## 2 Reliability Analysis Based on the AdaBoost Surrogate Model
### Adaptive AdaBoost Algorithm
#### 2.2.2 AdaBoost Algorithm  
First, the AdaBoost algorithm builds the first simple classifier, \(h_{1}\), based on the original data distribution. Then, Bootstrap produces \(T\) classifiers that are interconnected. Finally, these different classifiers are combined to form a stronger final classifier (strong classifier).  
In theory, as long as each weak classifier's classification ability is better than random guessing (a classification accuracy rate greater than 0.5), the error rate of the strong classifier will converge zero as the number of weak classifiers approaches infinity. Within the  
Figure 1: Flowchart of expanded sample variance and adaptive iteration method.  
framework of the AdaBoost algorithm, different training sets are generated by adjusting the weights of each sample. At the beginning, the weight of each sample is the same, and basic classifier, \(h_{1}(\mathbf{x})\), is trained under this sample distribution. The weights of the samples exhibiting poor \(h_{1}(\mathbf{x})\) scores are elevated, while the weights of properly classified samples are decreased to highlight the incorrect samples. Meanwhile, \(h_{1}(\mathbf{x})\) is assigned a weight according to its error, representing its importance; the lower the error rate, the greater the corresponding weight. The basic classifier is trained again using the new sample weight to obtain the basic classifier, \(h_{2}(\mathbf{x})\), and its weight. Therefore, by analogy, after \(T\) rounds, \(T\) fundamental classifiers and their associated weights can be derived. Finally, \(T\) basic classifiers are added according to the previously calculated weights to obtain the final desired strong classifier.  
Let us assume that the training set sample is denoted as \(T=\{E_{11},E_{12},G_{12},G_{13},X_{c},X_{t},\)\(Y_{c},Y_{t},S_{12},S_{13},\text{d}\}\), and the output weight of the training set in the \(k\)-\(th\) weak learner is \(D(k)=(w_{k1},w_{k2},\ldots w_{km});w_{1i}=\frac{1}{m};i=1,2\ldots m\).  
Let us take a binary classification problem as an example. If the output is \(\{0,1\}\), the weighted error rate of the \(k\)-\(th\) weak classifier, \(G_{k}(x)\), on the training set is \(e_{k}=P(G_{k}(x_{i})\
eq y_{i})=\sum\limits_{i=1}^{m}w_{ki}l(G_{k}(x_{i})\
eq y _{i})\).  
The weight coefficient of the \(k\)-\(th\) weak classifier,\(G_{k}(x)\), is denoted as \(\alpha_{k}=\frac{1}{2}\log\frac{1-e_{k}}{e_{k}}\). An increase in the classification error rate, \(e_{k}\), is associated with a decrease in the corresponding weak coefficient, \(\alpha_{k}\), of the classifier weight. Assuming that the sample set-weight coefficient for the \(k\)-\(th\) weak classifier is \(D(k)=(w_{k1},w_{k2},\ldots w_{km})\), then the sample set-weight coefficient of the corresponding \(k+1\) weak classifier is \(w_{k+1,i}=\frac{w_{ki}}{Z_{k}}\exp(-\alpha_{k}y_{i}G_{k}(x_{i}))\), where \(Z_{K}\) represents a normalization factor, \(Z_{K}=\sum\limits_{i=1}^{m}w_{ki}\exp(-\alpha_{k}y_{i}G_{k}(x_{i}))\). It is apparent from the \(w_{k+1}\) calculation formula that if the \(i\)-\(th\) sample is classified incorrectly, \(y_{i}G_{k}(x_{i})<0\), resulting in the increase in the weight of the sample in the \(k+1\) weak classifier. Should the classification be deemed accurate, the weight in the \(k+1\) weak classifier decreases.  
The weighted voting method is adopted for AdaBoost classification, and the ultimate strong classifier can be expressed as follows: \(f(x)=sign(\sum\limits_{k=1}^{K}\alpha_{k}G_{k}(x))\).