## 4 MLE MMM Modeling
### MLE MMM Modeling  
The six MMM parameters above contain damage information from different levels. On the basis of the experiment data, these characteristic values are divided into three defect levels, which are normal state, hidden crack state and macroscopic crack state. Considering X-ray detection was taken as the basis comparison of MMM testing experiments, the three defect levels are defined, compared with the current quantitative standards of X-ray detection NB/T 47013.2-2015 and SY/T 4109-2020. In the X-ray detection standards, level I is defined as normal and no defects, level II is defined as the defect length is less than T/3, and level III is specified as the defect length is less than 2T/3, in which T is the nominal thickness of the base metal. The normal state in this paper is consistent with level I of the X-ray detection standard. The hidden crack state, where cracks are invisible to the naked eye and less than 2 mm (T/3) in length, is equivalent to level II of the X-ray detection standard. The macroscopic crack state, where cracks are visible to the naked eye and greater \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\Delta H_{p}(x)\) & \(G_{x}{}^{\mathrm{max}}\) & \(Z_{x}{}^{\mathrm{max}}\) & \(\Delta H_{p}(y)\) & \(G_{y}{}^{\mathrm{max}}\) & \(Z_{y}{}^{\mathrm{max}}\) \\ \hline
8.046 & 1.9236 & 0.1421 & 6.9817 & 1.8659 & 0.2482 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of optimized bandwidth. than 2 mm in length, corresponds to level III in the X-ray detection standard. Here, 2 mm only refers to the length of the fatigue crack generated from the end of the prefabricated lack of penetration. The classification standard in this paper is stricter than the X-ray detection standard in order to reflect the advantages of MMM in early damage detection. From the view of MLE [25], if the total \(X\) is discrete, its distribution law is \(P[X=x]=p(x;\theta)\), \(\theta\in\)\(\in\)\(\Theta\). Here, \(\theta\) is the estimated parameter, \(\Theta\) is the possible range of \(\theta\). \(x_{1}\), \(x_{2}\),..., \(x_{n}\) are set as sample values of \(X_{1}\), \(X_{2}\),..., \(X_{n}\). Then, the probability of the incidents \(\{X_{1}=x_{1}\), \(X_{2}=x_{2}\),..., \(X_{n}=x_{n}\}\) is  
\[L(\theta)=L(x_{1},x_{2},..,x_{n};\theta)=\prod_{i=1}^{n}p(x_{i};\theta),\theta\in\Theta \tag{9}\]  
where \(L(\theta)\) is the likelihood function. The MLE estimation models based on MMM characteristic parameters are:  
\[L_{x}=\prod_{i=1}^{n}p_{x}(x_{i})=p\big{(}\Delta H_{p}(x)\big{)}\cdot p(G_{x} ^{\max})\cdot p(Z_{x}^{\max}) \tag{10}\]  
\[L_{y}=\prod_{i=1}^{n}p_{y}(x_{i})=p\big{(}\Delta H_{p}(y)\big{)}\cdot p\Big{(} G_{y}^{\max}\Big{)}\cdot p\Big{(}Z_{y}^{\max}\Big{)} \tag{11}\]  
where \(L_{x}\) and \(L_{y}\) are the MLE values of the tangential and normal signals, respectively. \(p_{x}(x_{i})\) and \(p_{y}(x_{i})\) are the probability of \(x_{i}\) based on optimized bandwidth kernel density estimation, respectively. The MLE probabilities of three defect levels, which are, normal state, hidden crack and macroscopic crack, are shown in Figure 6. The mathematical expectation values of the MLE probability are calculated through mathematical statistics. In this paper, 100 specimens were divided into 10 groups during the fatigue tensile experiments. Each specimen was scanned and the MMM signals were recorded every 20 kilocycles, and 6 characteristic values of each signal curve were extracted. On average, more than 50 sets of data were recorded for each specimen. In total, over 30,000 characteristic values were recorded, along with the corresponding fatigue cycles and defect levels. The Gauss kernel probability density functions of the three defect levels were established based on the database of 30,000 characteristic values. Then, the mathematical expectation values of MLE probability in Table 4 were obtained by integrating the Gauss kernel probability density functions for each defect level. Figure 6: MLE probability of three defect levels.