## 3 The Time-Varying Global Reliability Analysis Based on the HM-IMEM
### The High-Order Moments-Based Improved Maximum Entropy Method  
#### The Estimation of the High-Order Moments Based on the GF-Discrepancy-Based Point Selection Method  
Recently, there was a growing interest in the GF-discrepancy-based point selection method [32] due to its balanced distribution of point sets and effective control of error limits. Therefore, to address the issue of inadequate accuracy of high-order moments in the dimension reduction integral method for complex systems, the GF-discrepancy-based point selection method is introduced in this paper.  
Firstly, a brief description of the theory and implementation of the GF-discrepancy-based point selection method is provided. Supposing the representative point set corresponding to the random vector \(\mathbf{X}\) is noted as \(\{x_{q},q=1,2,\,\ldots\,N_{T}\}\), and the GF-discrepancy \(D_{\text{GF}}\) corresponding to the representative point set can be defined as  
\[D_{\text{GF}}=\max_{1\leq i\leq n}\left\{\sup\lvert\underline{E}_{i}(x)-F_{i}( x)\rvert\right\} \tag{9}\]  
in which \(F_{i}(x)\) is the marginal cumulative probability distribution function of \(X_{i}\) and \(\underline{E}_{i}(x)\) is the empirical cumulative probability distribution function of \(X_{i}\), namely  
\[\underline{E}_{i}(x)=\sum_{q=1}^{N_{T}}w_{q}\cdot I(x_{q,i}\leq x) \tag{10}\]  
\[w_{q}=\int_{\Omega_{q}}\rho_{X}(x)\text{dx} \tag{11}\]  
where \(I(\cdot)\) is the indication function, and \(w_{q}\) is the weight coefficient and also called the probability of assignment, which can be estimated by the probability measure in a set of representative regions. For a given representative point set, its representative regions can be adopted by Voronoi cell-based partition technique [33].  
In this paper, using the procedure of GF-discrepancy-based point selection method shown in Equations (9)-(11), the Sobol sequence [34] is employed to generate \(n\)-dimensional independent point set \(I_{n}=\{u_{q,2}=(u_{q,1},\,u_{q,2},\,\ldots,\,u_{q,n}),\,q=1,\,2,\,\ldots,\,N_{T}\}\), which is uniformly distributed in \([0,\,1]^{n}\). The initial point set \(x_{q,l}\) corresponding to \(\mathbf{X}\) can be generated as follows:  
\[x_{q,i}=F_{i}^{-1}\left(u_{q,i}\right),i=1,\ldots,n,\,q=1,\ldots,N_{T} \tag{12}\]  
where \(F_{i}^{-1}(\cdot)\) is the inverse function of \(F_{i}\left(\cdot\right)\).  
To minimize the GF-discrepancy of the selected point set, \(\theta_{q,i}\) shown in Equation (12) is reconstructed twice as follows:  
\[x^{\prime}_{q,i}=F_{i}^{-1}\left(\sum_{k=1}^{N_{T}}\left(\frac{I(x_{k,i}\leq x _{q,i})}{N_{T}}\right)+\frac{1}{2}\cdot\frac{1}{N_{T}}\right) \tag{13a}\] \[x^{\prime\prime}_{q,i}=F_{i}^{-1}\left(\sum_{k=1}^{N_{T}}\left(w_{k}\cdot I \left(x^{\prime}_{k,i}\leq x^{\prime}_{q,i}\right)\right)+\frac{1}{2}w_{q}\right) \tag{13b}\]  
where \(x^{\prime}_{q,i}\) and \(x^{\prime\prime}_{q,l}\) denote the reconstructed point sets.  
The point set \(x^{\prime\prime}_{q,i}\) and weight coefficient \(w_{q}\) obtained by GF-discrepancy-based point selection method are used to calculate raw moments \(M_{z,k}\) of the performance function, namely.  
\[M_{Z,k}=E\left(Z^{k}\right)=\sum_{q=1}^{N_{T}}w_{q}\left(Z\left(x^{\prime}_{q }\right)\right)^{k} \tag{14}\]