## 2 Detection Method of Microseisms in Coal Mines
### Microsesium Detection Method in Coal Mine  
#### 2.2.1 Regression Solution of Microseism Detection Signal in Coal Mine  
Support vector machine (SVM) is a machine learning method based on the Vapnik-Chervonenkis (VC) dimension theory of statistical learning theory and the structural risk principle [14] first proposed by the research group led by [PERSON] in 1995, which can be used for pattern classification and nonlinear regression [15]. The SVM method is used to \"raise the dimension\" of the spatiotemporal feature sample points of the coal mine microseisms obtained above, that is to map the nonlinear spatiotemporal feature data of coal mine microseisms in this space to a linearly separable high-dimensional space, so as to solve the high-dimensional nonlinear problem in the sample space [8]. The support vector machine method is applied to regression problems, which is called support vector regression (SVR). The temporal and spatial characteristics of coal mine microseisms were extracted in the previous article. In order to detect coal mine microseisms, it is necessary to combine the coal mine microseisms' characteristics with the support vector machine method, so as to realize the detection of coal mine microseisms. The specific method is to set the \(n\)'s training set of the time-space characteristics of microseisms in coal mines \(X=\left[x_{1},x_{2},\cdots,x_{n}\right]\), and an output vector with coal mine microseism detection results \(Y=\left[y_{1},y_{2},\cdots,y_{n}\right]\); the input vector \(x\) through nonlinear mapping \(\phi\big{(}P\big{(}s_{j}|s_{i}\big{)}\big{)}\) mapsto a high-dimensional feature space, and then, construct an optimized linear regression function in this feature space:  
\[f(x)=w\phi\big{(}P\big{(}s_{j}|s_{i}\big{)}\big{)}+C_{k}b \tag{8}\]  
where \(f(x)\) is used to obtain the regression function of microseism detection in coal mines; \(w\) is the weight vector; \(b\) is an offset item. Taking the above time-space characteristics of coal mine microseisms as constraints, ignoring the error of the data points relative to the nonlinear-feature-space-fitting curves, and introducing a relaxation variable, i.e., error tolerance \(\lambda_{i}\) and \(\lambda_{i}^{*}\), then the constrained optimization problem of Equation (8) is concretely expressed as:  
\[\begin{split}\underset{w,b,\lambda_{i},\lambda_{i}^{*}}{\text{ minize}}\bigg{\{}\frac{1}{2}\|w\|^{2}+D\sum\limits_{i=1}^{n}\big{(}\lambda_{i}+ \lambda_{i}^{*}\big{)}\bigg{\}}\\ \text{subject to}\bigg{\{} \begin{array}{l}y_{i}-w\phi(x_{i})-b \leq\varepsilon+\lambda_{i}\\ w\phi(x_{i})+b-y_{i}\leq\varepsilon+\lambda_{i},i=1,2,\cdots,n\\ \lambda_{i}\geq 0,\lambda_{i}^{*}\geq 0\end{array}  \tag{9}\]  
where \(\frac{1}{2}\|w\|^{2}\) is a rule item; \(D\) is the penalty factor; \(\varepsilon\) is the precision parameter. The constraint optimization problem of the coal mine microseism in Equation (9) is transformed into a dual-problem by introducing a Lagrange function, and the solution of Equation (8) is obtained by solving the dual-problem, which is the solution of the coal mine microseism detection results:  
\[f^{\prime}(x)=\sum\limits_{i=1}^{n}\underset{w,b,\lambda_{i},\lambda_{i}^{*}}{ \text{minize}}\,f(x)(a_{i}-a_{i}^{*})\,K(x,x_{i})+b \tag{10}\]  
where \(a_{i},a_{i}^{*}\) is the Lagrange multiplier, \(a_{i},a_{i}^{*}\in[0,D]\), and \(K(x,x_{i})\) is a kernel function. Since the radial basis function is applicable regardless of the sample size and dimension and has a wide convergence domain, there is only one parameter to be optimized. \(g\)'s algorithm is simple and easy to calculate, so the radial basis kernel function is selected for calculation. For the SVR of the RBF kernel function, penalty factor \(D\) and nuclear parameters \(g\) are the main parameter that affects the performance of SVR. The accuracy and reliability of the SVR prediction results depend on \((D,g)\)'s best choice [16].