## 3 Results and Analysis
### SVM-Based Prediction Model of Improved Loess Permeability
#### 3.3.1 Construction of the Model  
Support vector machine regression SVR is an application of Support Vector Machines in the field of function regression, which is a further extension of using a support vector machine SVM (Support Vector Machine) to solve the regression fitting problem. First, by analyzing several soil parameters, the indicators which are more closely related to the  
Figure 12: Degree of association of each influence factor.  
predicted quantity (i.e., the permeability coefficient) are selected. In this paper, they are admixture type, C\({}_{c}\), d\({}_{60}\), and dry density; then, the selected multiple indicators are used as the input of the SVR model, and the predicted quantity is used as the output of the SVR model. An SVR prediction model with a multivariate input and univariate output is established. The specific establishment process is as follows:  
Suppose the number of training samples is \(l\). Then, the set of training data is \((x_{i},y_{i})\), \(x_{i}=x_{1},x_{2}\ldots,l\)], where \(x_{i}\) is the input vector of the \(i\)th training sample. \(x_{i}=\left[x_{i}^{1},x_{i}^{2},\cdots,x_{i}^{N}\right]^{T}\) represents the N training parameters of the \(i\)th training sample, and \(y_{i}\) is the penetration coefficient of the ith training sample. Based on the mapping from the low-dimensional space to the high-dimensional feature space, the following linear regression function is considered [22]:  
\[f(x)=w\Phi(x)+b \tag{1}\]  
where the following hold: \(w\)--high-dimensional feature space weight vector; \(\Phi(x)\)--nonlinear mapping function; \(b\)--threshold value.  
To represent the effectiveness of the regression, the \(\varepsilon\) linear insensitive loss function is introduced, as shown in the following equation:  
\[L(f(x),y,\varepsilon)=\left\{\begin{array}{cc}0&\mid y-f(x)\mid\leq \varepsilon\\ \mid y-f(x)\mid-\varepsilon&\mid y-f(x)\mid>\varepsilon\end{array}\right. \tag{2}\]  
where the following hold: \(f(x)\)--the predicted value of the improved loess permeability coefficient returned by the regression function; \(y\)--the corresponding true value of the improved loess permeability coefficient.  
If the difference between \(f(x)\) and \(y\) is less than or equal to \(\varepsilon\), the loss value is equal to 0.  
Next, the optimization problem is constructed and solved by selecting the appropriate kernel function \(k\) (\(x\), \(x^{\prime}\)) and the appropriate penalty parameter \(c\):  
\[\left\{\begin{array}{l}\max Q(\alpha)=\sum\limits_{i=1}^{l}\alpha_{i}-\frac{ 1}{2}\sum\limits_{i=1}^{l}\sum\limits_{j=1}^{l}\alpha_{i}\alpha_{i}y_{i}y_{j}K \big{(}x_{i},x_{j}\big{)}\\ \text{s.t.}\left\{\begin{array}{l}\sum\limits_{i=1}^{l}\alpha_{i}y_{i}=0,i=1,2\ldots,l\\ 0\leq\alpha_{i}\leq c\end{array}\right.\end{array}\right. \tag{3}\]  
where \(c\) is the penalty factor. A larger value of \(c\) indicates a greater penalty for training errors exceeding \(\varepsilon\). \(\varepsilon\) specifies the error requirement of the regression function, and a smaller \(\varepsilon\) means the smaller the error of the regression function. Figure 13 graphically depicts a typical SVR. A \"spacing band\" is created on both sides of the linear function, and the spacing of this \"spacing band\" is \(\varepsilon\). The data points in the \"spacing band\" are penalized.  
By introducing the Lagrange function and transforming it into its dual form, \(\alpha\) and \(\alpha\)* can be found. Assuming that the optimal solutions of \(\alpha\) and \(\alpha\)* are \(\alpha=\left[\alpha_{1},\alpha_{2},\cdots,\alpha_{i}\right]\), \(\alpha^{*}=\left[\alpha_{1}^{*},\alpha_{2}^{*},\cdots,\alpha_{i}^{*}\right]\), respectively, the weight vector and threshold can be further derived. Thus, the regression function is as in Equation (4).  
\[f(x)=w\Phi(x)+b=\sum\
olimits_{i=1}(\alpha_{i}-\alpha_{i}^{*})\Phi(x_{i})\Phi (x)+b=\sum\limits_{i=1}^{i}(\alpha_{i}-\alpha_{i}^{*})K(x_{i},x)+b \tag{4}\]  
where \(f(x)\) is the permeability coefficient of the improved loess, and \(x\) is the associated soil parameters input.