## 4 Fault Identification
### States Identification Based on Support Vector Machine  
In Table 1, there are obvious differences in the infrared feature for the mean, std, and D, and setting a threshold seems to be an effective way to identify the conductive state types of the copper rod. However, when we apply this method to the obtained samples, the threshold identifying method cannot achieve an adequate accuracy. In general, SVM is more suitable for applications to limited samples and multidimensional image features [23; 24]. SVM is a classical machine learning model with strict mathematical proof and strong interpretability, which has an excellent generalization ability on limited samples and has been widely used in infrared image recognition [25]. For fault identification in the conductive copper rod, SVM can reasonably predict and estimate the unknown dependency between the infrared feature input and fault label output, which tries to search for the best separating hyperplane in the infrared feature space. This paper uses the soft margin support vector machine with RBF (radial basis function) to accomplish the fault identification task. SVM can tolerate some misclassification samples and avoid overfitting to a certain extent [26]. The original constraint problem for soft margin support vector machines is defined as follows:  
\[\left\{ \begin{array}{cccc}&&\min\frac{1}{2}\|w\|^{2}+C\sum\limits_{i=1}^{m} \xi_{i}\\ &s.t.&y_{i}(w^{T}x_{i}+b)\geq 1-\xi_{i},\quad\xi_{i}\geq 0,\quad i=1,2,\ldots,m \end{array} \right. \tag{13}\]  
where \(x\) is the input sample feature vector, \(y\) is the classification label, \(w\) is the weight distribution of SVM through iterative calculation, \(\xi_{i}\)s the relaxation variable introduced for each sample point, \(C\) is the adjustable penalty coefficient, and \(a\) is the Lagrange coefficient. The dual problem of soft margin support vector machines is defined as follows:  
\[\left\{ \begin{array}{cccc}&&\max\sum\limits_{i=1}^{m}a_{i}-\frac{1}{2}\sum \limits_{i=1}^{m}\sum\limits_{j=1}^{m}a_{i}a_{j}y_{i}y_{j}({x_{i}}^{T}x_{j})\\ &s.t.&\sum\limits_{i=1}^{n}a_{i}y_{i}=0,\quad 0\leq a_{i}\leq C,\quad i=1,2, \ldots,m\end{array} \right. \tag{14}\]  
As a kernel function, with the characteristics of less super-parameters, less calculation, and a strong learning ability, RBF can not only implicitly map sample data points to an infinite dimensional space but also calculate the product of high-dimensional feature vectors in a low-dimensional feature space. RBF is defined as follows:  
\[\kappa(x_{i},x_{j})=\exp(-\gamma\big{\|}x_{i}-x_{j}\big{\|}^{2}),\quad\gamma= \frac{1}{2\sigma^{2}}\ \ >0 \tag{15}\]  
In the SVM classification model, the parameter \(\gamma\) of RBF and the penalty coefficient \(C\) of the soft interval SVM need to be adjusted. The parameter \(\gamma\) affects the distribution of the sample data in the high-dimensional feature space. Moreover, the penalty coefficient \(C\) is used to balance the size of the classification margin and accuracy of the model. To obtain an SVM model with a strong generalization ability, in this paper, the grid search method and k-fold cross validation method are used to identify the optimal super-parameter combination. Figure 15 shows the accuracy distribution map of the SVM model with the 5-fold cross validation method under different super parameter combinations. When \(C^{*}=1.4\) and \(\gamma^{*}=0.1\), the classification accuracy of SVM reaches 92.1%. Finally, this paper selects the above optimal hyper-parameter combination to verify the performance of the SVM model for state identification using the obtained samples. Each state type contains 40 sample data, in which 30 are selected as training sets. The remaining samples form the test sets. After training, the classification accuracy of SVM reaches 90% using the test sets. Table 2 and Figure 16 show the classification results of the models using the test sets. Only three copper rod samples under the abnormal heating condition are misclassified.