2.1 PCA

PCA is a traditional statistical analysis method primarily suitable for processing high-dimensional data with a strong correlation between variables (Ren & Yu, 2011). The core concept of PCA involves forming linear combinations of the original indicators to create a set of new, independent, and comprehensive indicators (Duan et al., 2020). These new indicators contain most of the original information, thereby achieving the objective of dimension reduction. The algorithm is outlined as follows (Wu et al., 2019):

Where lnn is for variable coefficient. Equations (1) and (2) satisfy the following conditions: the sum of the squared coefficients in each equation is 1, and any two principal components are independent and unrelated to each other.

The steps to solve the principal component from the covariance matrix are as follows:

Step 3: Calculate the eigenvalues and eigenvectors of the Pearson correlation coefficient. The eigenvalues are denoted as ${\mathit{�}}_1,{\mathit{�}}_2,\dots ,{\mathit{�}}_{�}$ , satisfying ${\mathit{�}}_{�}\ge 0(�=1,2,\dots ,�)$ , and the unitized eigenvectors corresponding to the eigenvalues are denoted as ${\mathit{�}}_1,{\mathit{�}}_2,\dots ,{\mathit{�}}_{�}$ .

In the above equations, ${\mathit{�}}_{�}$ is the variance contribution ratio for the sth principal component and ${\mathit{�}}_{\mathrm{sum}�}$ is the cumulative contribution ratio of the previous $�$ principal components.

2.2 GRU neural network

The GRU neural network, first proposed by Chung et al., has demonstrated excellent performance in time-series forecasting and found widespread applications in transportation, power engineering, and other fields (Chung et al., 2014; Yue et al., 2021). The architecture of the GRU neural network is depicted in Figure 1.

In the above equations, ${\mathit{�}}_{�}$ and ${\mathit{�}}_{�}$ represent the output of the reset gate and update gate, respectively; $\mathit{�}$ represents the sigmoid function; ${\mathit{ℎ}}_{�-1}$ denotes the hidden state of the neurons at the previous moment; ${\mathit{�}}_{�}$ , ${\mathit{�}}_{�}$ , and ${\mathit{�}}_{�}$ represent the weight matrices of the update gate, reset gate, and temporary output, respectively; ${\mathit{�}}_{�}$ , ${\mathit{�}}_{�}$ , and ${\mathit{�}}_{ℎ}$ are bias functions; ${\mathit{�}}_{�}$ is the input value of the GRU neural network at the present moment; and ${\widetilde{\mathit{ℎ}}}_{�}$ and ${\mathit{ℎ}}_{�}$ refer to the vectors of information that provide temporary unit state and hidden layer output at the moment of $�$ , respectively.

2.3 PCA–GRU neural network

To extract the corresponding principal components from raw data, PCA dimension reduction processing is performed using SPSS software. Subsequently, the PCA–GRU neural network is constructed, using the principal components as the input layer for the GRU neural network. The model construction process of the PCA–GRU neural network is illustrated in Figure 2.

As depicted in Figure 2, the feature sequence obtained through PCA dimension reduction serves as the input for the GRU model. Typically, 80% of the data postdimensionality reduction is allocated to the training set and the remaining 20% is allocated to the test set. The data are subjected to normalization, and the model's internal training facilitates denormalized output.

3.1 Project overview

In this study, the existing data in the TBM project of Lanzhou Water Source Construction (Jiang, 2020; Yan & Jiang, 2020) were used to verify the effectiveness of the PCA–GRU model in predicting the rock mass classification. The excavation diameter of the tunnel is 5.46 m, and the final diameter with lining is 4.60 m (Yan & Jiang, 2020). The length of the TBM construction section for the tunnel is approximately 27.0 km. The tunnel is excavated using two double-shield TBMs combined with drilling and blasting in certain segments (Yan & Jiang, 2020). Among the excavation sections, the length of the TBM1 construction section is 12.426 km, while that of the TBM2 construction section is about 13.259 km. The main design parameters of the two double-shield TBMs are shown in Table 1.

According to the HC method, the rock mass classification for the TBM1 construction section indicates that the predominant rock mass type is Class II, approximately 48% (Yan & Jiang, 2020). Following this, Class III and Class IV account for approximately 25% and 23%, respectively, while Class V is less prevalent, making up around 4%. The percentage distribution of the rock mass classes for the Lanzhou Water Source Construction TBM project is shown in Figure 3. Class III rock masses are primarily characterized by poor integrity, whereas Class II rock masses show comparatively good integrity. Class IV and V rock masses are typically fragmented and demonstrate extremely poor integrity.

3.2 Data collection

In this study, three different classifications of rock mass—Class II, Class III, and Class IV—were utilized (Jiang, 2020). The initial data set collection included nine parameters: uniaxial compressive strength (UCS), Brazilian tensile strength (BTS), rock integrity index (Kv), rock wear resistance index (CAI), deformation modulus (E0), Poisson's ratio (μ), net tunneling rate (PR), rotation speed (RPM), and cutterhead thrust (TF). The selection of these parameters was guided by principles of parameter availability, model complexity, and their frequency of use in previous research (Jiang, 2020). The final sample data set consisted of 150 carefully selected sets of representative engineering measured data. The geological daily report of the tunnel construction site was used to extract tunneling parameters such as TF, RPM, and PR. Average values were then computed for various tunneling sections.

The statistical description of the input variables in the sample data set is presented in Table 2. The percentage distribution of rock mass classes in the sample data set is shown in Figure 4. The percentage distribution of different lithologies in the sample data set is shown in Figure 5. Violin with box depicting the input variables for three rock mass classifications in the sample data set is shown in Figure 6.

The training data set shows variation in the percentages of different rock mass classes, as depicted in Figure 4. The percentages of various rock mass classes in the test data set, however, correspond to the real engineering situation shown in Figure 4. The correspondence between rock mass classes in the test data set and the actual engineering scenario is advantageous for a more thorough examination of the model's predictive performance. As shown in Figure 5, there are variations in the percentages of different lithologies in the training data set. However, the percentages of the test data set for various lithologies remain unchanged. It is valuable to assess the model's predictive performance on rock mass classes based on the consistency across various lithologies. As depicted in Figure 6, violin with box depicting input variables for three rock mass classifications in the sample data shows significant variation among the three rock mass classifications. However, from Figure 6a,c,e,f,g,h,i, it becomes evident that there is an overlap in certain input variables between Class III and Class IV rock masses. This overlap may affect the predictive performance of the subsequent model for Class III and Class IV rock masses.

3.3 PCA dimension reduction

In the above equation, ${�}_{�}=\frac{�-\overline{�}}{�}$ represents the normalized variable value; $�$ denotes the raw variable value; $\overline{�}$ is the average of the data for this set; and $�$ is the standardized deviation of the data for this set.

The standardized data were subjected to PCA using SPSS software. The correlation coefficients between the various input variables are depicted in Figure 7. Additionally, Table 3 presents the cumulative contribution ratio and the principal component contribution ratio for each principal component. According to experience, a correlation coefficient reaching 0.5 indicates a particularly strong correlation between input variables, suggesting potential mutual influence. From Figure 7, it is evident that many correlation coefficients among the input variables exceed 0.5, potentially influencing the prediction results of rock mass classification.

The first two principal components have a cumulative contribution ratio of 88.37%, as indicated in Table 3. This means that the first two principal components together account for 88.37% of the original data, satisfying the requirements since their respective eigenvalues are greater than 1. Therefore, these two principal components are selected for further analysis. Table 4 illustrates the variance extraction ratio of raw variable information by the first two principal components.

As depicted in Table 4, the first and second principal components effectively capture the majority of information from the raw data. The first principal component is primarily associated with UCS, BTS, Kv, CAI, E0, μ, PR, RPM, and TF, while the second principal component is mainly related to Kv and μ. Following PCA preprocessing, the coefficients of component scores for each principal component are obtained, as presented in Table 5. The individual values for each principal component score are determined through a linear combination of the standardized raw data values with their respective principal component score coefficients, as outlined in Table 6.

Given the slow computational time associated with the PCA dimension reduction technique using SPSS software, this study endeavors to improve the PCA–GRU model. These improvements are designed to expedite the prediction process for rock mass classification, ensuring timely and accurate results. To streamline the preprocessing steps for rock mass classification, a MATLAB program has been developed.

The program seamlessly integrates standardization and PCA dimension reduction techniques, facilitating the extraction of principal components F1 and F2 directly from the raw data. These processed principal components are subsequently input into the GRU model for computation, eliminating the need for cumbersome intermediate steps.

3.4 Performance indices

The key evaluation metrics, namely, accuracy (A), precision (P), and recall (R), are frequently used to assess the predictive performance of models. The accuracy (A) metric evaluates the proportion of correctly predicted outcomes among all observations made by the models. Precision (P) measures how many of the positive predictions were correct, emphasizing the accuracy of positive classifications. On the other hand, recall (R) highlights the models' sensitivity to positive instances by measuring the percentage of real positive instances that the models have correctly identified.

4.1 Selection of hyperparameters

The main hyperparameters for the PCA–GRU model, the GRU model, and the LSTM model include the hidden layer, dropout layer, fully connected layer, number of neurons, maximum number of training cycles, minimum batch size, dropout rate, initial learning rate, learning decay factor, and gradient threshold. The RF model is characterized by two hyperparameters: decision trees (DT) and minimum leaf size (LFmin). Similarly, the SVM model has two hyperparameters, namely, the penalty factor (C) and the radial basis function gamma (g).

The quantity of neurons and the number of hidden layers in the PCA–GRU, GRU, and LSTM models can affect how well the models predict outcomes. As indicated in Table 7, the aforementioned hyperparameters were adjusted. Additionally, a layer of dropouts with a ratio of 0.2 was incorporated into the models to prevent overfitting. A maximum of 1000 training epochs was set for the models, with a minimum batch size of 100. The fixed values were 0.2 for the learning decay factor, 1 for the gradient threshold, and 0.01 for the initial learning ratio. The fully connected layer was limited to 1, with four neurons, to simplify the adjustment of hyperparameters.

The minimum leaf size (LFmin) and the number of DT in the RF model can affect the model's ability to predict outcomes. The radial basis function parameter (g) and the penalty factor (C) in the SVM model may have an effect on the model's predictive ability. As a result, the hyperparameters mentioned above are modified as shown in Table 7.

As Figure 8 illustrates, the PCA–GRU model selects hyperparameter 6 with the best predictive performance, while the GRU and LSTM models select hyperparameter 7 with the best performance. By selecting hyperparameter 4, the RF model's predictive performance is at its best. The SVM model's predictive ability is not significantly affected by changes to its hyperparameters. The optimal hyperparameters are then determined by taking the values that correspond to each model's best predictive performance in the previously mentioned situations.

Figure 9 shows the computational time of the PCA–GRU model for different hyperparameters. The PCA–GRU model takes approximately 20 s, as depicted in Figure 9. This quick processing time enables rapid responses for the classification prediction of rock masses in TBM tunneling.

4.2 Prediction results of five models

The predictions for the rock mass classification by the PCA–GRU model were compared with those of the GRU, LSTM, RF, and SVM models based on optimal performance corresponding to the best hyperparameters mentioned earlier in this paper. Confusion matrices for rock mass classification test data of the PCA–GRU, GRU, LSTM, RF, and SVM models are presented in Figure 10. The radar chart comparing predictions for rock mass classification by the PCA–GRU, GRU, LSTM, RF, and SVM models is shown in Figure 11. The numbers 1, 2, and 3 in Figure 10 correspond to the Class II, III, and IV rock masses, respectively. The confusion matrix's diagonal elements indicate how many correct classifications the model achieves for every specific class. On the other hand, misclassification is indicated by the off-diagonal elements, which show differences between the predicted and real classes. These matrix elements can be used to compute a variety of performance metrics, such as accuracy (A), precision (P), and recall (R), to fully assess the efficacy of the model.

The PCA–GRU model predicts rock mass classification with A, MP, and MR of 0.9667, 0.963, and 0.9763, respectively, as illustrated in Figure 11. These values are significantly higher than those of the GRU, LSTM, RF, and SVM models. The prediction results indicate that the PCA–GRU model outperforms the GRU, LSTM, RF, and SVM models in terms of overall performance in predicting the rock mass classification.

The PCA–GRU model shows high precision (P) and recall (R) in predicting rock masses of Class II, III, and IV, as illustrated in Figure 10. On the other hand, the GRU, LSTM, RF, and SVM models show lower precision (P) for Class III rock masses and lower recall (R) for Class IV rock masses. Certain input variables overlap, explaining why models like GRU, LSTM, RF, and SVM have lower recall (R) in Class IV rock mass prediction and lower precision (P) in Class III prediction. Among these variables are UCS, CAI, E0, μ, PR, RPM, and TF in relation to these two rock mass classes. The models may incorrectly classify Class IV rock masses as Class III due to this overlap.

The PCA dimension reduction technique has reduced the nine raw input variables to two comprehensive indicator variables that accurately represent the original data's features. This aids in eliminating input variable overlap in the distribution of Class III and IV rock masses. As a result, Class II, III, and IV rock masses can be successfully identified using the PCA–GRU model. The prediction outcomes demonstrate how effectively the PCA–GRU model predicts the rock mass classification of TBM projects.

4.3 Random sampling analysis

The model's ability to generalize may be impacted by the uneven distribution of various rock mass classes and lithologies, which change in percentage between the training and test data sets. To test the model's generalization, the ratio of the training data set to the test data set is maintained at 8 : 2, and both the training and test data sets are shuffled at random. Ten randomly shuffled data sets are used to compare the predictions so as to improve the reliability of the results. Figure 12 shows the model predictions obtained from 10 data sets that are shuffled at random.

The predictive performance metrics A, MP, and MR of the PCA–GRU model are clearly higher than those of the GRU, LSTM, RF, and SVM models, as shown in Figure 12. The results of the research suggest that the PCA–GRU model performs better in terms of generalization than the GRU, LSTM, RF, and SVM models. This makes it more appropriate for conditions where the distribution of various rock mass classes and lithologies change in percentage.