# A Novel Fault Diagnosis Method for TE Process Based on Optimal Extreme Learning Machine
## 2 Materials and Methods
### Coyote Optimization Algorithm
#### 2.2.1 Algorithm Flow  
The following is how COA replicates the birth, development, death, and movement of coyote populations. Step 1: Set the number of coyote groups \(N_{p}\), the number of coyote individuals per group \(N_{c}\), the dimension \(D\), termination condition nfevalmax, and other parameters. Step 2: Initialize the coyote pack at random; the \(c\)th individual inside the \(p\)th pack at time \(t\) is defined as  
\[SoC_{c,j}^{p,t}=lb_{j}+r_{j}(ub_{j}-lb_{j}) \tag{1}\]  
\[SoC_{c}^{p,t}=(SoC_{c,1}^{p,t},SoC_{c,2}^{p,t},\dots,SoC_{c,D}^{p,t}) \tag{2}\]where \(ub_{j}\) and \(lb_{j}\) are the \(j\)th dimension's upper and lower bounds, respectively, and \(r_{j}\) is a randomly generated real number in the range [0, 1]. Step 3: Assess the fitness of coyotes. \[Fit_{c}^{p,t}=FitFunc(SoC_{c}^{p,t}) \tag{3}\]  
Step 4: Coyotes may split away from their original pack or are banished, resulting in a splinter group.A population shift's probability is defined as  
\[P_{e}=0.005N_{c}^{2},N_{c}\leqslant 14 \tag{4}\]  
Step 5: Find the head coyote in the present pack \(alpha^{p,t}\),and calculate the current coyote cultural trend of the coyote pack \(clut^{p,t}\)  
\[alpha^{p,t}=\{SoC_{c}^{p,t}||arg_{\{c=1,2,...N_{c}\}}minFitFunc(SoC_{c}^{p,t})\} \tag{5}\]  
\[clut_{j}^{p,t}= \begin{cases}O_{\frac{N_{c}+1}{2},j^{t}}^{p,t}N_{c}\text{ {is odd}}\\ O_{\frac{N_{c}}{2},j}^{p,t}+O_{\frac{N_{c}+1}{2},j^{t}}^{p,t}\text{ otherwise}\end{cases}  \tag{6}\]  
where \(O_{\frac{N_{c}+1}{2},j}^{p,t}\) denotes that when \(N_{c}\) is odd, the median of all \(p\)-groups within the \(p\)-group at moment \(t\) the median of the \(j\)th dimensional variable of coyotes. Step 6: In genetics, birth and death events are modeled. The birth of a baby coyote (\(pup^{p,t}\)) was written as a mixture of the social standing of both parents (randomly picked) plus environmental factors, and coyote age (in years) was written as \(age_{c}^{p,t}\). \[pup_{j}^{p,t}= \begin{cases}SoC_{m_{1},j}^{p,t},rand_{j}<P_{s}\text{ \ or \ }j=j_{1}\\ SoC_{m_{1},j}^{p,t},rand_{j}<P_{s}+P_{\alpha}\text{ \ or \ }j=j_{2}\\ R_{j},rand_{j},\text{otherwise}\end{cases}  \tag{7}\]  
where \(m_{1}\), \(m_{2}\) are random coyotes from within the \(p\)-pack, \(j_{1}\), \(j_{2}\) are two random dimensions of the problem, and \(R_{j}\), \(rand_{j}\) are random numbers within [0, 1] generated by uniform probability. The discrete probability (\(P_{s}\)) and the association probability (\(P_{a}\)) affect the cultural diversity of individuals in a coyote pack, defined as  
\[P_{s}=\frac{1}{D},P_{a}=\frac{1-P_{s}}{2} \tag{8}\]  
Assume that \(\omega\) means that the coyotes in the group are not as well adapted as the pups, and \(\phi\) is the number of coyotes in the group at the time. if \(\phi\) is 1 and \(\omega\) holds, i.e., the number of coyotes in a group is If \(\phi\) is 1 and \(\omega\) holds, i.e., the number of coyotes in a group is 1 and the adaptability of pups is higher than that of only one coyote, then the pups survive. If \(\phi\) is greater than 1 and \(\omega\) holds, the pups survive and the only coyote in the group dies; if \(\phi\) is greater than 1 and \(\omega\) holds, the pups survive and the only coyote in the group dies. In addition, \(\omega\) holds, the pup survives and the oldest coyote in the group dies; In all other cases, the pups died. Step 7: Calculate the effect of cultural trends in the head wolf and the pack on the renewal of individuals within the coyote pack corresponding to the current moment \(\delta_{1}\),\(\delta_{2}\), with  
\[\delta_{1}=alpha^{p,t}-SoC_{cr1}^{p,t}\text{ \, \ }\delta_{2}=cult^{p,t}-SoC_{cr2}^{p,t} \tag{9}\]  
where \(cr1\) and \(cr2\) are the current pack's random coyotes, respectively. Figure 2: Framework of the proposed fault diagnosis method.