## 2 The Proposed Prediction Framework
### Introduction of Support Vector Machine  
A support vector machine (SVM) is a supervised machine learning technique used to separate classes [30; 31; 32; 35; 36]. SVM is widely used in tunneling research, such as ultimate strength prediction [37], geological prediction [38], and TBM performance prediction [39]. Similar to the logistic regression method, SVM is based on the linear function \(\omega^{T}x+b\), but the output of SVM is a classification judgement, rather than a probability. If \(\omega^{T}x+b\) is positive, SVM prediction belongs to the positive class; If \(\omega^{T}x+b\) is negative, SVM prediction belongs to the negative class. Because SVM is a linear classification model and the cutter replacement evaluation criterion is nonlinear with respect to tunneling data, classification using SVM directly will obtain poor results. The most important reason for the current wide application of SVM is the use of the kernel technique. The basis of the kernel technique is that many machine learning algorithms can be written as dot products between samples. For example, a linear function in SVM can be rewritten as:  
\[\omega^{T}x+b=b+\sum_{i=1}^{m}\alpha_{i}x^{T}x^{(i)} \tag{4}\]  
where, \(x^{(i)}\) is a training sample of TBM operational data and prior geological information, and \(\alpha\) is coefficient vector. After replacing \(x\) with the output of eigenfunction \(\varnothing(x)\) and replacing the dot product by the kernel function \(k(x,x^{(i)})=\varnothing(x)\cdot\varnothing\Big{(}x^{(i)}\Big{)}\), we can use the following function to make the cutter replacement prediction \(f(x)\):  
\[f(x)=b+\sum_{i}\alpha_{i}k\Big{(}x^{T}x^{(i)}\Big{)} \tag{5}\]  
This function is non-linear about \(x\), but it is linear about \(\varnothing(x)\). The relationship between \(\alpha\) and \(f(x)\) is also linear. \(f(x)\) preprocesses all the inputs with \(\varnothing(x)\), and then learns the linear model in the new transformation space. The kernel technique enables SVM to learn nonlinear models (functions of \(x\)) using convex optimization techniques that ensure effective convergence. Because \(\varnothing\) is fixed, only \(\alpha\) needs optimizing, which means that the optimization algorithm can treat the decision functions as linear functions in different spaces and build hyperplanes to separate the data sets. In our paper, a Bayesian optimization algorithm is used to optimize hyper-parameters. In most cases, even though \(\varnothing(x)\) is hard to calculate, \(k(x,x^{\prime})\) is easy to calculate. The Gaussian kernel, also known as the radial basis function (RBF), is provided by:  
\[k\big{(}x,x^{\prime}\big{)}=exp\bigg{(}-\frac{1}{2}\big{(}x-x^{\prime}\big{)} ^{T}\sum(x-x^{\prime}\big{)}^{-1}\bigg{)} \tag{6}\]  
where \(\sum\) is the covariance of each feature in the observations, a p-dimensional matrix. Because \(\sum\) is ball-shaped, the kernel function of TBM operational data and prior geological information is:  
\[k\big{(}x,x^{\prime}\big{)}=exp\bigg{(}-\frac{\|x-x^{\prime}\|^{2}}{2\sigma^{ 2}}\bigg{)},\ \sigma_{j}=\sigma,\forall j \tag{7}\]  
where \(\sigma_{j}\) is the characteristic length scale of the \(j\)th feature. Assuming that the value of the category label is 0 or 1, which represents negative samples (need replacement) and positive samples (normal operation), the optimal classification hyperplane established by a sample \(y_{i}\) for SVM conforms to the following formula:  
\[y_{i}\Big{(}W^{T}X+b\Big{)}>0,\ \ i=1,2,\ldots,n \tag{8}\]where  
\[\widetilde{\gamma}=y_{i}\left(W^{T}X+b\right)\,,i=1,2,\ldots,n \tag{9}\]  
is called function distance. To solve the coefficients, the objective function of SVM optimization is:  
\[\max\left(\gamma=\frac{\left|W^{T}X+b\right|}{\left\|\omega\right\|}\right) \tag{10}\]  
When \(\widetilde{\gamma}=1\), the distance between the support vector and the optimal classification hyperplane is \(\frac{1}{\left\|\omega\right\|}\) and the distance between two support vectors is \(\frac{2}{\left\|\omega\right\|}\).