# An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network
## 3 Calculation Theory of Improved BP-MCS Method
### Construction of Sample Points  
For one-dimensional normal distribution, \([\mu-5\sigma,\;\mu+5\sigma]\) contains almost all sample points in the whole design space. Therefore, when sampling low-dimensional problems (the number of variables \(m<4\)) by the random integer method, the sampling range can be defined as all integer points within \([-5,5]\), and then the sample points can be constructed according to the mean \(\mu\) and standard deviation of variables \(\sigma\), that is:  
\[x_{i}=\mu_{i}+a\sigma_{i} \tag{7}\]  
where: \(a\) is the integer value extracted by random integer method. Since \(a\) is an integer point in \([-5,5]\), the sample points constructed are very evenly distributed in the global design space. However, with the increase of variable dimension, the agglomeration effect of a large number of sample points to the center point becomes more and more obvious. At this time, it is necessary to explore the sampling frequency and extraction range of the random integer method. Considering that the random integer method used to construct sample points is similar to the orthogonal experiment method with multi-factor and multi-level, when the sampling range of random integer method is \([-3,3]\), the orthogonal experiment design with multi-factor 7 level can be carried out. Based on the number of test groups used in the orthogonal test method, when the variable number \(m\) is 2-8, the orthogonal test table can be L\({}_{49}\)(\(7^{8}\)), and the number of test groups is 49. Taking the three-dimensional standard normal distribution as the research object, 49 groups of sample points were sampled by the random integer method and orthogonal test method, respectively, and the distribution of sample points is shown in Figure 2. Figure 2 shows that the spatial distribution of the sample points adopted by the orthogonal test method has the characteristics of stratification, and each factor is orthogonal in pairs and each level has the same number of occurrences. Therefore,it is found that the weights are discontinuous and have the characteristics of hierarchical distribution when calculated by Equation (6). However, the sample points adopted by the random integer method effectively avoid this phenomenon, and the sample points are evenly distributed in the design space and the weights are continuously distributed, which shows that the random integer method is essentially an optimization of the orthogonal test method. The orthogonal test method mainly selects representative sample points to reflect the global design space, so the number of experimental groups can be used as the sampling times of the random integer method. The orthogonal tests of different variable dimensions are shown in Table 1. When the number of variables \(m\) is 9, the orthogonal test table can be L\({}_{64}\)(8\({}^{9}\)), and then the pseudo-horizontal method can be used to replace the excess level with the existing level. Considering that the sampling times of the random integer method are related to the dimensionality of random variables, and the number of experimental groups of the orthogonal test method should be included, the sampling times of random integer method can be \(10\times m\) groups. Since normal distribution can be normalized into standard normal distribution, this paper takes \(m\)-dimensional standard normal distribution as the research object and uses MCS to carry out random sampling of different orders of magnitude. The obtained sample point weight orders are shown in Table 2. At the same time, the random integer method is used to carry out the statistics of weight orders in different extraction ranges, and the specific results are shown in Table 3.  
As can be seen from Tables 2 and 3, when using the random integer method to construct sample points, for low-dimensional problems (\(m<4\)), the weight order of sample points constructed in \([-5,5]\) can reach the minimum order of magnitude required when MCS is used to extract \(10^{7}\) times.