# TBM Enclosure Rock Grade Prediction Method Based on Multi-Source Feature Fusion
## 2 Methodologies
### Local Causality Matrix Construction Method for Multi-Source Data Based on Direct-LiNGAM
In order to consider the causal relationship between multi-source data, we construct a novel hybrid neural network (TRNet) based on several algorithms mentioned in the previous sections. As demonstrated in Figure 5, the TRNet model represents a pioneering deep learning architecture that is founded on a CNN and KAN. It comprises an input layer, a feature coding module (CNN), a feature fusion module (KAN), a regularisation module (Dropout), and a classifier module. Figure 4: Local causality embedding strategy. Initially, the input data consists of multi-source feature vectors represented as a matrix \(X\in\mathbb{R}^{N\times d}\), where \(N\) signifies the number of samples and \(d\) encompasses the feature dimension (comprising HSP-coded data, rock fragmentation index, TBM boring parameter data, and 3D spatial coordinate information). This data is represented as a matrix \(X\), where each row represents a mileage point and each column represents a feature. The distance span of the input data is denoted as \(T\), and the number of features is represented by \(X\). The input matrix \(X\) can be expressed as follows:  
\[X= \begin{bmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,X}\\ x_{2,1}&x_{2,2}&\cdots&x_{2,X}\\ \vdots&\vdots&\ddots&\vdots\\ x_{T,1}&x_{T,2}&\cdots&x_{T,X}\end{bmatrix}  \tag{9}\]  
where \(x_{i,j}\) denotes the \(i\)th eigenvalue of the \(t\)th mileage point. The Direct-LiNGAM method is utilised to compute the causal adjacency matrix between the input features \(A\). The size of matrix \(A\) is \(X\times X\) and its element \(a_{i,j}\) denotes the causal strength of the \(i\)th feature to the \(j\)th feature. The matrix \(A\) can be expressed as follows:  
\[A= \begin{bmatrix}a_{1,1}&a_{1,2}&\cdots&a_{1,X}\\ a_{2,1}&a_{2,2}&\cdots&a_{2,X}\\ \vdots&\vdots&\ddots&\vdots\\ a_{X,1}&a_{X,2}&\cdots&a_{X,X}\end{bmatrix}  \tag{10}\]  
where \(a_{i,j}\) indicates a stronger causal relationship. The local causality matrix \(C\) is then generated by multiplying the causal adjacency matrix \(A\) with the input matrix \(X\). The size of matrix \(C\) is \(T\times X\), and its elements \(c_{t,j}\) denote the local causality strength of the \(j\)th feature at the \(t\)th time point. The expression for the matrix \(C\) is as follows:  
\[C=X\cdot A \tag{11}\]  
where each row of \(C\) represents a local causal feature at one point in time. The original input matrix \(X\) is spliced with the local causality matrix \(C\) to form the final input matrix \(X_{input}\). The size of the matrix \(X_{input}\) is \(T\times 2X\) and it can be expressed as  
\[X_{\text{input}}=\mathsf{Concat}(X,C) \tag{12}\]  
where the \(\mathsf{Concat}\) function denotes splicing along the feature dimensions. Subsequently, two-layer one-dimensional convolution (1D Conv) was used to achieve feature space mapping and local pattern extraction, and ReLU was used for all activation functions:  
\[H^{(1)}=ReLU(W_{1}*X_{input}+b_{1}) \tag{13}\]  
These include 32 filters (kernel size \(3\times 3\)) with output dimension \(d\_{\text{model}}/2\). The use of ReLU has been demonstrated to enhance feature sparsity and reduce ineffective signal transmission through its unilateral suppression properties. \[H^{(2)}=ReLU(W_{2}*H^{(1)}+b_{2}) \tag{14}\]  
These include 64 filters (kernel size \(3\times 3\)) with output dimension \(d\_{\text{model}}/2\). The dual ReLU design avoids introducing additional hyperparameter complexity while preserving feature expressiveness. The output of the activation layer is then connected to a stacked KAN network. The KAN decomposition, as outlined by the Kolmogorov-Arnold theorem, involves the transformation of multivariate continuous functions into combinations of univariate functions and additions. This process enables the fusion of features and the reduction of dimensionality through the utilisation of a learnable B-spline function.