# Improved Bootstrap Method Based on RBF Neural Network for Reliability Assessment
## 5 Example Analysis  
Retrieve the maintenance records for seven CNC machines (designated as K1, K2,..., K7) operating under similar conditions within a single factory, spanning three years, to acquire 61 instances of failure data for this specific model (Table 3). Point and interval estimation of the shape and scale parameters are performed using the maximum likelihood estimation method, bootstrap method, RBF + bootstrap method, and modified RBF + bootstrap method. 1. Using the maximum likelihood estimation to estimate the parameters of the Weibull distribution for the original data yields \(\hat{m}=1.2694\) and \(\hat{\eta}=1204.7\), and the reliability function is as follows: \[R(t)=\exp\left[-\bigg{(}\frac{t}{1204.7}\bigg{)}^{1.2694}\right].\] (25) Then, \(MTBF\) is \[MTBF=\int_{0}^{\infty}R(t)dt=\int_{0}^{\infty}\exp\left[-\bigg{(}\frac{t}{120 4.7}\bigg{)}^{1.2694}\right]dt=1118.20\;\text{h}.\] (26)
2. The conventional bootstrap method is used to expand the original data, and sampling is performed 1000 times, resulting in the expanded samples \(X^{*}(k)=\big{[}x_{k1}^{*},x_{k2}^{*},\ldots,x_{k61}^{*}\big{]}\), \((k=1,2,\ldots,1000)\). The overall distribution of \(X^{*}\) is shown in Figure 8a. Solving Equation (14) yields \(\big{[}\hat{m}_{k}^{*},\hat{\eta}_{k}^{*}\big{]},k=1,2,\ldots,1000\), and the average estimate is the following:  
\[\left\{ \begin{array}{l}\overline{m}^{*}=\frac{1}{1000}\sum\limits_{k=1}^{10 00}\widehat{m}_{k}^{*}=1.3199\\ \overline{\eta}^{*}=\frac{1}{1000}\sum\limits_{k=1}^{1000}\widehat{\eta}_{k}^ {*}=1208.4\end{array} \right. \tag{27}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number** & \multicolumn{6}{c}{**Time between Failures (h)**} \\ \hline
**K1** & 63.5 & 215.5 & 302 & 639.5 & 945.5 & 1264.25 \\ \hline  & 2332.5 & 2591.5 & 2894 & & & \\ \hline
**K2** & 178 & 318.08 & 374.5 & 645.5 & 1240.42 & 1246.58 \\ \hline  & 1337 & 1419.5 & 2154 & & & \\ \hline
**K3** & 215.3 & 230.17 & 837.33 & 838.67 & 1017.27 & 1486 \\ \hline  & 2491.17 & 2842.33 & & & & \\ \hline
**K4** & 537.25 & 862.38 & 953.67 & 1027.67 & 1045.5 & 1274 \\ \hline  & 1584 & 2449.25 & 3062.08 & & & \\ \hline
**K5** & 194 & 271.5 & 399 & 913 & 1040 & 1873.5 \\ \hline  & 2304.5 & 3062.5 & & & & \\ \hline
**K6** & 141.5 & 239.5 & 241.83 & 397.67 & 454.5 & 1382.5 \\ \hline  & 2027.5 & 2312 & 2591.83 & & & \\ \hline
**K7** & 153.5 & 184 & 186 & 409 & 639 & 655.5 \\ \hline  & 686 & 1037 & 1375 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Equipment failure data. The parameter distribution obtained by solving Equation (18) for \(MTBF_{k}\) is shown in Figure 8b. A mean value of \(MTBF_{mean}=1114.97\) h is obtained, and the 95% confidence interval of \(MTBF\) is (1099.51, 1130.47), using the bias correction method. 3. The original data are expanded using the conventional bootstrap method and the improved bootstrap combined with the RBF neural network method. The \"newrb\" function in MATLAB (v2018b, MathWorks, Inc., Natick, MA, USA) is used to construct the RBF radial basis neural network. The network performance targets, expansion constants, and number of neurons, respectively, are set as \([\text{goal, spread,maxNeuron}]=[0,1,25]\). The calculation process is shown in Figure 7, and the results converge to yield the expanded samples of \(X\), \(X^{*}(k)=\left[x_{k1}^{*},x_{k2}^{*},\ldots,x_{k61}^{*}\right]\), and \((k=1,2,\ldots,1000)\). The average estimates of the Weibull parameters obtained using the conventional bootstrap method + RBF neural network are \([\overline{m}_{TR}*,\overline{\eta}_{TR}*]=[1.2773,1206.5]\), with \(MTBF_{TR}=1118.32\) h. The 95% confidence interval of \(MTBF_{TR}\) is (1114.37, 1121.86), using the bias correction method. The average estimates of the Weibull parameters obtained using the improved bootstrap method + RBF neural network are \([\overline{m}_{BR}*,\overline{\eta}_{BR}*]=[1.2742,1168.2]\), with \(MTBF_{BR}=1083.41\) h. The 95% confidence interval of the \(MTBF_{BR}\) obtained using the bias correction method is (1080.13, 1089.15).