{l}\zeta_{i,j}=\frac{\lambda_{i,2}}{2}+(-1)^{3-l}\sqrt{ n+\frac{\lambda_{i,2}^{2}}{4}}\\ w_{i,j}=\frac{1}{n}\frac{(-1)^{l}\zeta_{i,j-l}}{\zeta_{i,1}-\zeta_{i,2}}=\frac{ 1}{2n}\frac{(-1)^{l}\zeta_{i,j-l}}{\sqrt{n+\left(\lambda_{i,3}/2\right)^{2}}} \end{array} \right.\ l=1,2\ \mathrm{j}=1,2 \tag{7}\]  
According to Equations (5)-(6), the moment of \(Y\) can be estimated more accurately when more points are taken into propagation. [PERSON] et al. [18] proposed an improved 3 PEM scheme by adding the new pair of estimate points to [PERSON]'s 3 PEM [16]. [PERSON]'s computation of the sample points' locations may lead to imaginary solutions, especially when the input variable has a skewed distribution. So [PERSON]'s technique is not always appropriate for practical civil engineering since the randomness of a structure probably follows a non-negative distribution. That is why a novel 4 PEM is proposed in which a new pair of points are added to [PERSON]'s 2 PEM. This new method not only keeps the advantages of 2 PEM, which easily yields real solutions and with relatively little computation, but also increases the accuracy of 2 PEM. Moreover, only the first three moments of the input parameters are necessary for the 4 PEM, while [PERSON]'s [18] and [PERSON]'s [16] methods need the first four. Substitute m = 2 and r = 1, 2 to Equation (5) and let all the weights take the same value, then a pair of new solutions of Equation (5) can be obtained as [18]  
\[\left\{ \begin{array}{l}\zeta_{i,3}=-\zeta_{i,4}=\sqrt{n}\\ w_{i,3}=w_{i,4}=\frac{1}{2n}\end{array} \right. \tag{8}\]  
The proposed 4 PEM is the combination of the new pair of points \(\zeta_{i,3},\zeta_{i,4}\) and the former two points \(\zeta_{i,1},\zeta_{i,2}\), thus, the corresponding weights will conflict with the inherent condition, which is \(\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}w_{i,j}=1\). In this regard, all the weights for these four points need to be reduced by half and the demonstration of the modified 4 PEM is  
\[\left\{