# A Novel Fault Diagnosis Method for TE Process Based on Optimal Extreme Learning Machine
## 2 Materials and Methods
### Coyote Optimization Algorithm
#### 2.2.2 Bernoulli Shift Coyote Optimization Algorithm  
This paper improves the COA algorithm in terms of chaotic sequences, adding Bernoulli shift chaotic interference, shifting some individuals in the population to generate new individuals, and increasing the population's diversity in order to improve the algorithm's convergence speed, avoid the algorithm falling into local optimum as much as possible, and increase the population's diversity. The Bernoulli shift coyote optimization method BCOA is suggested, and the algorithm's specific implementation is detailed in detail with stages.  
The birth of a coyote is the single circumstance that affects the population's diversity in the COA algorithm, and individual coyote genes may be inherited from one parent or created at random. The search for optimal solutions is limited as a result, and population variety is not guaranteed. Before the population change operation, a chaotic Bernoulli shift is established, and the \(1/N_{c}\) people with low fitness at the time are replaced as beginning values into the Bernoulli shift to produce new individuals and replace them.Simultaneously, the execution probability \(r\) for the chaotic interference mechanism is determined and integrated with the standard weight lowering approach to define it as a linear decreasing function, taking into account the balance between global and local performance of the algorithm. The Bernoulli shift, according to the literature [50; 51], has improved traversal uniformity and optimization search efficiency, and its formulation is  
\[x_{n+1}=(2x_{n})\ mod\ 1 \tag{12}\]  
Set the threshold \(R\), calculate the chaotic perturbation execution probability \(r\), and have  
\[r=r_{max}-(r_{max}-r_{min})\frac{k}{T_{max}} \tag{13}\]  
where \(T_{max}\) is the maximum number of iterations. The detection mechanism is used to find \(1/N_{c}\) poorly adapted individuals are substituted into the Tent mapping as initial values and an equal number of new individuals are generated to replace the original individuals.  
Since the range of \(\overline{SoC}_{c,j}^{t}\) in the mapping is between [0, 1], and the COA algorithm The individual \(SoC_{c,j}^{p,t}\) in the mapping is different from the individual xp in the COA algorithm, so a variable conversion is needed.  
\[\overline{SoC}_{c,j}^{t}=\frac{SoC_{c,j}^{p,t}-lb_{j}}{ub_{j}-lb_{j}},j=1,2, \ldots,D. \tag{14}\]where \(ub_{j},lb_{j}\) are the upper and lower bounds of the \(j\)th dimensional variable of the \(p\)-group at time \(t\), respectively, \(SoC_{c,j}^{p,t}\) is the variable in the \(j\)th dimension of the individual at time \(t\), \(\overline{SoC}_{c,j}^{t}\) is the dimensional variable of the individual at time \(t\) is the dimensional variable corresponding to the \(j\)th dimension of the individual at time \(t\) after the transformation of the Tent mapping.  
Use the Tent mapping expression to turn Equation (14) into a chaotic sequence of variables column (\(i=1,2,\ldots,N_{c},m=1,2,\ldots,C_{max}\)), where \(C_{max}\) is the maximum number of iterations of the chaotic search.  
Using the following equation to map \(SoC_{c}(m)\) to the original solution space, a new individual is generated \(newSoC_{c}\)  
\[newSoC_{c,j}^{p,t}=SoC_{c,j}^{p,t}+\frac{lb_{j}-ub_{j}}{2}(2\hat{\pi}_{i,j}(m) -1) \tag{15}\]