# Improved Bootstrap Method Based on RBF Neural Network for Reliability Assessment
## 3 Bootstrap Methodology and Its Improvement
### Improved Bootstrap Data Expansion Method Based on RBF Neural Network  
The methodology of the bootstrap data expansion method based on the RBF neural network is depicted in Figure 7.  
Figure 6: Schematic of the RBF neural network.  
First, the empirical distribution function expressed in Equation (5) is used to obtain the original sample dataset \(X=[x_{1},x_{2},...,x_{n}]\). The RBF neural network is trained based on the original sample dataset \(X\) and the set of empirical distribution values \(F\). Notably, the effectiveness of this method has been demonstrated in [10]. Setting \(u=5\), the modified empirical distribution function in Equation (6) is then used to generate a set of empirical distributions based on the original sample dataset \(F^{\prime}=F^{\prime}_{n}(x_{1})\), \(F^{\prime}_{n}(x_{2}),\ldots,F^{\prime}_{n}(x_{n})\). As the RBF neural network produces more reliable outputs for inputs which are close to the training samples [25], a neighborhood function \(R_{i}\) is introduced based on the input set. The expanded sample dataset \(X^{*}\) is then obtained using the input set \(S_{j}\) based on \(R_{i}\). The specific implementation steps are as follows.  
1. The original samples \(X=[x_{1},x_{2},...,x_{n}]\) are sorted in descending order to obtain the order statistic of the sample \(X^{\prime}=\Big{[}x_{(1)},x_{(2)},\ldots,x_{(n)}\Big{]}\), where \(x_{(1)}=\underset{1\leq i\leq n}{\min}x_{i}\), \(\ldots\), \(x_{(n)}=\underset{1\leq i\leq n}{\max}x_{i}\). Substituting \(X\) into Equation (5) yields the set of empirical distribution values of \(X\), \(F=[F_{n}(x_{1}),F_{n}(x_{2}),\ldots,F_{n}(x_{n})]\).  
Figure 7: Flowchart of the RBF + bootstrap approach.  
2. RBF neural network training: The RBF neural network is trained by considering \(F_{n}(x_{i})(i=1,2,\ldots,n)\) as the input and \(x_{i}\) as the output of the network \(o_{j}\). The Gaussian radial basis function is used in the network, as shown in Equation (8).
3. The neighborhood function \(R_{i}\) of the set of \(X\) empirical distributions \(F\) is introduced. The input set \(S\) of the RBF neural network is then obtained, and \(X\) is substituted into Equation (6) to obtain the set of \(X\)-corrected empirical distribution values \(F^{\prime}=F^{\prime}_{n}(x_{1}),F^{\prime}_{n}(x_{2}),\ldots,F^{\prime}_{n}(x _{n})\). Let \(f_{i}=F^{\prime}_{n}\big{(}x_{(i)}\big{)}\), and the neighborhood function \(R_{i}\) be \[\left\{\begin{array}{l}R_{1}=\left[f_{1}-\frac{f_{2}-f_{1}}{r},f_{1}+\frac{ f_{2}-f_{1}}{r}\right]\\ R_{i}=\left[f_{i}-\frac{f_{i}-f_{1}}{r},f_{i}+\frac{f_{i}+f_{i}-f_{i}}{r} \right],i=2,\ldots,n-1\\ R_{n}=\left[f_{n}-\frac{f_{n}-f_{n-1}}{r},f_{n}+\frac{f_{n}-f_{n-1}}{r} \right]\end{array}\right.,\] (10) where \(r\) is the neighborhood parameter (\(r\geq 2\)). The input set \(S=[s_{1},s_{2},\ldots s_{n}]\) of the RBF neural network is generated sequentially from the uniform distribution of each neighborhood \(U[R_{i}]\), where \(s_{j}\sim U\big{[}R_{j}\big{]}\).
4. The input set \(S\) is fed into the RBF neural network to obtain the expanded sample \(X^{*}\). The elements of \(S\) are input into the RBF neural network sequentially. When the input is \(s_{j}\), the output is \[o_{j}=w_{1}\cdot h_{1}\big{(}s_{j}\big{)}+w_{2}\cdot h_{2}\big{(}s_{j}\big{)}+ \ldots+w_{m}\cdot h_{m}\big{(}s_{j}\big{)}.\] (11) The set \(X^{*}\) consisting of the RBF neural network outputs \(o_{j}\) is the augmented sample of \(X\).
5. Steps (3) and (4) are repeated \(N\) times to obtain the expanded sample \(X^{*}(k)=\big{[}x_{k1}^{*},x_{k2}^{*},\ldots,x_{kn}^{*}\big{]},(k=1,2,\ldots,N)\) of \(X\).