# Research on a Coal Seam Gas Content Prediction Method Based on an Improved Extreme Learning Machine
## 2 Development of Prediction Models for Coal Seam Gas Content
### Development of a Gas Content Prediction Model Based on the GASA-BPNN Algorithm
If so, then terminate the process and proceed to steps (7)-(11). Otherwise, increment \(K\) by 1 and proceed to step (12). 7. BPNN Initialization: The solution obtained from step (6) is assigned to the BPNN. The maximum number of training iterations is set to H = 800, and the iteration counter is initialized to \(S=0\). 8. The input dataset is processed by propagating the data forward through the layers of the BPNN. The input for the _f_th neuron in the hidden layer is computed using Equation (3), and the input for the _j_th output node is computed using Equation (4). \[\gamma_{f}=\sum_{v=1}^{5}v_{vf}x_{v}\] (3) where \(\gamma_{f}\) represents the input of the hidden layer neuron \(f\) and \(v_{vf}\) represents the weight of the connection between the input layer neuron \(v\) and the hidden layer neuron \(f\). \[\beta_{j}=\sum_{f=1}^{12}w_{\beta f}b_{f}\] (4) where \(\beta_{j}\) is the input of output node \(j\); \(w_{f}\) is the weight of the connection between intermediate layer node \(f\) and output layer node \(j\); and \(b_{f}\) is the output of intermediate layer node \(f\). 9. The BPNN weights and thresholds are updated by backpropagating the MSE to each node, and the parameters are adjusted using the error-adaptive learning rate gradient descent method described in Equation (5). If the error approaches the target value with less fluctuation, then the model training direction is correct, and the learning rate can be increased. However, if the error increases beyond the allowed range, then the model training direction is incorrect, and the learning rate should be reduced. \[\mu(S+1)= \begin{cases}S_{in}\mu(S),E(S+1)<E(S)\\ S_{de}\mu(S),E(S+1)>E(S)\end{cases} \] (5) where \(\mu(S+1)\) is the learning rate when the number of iterations is \(S\)+1; \(\mu(S)\) is the learning rate when the number of iterations is \(S\); \(S_{in}\) is the incremental coefficient; \(S_{de}\) is the decremental coefficient; \(E(S+1)\) is the error when the number of iterations is \(S+1\); and \(E(S)\) is the error when the number of iterations is \(S\). Figure 8: Schematic diagram of the sample division. 10. BPNN stopping criterion: Verify whether the termination criterion has been met, where the minimum error is defined as the stopping condition. If the criterion is satisfied, then the network completes its learning phase, and the model is established, advancing to step (11). Otherwise, examine if H is equal to \(S\). If true, then return to step (7). Otherwise, increase \(S\) by one and go back to step (8). 11. The performance evaluation of the model involves testing the model on the test set. The evaluation metrics used to assess the model's performance are the MSE, iteration number, and relative prediction error. 12. The selection process in the GA involves the use of the roulette selection method, which serves to screen individuals based on their fitness values. Specifically, for an individual \(x_{u}\) with a fitness value of \(f_{u}\), the probability of \(x_{u}\) being selected is given by \(p_{u}=f_{u}/\sum_{j=1}^{N}f_{u}\). 13. The process of GA crossover involves randomly selecting two individuals, \(x_{1}\) and \(x_{2}\), from the population P\({}_{K}\), and generating new individuals, \(x_{1}^{\prime}\) and \(x_{2}^{\prime}\), through the arithmetic crossover operation outlined in Equation (6). \[ \begin{cases}x_{1}^{\prime}=\lambda_{1}x_{1}+\lambda_{2}x_{2}\\ x_{2}^{\prime}=\lambda_{1}x_{2}+\lambda_{2}x_{1}\end{cases} \] (6) where \(\lambda_{1}+\lambda_{2}=1\), \(\lambda_{1}>0\), and \(\lambda_{2}>0\). 14. For the GA variation, an individual genotype X = x\({}_{1}\)x\({}_{2}\)...\(x_{b}\)...\(x_{s}\) is randomly selected, and the genetic operation is performed at the mutation point \(x_{b}\) using Equation (7). \[x_{b}^{\prime}= \begin{cases}x_{b}+\Delta(K,UB-x_{b}),r\leq 0.5\\ x_{b}-\Delta(K,x_{b}-LB),r>0.5\end{cases} \] (7) Here, UB and LB denote the upper and lower boundary values of the variable \(x_{b}\), respectively. \(r\) represents a random number, and \(K\) denotes the evolutionary algebra.