## 3 Methodology  
### Iajs-Hkrvm Model  
#### 3.1.1 Hybrid Kernel Relevance Vector Machine  
The RVM model is a supervised learning method based on a sparse Bayesian framework. Compared with SVM, the RVM model has stronger sparsity and can ensure a good generalization performance with fewer training samples, and its kernel function does not need to meet Mercer conditions. The principle of the RVM model is as follows: the sample dataset obtained by numerical simulation is \(S=\{\mathbf{x}_{n},\,\mathbf{t}_{n}\mid\,n=1,2,\ldots,N\}\), where \(N\) is the number of samples in the dataset, \(\mathbf{x}_{n}\) is the input vector, and \(\mathbf{t}_{n}\) is the output vector. Figure 1: Research framework. Assuming that the prediction target is an independently distributed data sample containing noise, the prediction target is shown in Equation (1). \[\left\{ \begin{array}{l}t_{n}=y(x_{n};\mathbf{\omega})+\xi_{n}\\ y(\mathbf{x};\mathbf{\omega})=\sum\limits_{n=1}^{N}\omega_{n}K(\mathbf{x},x_{n})+\omega_{0} \end{array} \right. \tag{1}\]  
where \(\mathbf{\omega}=\left\{\omega_{n}\right\}_{n=0}^{N}\) is the model weights vector, \(\xi_{n}\) is the independent Gaussian noise that satisfies distribution \(N\) (\(0\), \(\sigma^{2}\)), and \(K(\mathbf{x},x_{n})\) is the kernel function. The training process of the RVM model is described as follows: the initial values of the hyperparameters \(\alpha\) and \(\sigma^{2}\) are set in advance and calculated through several iterations until all parameters meet the convergence conditions. After the iteration is terminated, the hyperparameters are updated to \(\mathbf{\alpha}_{\text{MP}}\) and \(\sigma_{MP}^{2}\)[23]. When the sample to be predicted is \(x^{\star}\), the distribution of the prediction result is shown in Equation (2). \[p\left(t^{\star}|\mathbf{t},\mathbf{\alpha}_{MP},\sigma_{MP}^{2}\right)=\int p\left(t ^{\star}\left|\mathbf{\omega},\sigma_{MP}^{2}\right)p\left(\mathbf{\omega}\right|t, \mathbf{\alpha}_{MP},\sigma_{MP}^{2}\right)\text{d}\mathbf{\omega}=N\left(t^{\star} \left|y^{\star},\left(\sigma^{\star}\right)^{2}\right.\right) \tag{2}\]  
where \(y^{\star}=\mu^{\text{T}}\varphi(x^{\star})\), \((\sigma^{\star})^{2}\)= \(\sigma_{MP}^{2}+\varphi(x^{\star})^{\text{T}}\Sigma\varphi(x^{\star})\), \(\mu\) is the mean of the Gaussian distribution, \(\Sigma\) is the variance of the Gaussian distribution, \(\varphi\) is the matrix of the kernel function, \(\mathbf{\alpha}_{\text{MP}}\) and \(\sigma_{MP}^{2}\) are the hyperparameter estimate value corresponding to the maximum marginal likelihood function obtained by the maximum likelihood method, and \(y^{\star}\) is the prediction value. The choice of kernel function and kernel parameter of the RVM model will affect the prediction accuracy. When the sample data features are related to physical laws with deep nonlinear fluctuation characteristics, it is difficult for the single-kernel RVM model to obtain the deep nonlinear fluctuation features of the sample. The local resistance characteristics of water transmission tunnel maintenance ventilation are complicated, which are affected by the structural parameters, ventilation parameters, and other parameters, and are related to complex turbulent movement in the local area and have strong nonlinear fluctuation characteristics. Therefore, the single-kernel RVM model is difficult to be able to achieve the efficient and accurate prediction of the local resistance coefficient of maintenance ventilation. In this paper, a Gaussian kernel function with strong local learning ability and Sigmoid kernel function with strong global generalization ability are combined in a weighted way to establish the HKRVM model for predicting the local resistance coefficient of water transmission tunnel maintenance ventilation. The expression of the hybrid kernel function is shown in Equation (3).