# 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 this paper, we introduce the Direct-LiNGAM method as a means to consider causal relationships among multi-source data, thereby enhancing the model's ability to perceive the intrinsic connections of multi-source data and facilitating more effective multi-source feature fusion. The Direct-LiNGAM method is a causal discovery algorithm improved for LiNGAM, which is applicable to a larger number of variables [31; 32]. The method has also been extended to allow for latent variables, time series data, and feedback loops [33]. The method does not require a priori knowledge to obtain causal relationships between variables and has a stable computational complexity that is easy to converge [28]. The fundamental concept underpinning the Direct-LiNGAM algorithm can be summarised as follows: Initially, the exogenous variables are selected and incorporated into the causal order. Subsequently, the dataset is updated and the subsequent stage of variable selection is initiated. This process is repeated until the final causal order is obtained. The specific process and algorithm flow of selecting exogenous variables in this paper are as follows, for the case of multi-source data. Assuming a set of variables \(x\), the specific procedure is as follows: First, for each variable \(x_{j}\in U\), where \(U\) denotes the set of remaining variables, compute the residuals \(r_{i}^{(j)}\) from regressing \(x_{i}\in U\setminus\{x_{j}\}\) on \(x_{j}\). Then, using an ICA tool, estimate the residuals \(r_{i}^{(j)}\), and compute the kernel-based mutual independence measure, \(\widehat{M}_{kernel}\left(x_{j},r_{i}^{(j)}\right)\). From this, ob  
Figure 3: Schematic diagram of KAN network architecture. tain the total kernel-based independence measure \(T_{kernel}(x_{j};U)\), as defined by Equation (8), and identify the variable that minimises this quantity as the exogenous variable:  
\[T_{kernel}\big{(}x_{j};U\big{)}=\sum_{i\in U_{i+i}}\widehat{M}I_{kernel}\big{(} x_{j},r_{i}^{(j)}\big{)} \tag{7}\]  
\[x_{m}=argmin_{j\in U/K}T_{kernel}(x_{j};U) \tag{8}\]  
Step 1. Given an \(m\times n\) observation matrix \(X\), centre each variable (column-wise). Step 2. For each \(x_{j}\in U\), regress all other \(x_{i}\in U\setminus\{x_{j}\}\), and obtain residuals \(r_{i}^{(j)}\). Then compute the kernel-based independence measure \(\widehat{M}I_{kernel}\big{(}x_{j},r_{i}^{(j)}\big{)}\), and derive \(T_{kernel}(x_{j};U)\). Select the variable \(x_{m}\) that minimises this measure as the exogenous variable. Step 3. Remove \(x_{m}\) from the set \(U\) and record it in the causal ordering set \(K\). Step 4. \(x=r^{(m)}\), \(X=R^{(m)}\). Step 5. If the number of variables in \(U\) is greater than 1, return to Step 2. Otherwise, proceed to Step 6. Step 6. Append the final variable in \(U\) to the end of \(K\); this yields the full causal order. Step 7. Estimate the final adjacency matrix \(B\) using the identified causal order. Eventually, as demonstrated in Figure 4, the causal adjacency matrix of the input features is computed based on the Direct-LiNGAM method, which is subsequently multiplied by the original input vectors to finally obtain the local causality matrix. This matrix employs colour shades to indicate the strength of causal relationships between variables; the darker the shade, the stronger the relationship. This is carried out to help the model capture causal relationships and correlations between multiple sources of data. The local causality matrix is then spliced with the original input vectors, and the resultant matrix is used as inputs to the model. Neural Network (TRNet) for TBM Surrounding Rock Grade Prediction Based on Multi-Source Feature Fusion  
In order to achieve instant prediction of the surrounding rock state in front of the TBM tunnel face, the problem was transformed into a classification problem based on multi-source data.