## 3 Research Methodology  
### KPCA and GRA  
KPCA is an improved processing method of PCA. As for PCA, it only applies to linear data, and there is a significant error when dealing with nonlinear data. In contrast, KPCA can largely preserve the local structure of nonlinear data by introducing kernel functions. Meanwhile, PCA will map data points into a low-dimensional space when dealing with high-dimensional data, facing dimensional disaster and leading to the loss of distance information between data points. Therefore, KPCA is more advantageous than PCA [34]. The basic idea is to project the samples that are linearly indistinguishable in the low-dimensional space to the high-dimensional space and make them linearly distinguishable by the kernel function [35]. For a set of samples \(X=[x_{1},\,x_{2},\,\ldots,\,x_{n}]\in R^{m}\), the mapping function is  
\[\sum\limits_{i=1}^{n}\varphi(x_{i})=0(\varphi:R^{m}\to R^{k}(k\gg m)) \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Type of Pile** & **Parameter** & **Insertion Phase** & **Scooping Phase** & **Stopping Phase** & **Lifting Phase** & **Total** \\ \hline \multirow{2}{*}{Gravel} & Fuel consumption (mL) & 27.12 & 49.27 & 4.10 & 15.61 & 96.10 \\  & Percentage (\%) & 28.22 & 51.27 & 4.27 & 16.24 & 100.00 \\ \hline \multirow{2}{*}{Sand} & Fuel consumption (mL) & 15.97 & 41.15 & 6.45 & 24.68 & 88.26 \\  & Percentage (\%) & 20.51 & 52.83 & 8.29 & 31.68 & 100.00 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Fuel consumption.  
where \(\varphi(x_{i})\) is the mapping function, \(R^{m}\) is an m-dimensional vector, and \(R^{k}\) is a k-dimensional vector.  
Then the covariance matrix can be expressed as  
\[C=\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})\varphi(x_{i})^{T} \tag{5}\]  
Suppose the eigenvalue of \(C\) is \(\lambda\), and its corresponding eigenvector is \(v\), the result is  
\[\lambda v=Cv=(\frac{1}{n}\sum_{i=1}^{n}\varphi(x_{i})\varphi(x_{i})^{T})v= \frac{1}{n}\sum_{i=1}^{n}\,(\varphi(x_{i})^{T}v)\varphi(x_{i}) \tag{6}\]  
For \(\lambda v=Cv\), by multiplying each left and right by \(\varphi(x_{i})\), we can obtain \(\lambda\varphi(x_{i})v=Cv\varphi(x_{i})\). The introduced kernel function matrix can be expressed as \(K_{ij}=K(x_{i},\,x_{j})=\varphi(x_{i})\varphi(x_{j})\). Then, we can obtain \(n\lambda a=Ka\), where \(a\) is the eigenvector of \(K\). Then, the projection of \(x\) into the higher dimensional space can be expressed as  
\[v\varphi(x)=\sum_{i=1}^{n}a_{i}\varphi(x_{i})\varphi(x)=\sum_{i=1}^{n}a_{i}K (x_{i},x) \tag{7}\]  
In this study, the Gaussian kernel function is chosen as the kernel function, which is defined as follows:  
\[K(x,q)=\frac{\exp(-\|x-q\|^{2})}{\sigma^{2}} \tag{8}\]  
where \(\sigma\) is the function parameter, and \(q\) is the center of the function.  
GRA is an analytical method based on grey system theory, mainly used to study the strength of association between influencing factors. It converts the grey information between multiple factors into black-and-white information, then uses the magnitude of the grey relation coefficient value as the basis for evaluating the strength of the correlation. The specific steps are as follows.  
1. Select the parent sequence \(Y\) and the subsequence \(X\).
2. To eliminate the influence of the magnitude, the parent sequence and the subsequence are normalized, and the normalized parent sequence is recorded as \(Y^{\prime}\) and the subsequence is recorded as \(X^{\prime}\).
3. The formula is shown in Equation (9) to calculate the grey relation coefficient. \[\xi(i)=\frac{\min_{i}(\min|Y^{\prime}-X^{\prime}(i)|)+\rho\max_{i}(\max|Y^{ \prime}-X^{\prime}(i)|)}{|Y^{\prime}-X^{\prime}(i)|+\rho\max_{i}(\max|Y^{ \prime}-X^{\prime}(i)|)},\,\,\,i=1\cdots n\] (9)  
where \(\rho\) is the discriminant coefficient, which takes values from 0 to 1 and is taken as 0.5 in this paper.