Hybrid Random Forest-Based Models for Earth Pressure Balance Tunneling-Induced Ground Settlement Prediction

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This paper employs intelligent methods to forecast ground settlement caused by tunnel construction in order to provide a solid foundation for risk warning and risk management.

Abstract

Construction-induced ground settlement is a serious hazard in underground tunnel construction. Accurate ground settlement prediction has great significance in ensuring the surface building’s stability and human safety. To that end, 148 sets of data were collected from the Singapore Circle Line rail traffic project containing seven defining parameters to create a database for predicting ground settlement. These parameters are the tunnel depth (H), the tunnel advance rate (AR), the EPB earth pressure (EP), the mean SPTN value from the soil crown to the surface (Sm), the mean water content of the soil layer (MC), the mean modulus of elasticity of the soil layer (E), and the grout pressure used for injecting grout into the tail void (GP). Three hybrid models consisting of random forest (RF) and three types of meta-heuristics, Ant Lion Optimizier (ALO), Multi-Verse Optimizer (MVO), and Grasshopper Optimization Algorithm (GOA), were developed to predict ground settlement. Furthermore, the mean absolute error (MAE), the mean absolute percentage error (MAPE), the coefficient of determination (R²) and the root mean square error (RMSE) were used to assess predictive performance of the constructed models for predicting ground settlement. The evaluation results demonstrated that the GOA-RF with a population size of 10 has achieved the most outstanding predictive capability with the indices of MAE (Training set: 2.8224; Test set: 2.3507), MAPE (Training set: 40.5629; Test set: 38.5637), R² (Training set: 0.9487; Test set: 0.9282), and RMSE (Training set: 4.93; Test set: 3.1576). Finally, the sensitivity analysis results indicated that MC, AR, Sm, and GP have a significant impact on ground settlement prediction based on the GOA-RF model.

1. Introduction

Surface traffic cannot meet the development needs of a large city, therefore underground tunnel construction is gradually becoming the primary option for urban transportation projects [1]. Earth pressure balance (EPB) is a common method for underground construction in cities with dense surface buildings. Viewed from the construction perspective, the main construction area of an urban tunnel is concentrated in the soft layer under the dense building group in cities [2]. This will inevitably cause settlement due to the effects of two major factors: shallow depth and soft ground [3,4,5]. Once ground settlement occurs, building safety and other public utilities above the construction area are seriously threatened. Therefore, predicting ground settlement caused by tunneling is an essential part of the risk analysis of urban tunnel projects [6,7,8].

To provide an important foundation for assessing potential risks and adjusting construction plans [9], four methods are commonly used to assess ground settlement caused by tunnel construction, i.e., empirical [10,11], analytical [12], numerical [13,14], and artificial intelligence methods [15,16,17,18]. Although the empirical method has a longer history of application, which not only involves incomplete parameters, it also does not have the best prediction accuracy [19]. Some widely used empirical equations are listed in

Table 1. The empirical method, analytical method, and numerical method for ground settlement calculation.

Category	Method	Reference
Empirical methods	$S_{\max }=V_S/2.5i$	Well known
	$S=S_{\max }\times e^{(-x^2/2i^2)}$	[11]
	$S_{\max }=f(A/2R)$	[22]
	$V_S=\sqrt{2\pi }{S}_{\max }\times i_0$	[23]
	$i_1=0.386A+2.84$	[24]
	$S_{\max }=0.313\left(\frac{VL\times D^2}{i}\right)$	[25]
	$S_{\max }=0.785\left[{\gamma }_n\times \left(A+{\sigma }_s\right)\right]\cdot \left(\frac{D^2}{i\times E}\right)$	[26]
	$S_{\max }=0.01875(\frac{R^2}{i})\times e^{0.26\left[{\gamma }_n\times (Z+{\sigma }_s)-{\sigma }_T\right]/U_{cs}}$	[27]
Analytical Methods	The analytical solutions for ground settlement caused by tunnel construction in initially isotropic and homogeneous soils	[28]
	The analytical solution of the uniform elastic half-space tunnel is performed using the approximation method suggested by Sasageta [28].	[29]
Numerical Methods	FEM (PLAXIS 3D, ABAQUS)	[30,31,32,33,34]
	FDM (FLAC3D)	[35,36,37,38]
	DEM (3DEC)	[39,40,41,42,43]

Note: S = theoretical settlement (m); Smax = maximum settlement (m); i = point of inflexion (m); x = transverse horizontal distance from the tunnel center line (m); A = tunnel axis depth (m); D = equivalent tunnel excavation diameter (m); C = empirical coefficient; VL = volume loss; E = elasticity modulus; ${\gamma }_n$ = natural unit weight; ${\sigma }_s$ = surface surcharge (kPa); ${\sigma }_T$ = required working face support pressure (kPa); Ucs = undrained cohesion of soil.

. Subsequently, analytical, and numerical simulation methods were used to calculate ground settlement. In most real-world situations, the analytical method struggles to recreate the excavation process during the calculation; as a result, numerous simplifications were applied in the calculation process, leading to biased results [20]. Łodygowski and Sumelka [21] pointed out the shortcomings of numerical simulation methods in terms of timeliness and the problem of considering boundary difficulties. Therefore, avoiding the complex process of reverting tunnel construction in the calculation process and the method of exploring the relationship between numerous weakly correlated parameters becomes a challenge in ground settlement prediction. Table 1 displays a selection of studies pertaining to empirical, analytical, and numerical methods for calculating ground settlement.

Artificial intelligence methods have received focused attention due to the fact that they do not require prior experience as support, as well as having the ability to work with multiple influential variables. Samui [44] introduced a support vector machine (SVM) approach for predicting ground settlement in shallow soils and analyzed the importance of parameters affecting settlement prediction. Nejad et al. [45] found that artificial neural network (ANN) has the better performance for the pile foundation settlement prediction than other conventional artificial intelligence methods. Moreover, artificial intelligence techniques have been implemented in a vast array of fields [46,47,48,49,50,51].

Table 2. Intelligent method for ground settlement prediction.

Method	Target of Prediction	Database Capacity	Reference
ANN	Ground surface settlement	-	[15]
BPNN, SVM, GP	Ground surface settlement	230	[16]
RF	Ground movements	66	[19]
BPNN, LSTM, GRU, Conv1d	Ground surface settlement, Longitudinal settlement curve, Shield operational parameters	328	[20]
ANN	Ground surface settlement	-	[52]
ANN	Ground surface settlement	49	[53]
ANN	Ground surface settlement	142	[54]
BPNN, OLS	Ground surface settlement	432	[55]
ANN	Ground surface settlement	-	[56]
ANN, WNN	Ground surface settlement	49	[57]
ANN	Ground surface settlement	143	[58]
SVM	Ground surface settlement	-	[59]
RF	Ground surface settlement	49	[60]
MARS, ANN, RVM, SVM, GP	Ground surface settlement	148	[61]
SVM	Surrounding rock deformation	-	[62]
BPNN, RBF, GRNN	Ground surface settlement	30	[63]
BPNN, WNN, GRNN, ELM, SVM, RF	Ground surface settlement	200	[64]
BPNN, GRNN, ELM, SVM, RF	Ground surface settlement	236	[65]
RF, LSTM	Ground surface settlement	4249	[66]
ANFIS	Ground surface settlement	143	[67]
XGBoost, BPNN, SVM, MARS	Ground surface settlement	148	[68]

Note: ANN (Artificial Neural Network), ANFIS (Adaptive-Network-Based Fuzzy Inference System), BPNN (Back Propagation Neural Network), Conv1d (Convolutional Neural Network), ELM (Extreme Learning Machine), GP (Gaussian Process), GRNN (General Regression Neural Network), GRU (Gated Recurrent Unit), LSTM (Long Short-Term Memory), MARS (Multivariate Adaptive Regression Splines), OLS (Ordinary Least Squares), RBF (Radial Basis Function), RF (Random Forest), SVM (Support Vector Machine).

displays the application of different artificial intelligence methods to ground settlement prediction.

However, numerous shortcomings of some intelligent methods are gradually revealed in practical application. For example, it is difficult to determine the network structure of ANN models and have poor generalization performance [18]. SVM also has practical issues, such as high model complexity and significant limitations in kernel functions [18]. As a consequence, finding an appropriate prediction method for ground settlement brought on by tunnel excavation remains a critical issue.

Breiman [69] proposed an artificial intelligence approach called RF, which is an integrated model combining of tree predictors. This model has good generalization and is more robust in terms of noise control. Numerous scholars revealed the advantages of RF models in the field of geotechnical and tunneling engineering [70,71,72,73,74,75,76,77,78,79]. Dong et al. [80] compared the classification performance of ANN, SVM and RF models for predicting the rockburst intensity. The results showed that the misclassification rates of these models are 0.1, 0.2, and 0, respectively. Tao et al. [81] conducted a study on predicting the tunnel boring machine (TBM) penetration rate. The results showed that the RF regression algorithm has achieved a high accuracy prediction of the mean squared error (MSE) at 0.0504. Zhou et al. [18] investigated the geometric, geological, and operational parameters as input variables to train RF models for the ground settlement prediction. A good prediction accuracy has been achieved by means of valuable performance indices (R²: 0.93, RMSE: 15). However, few RF models with particularly good results were developed to predicting ground settlement during tunnel excavation. Motivated by the above considerations, the main goal of this paper is to establish some hybrid RF models based on the three types of meta-heuristics named, Ant Lion Optimizer (ALO), Multi-Verse Optimizer (MVO), and Grasshopper Optimization Algorithm (GOA). In practice, these metaheuristic techniques are meritocratic with regard to the RF hyperparameters and improve the predictive accuracy of the model.

This paper is organized as follows: the RF model and three used meta-heuristic algorithms are introduced in Section 2. Section 3 describes the relevant parameters, four performance evaluation indices and model development (Matlab 2021 was used in this paper to complete the model construction). Subsequently, the performance evaluation results of all hybrid RF models for the ground settlement prediction are displayed and discussed in Section 4. Finally, the main conclusions and limitations of this paper are summarized in Section 5.

2. Materials and Methods

2.1. Random Forest

RF is a class of multi-decision tree integration models previously proposed by Breiman [69]. There are two types of trees in RF, i.e., classification tree and regression tree. The latter is used to solve the regression problem including the ground settlement prediction during tunnel excavation. For the regression RF model, the training performed for each decision tree is based on a separate subset of data, even if a data point in the dataset changes, it only affects one decision tree and hardly affects all of them [82]. This is one of the reasons why the RF model is resistant to overfitting. The framework of RF regression can be represented by the following [69]:

• Multiple training sets are randomly generated using the Bootstrap resampling method.
• Each training subset generates a decision tree that will be split in an optimal way in a randomly selected set of attributes. The tree will grow to its maximum without being pruned.
• The above steps were repeated until the number of regression trees reached the upper limit set by the researchers.

2.2. Ant Lion Optimizier (ALO)

The Ant Lion Optimization (ALO) algorithm is a meta-heuristic approach proposed by Mirjalili [83] to solve the optimization problem. This algorithm was inspired by the predation strategy of ant lions. Ant lions in nature are good at building funnel-shaped traps to capture ants who fall into their traps. The ALO optimization algorithm simulates the predation strategy of ant lions to search for the optimal solution. The algorithm consists of five steps: simulating the random movement of ants, preparing the trap, trapping the ants, feeding, and reconfiguring the trap.

The irregular actions of ants in the search for food can be represented in the algorithm by the following equations:

$$G(l)=\left[0,\mathit{csm}(2r(l_1)-1),\mathit{csm}(2r(l_2)-1),.....,\mathit{csm}(2r(l_n)-1)\right]$$

(1)

$$r(l)=\{\begin{array}{c}1 , \mathit{if} \mathit{rand}>0.5\\ 0 , \mathit{if} \mathit{rand}\le 0.5\end{array}$$

(2)

where G(l) is the set of steps for ants to wander randomly, csm represents the cumulative sum, n represents the maximum number of iterations, l shows the step length of the ant movement (the current iteration in the algorithm), and r(l) is a random function, rand denotes a random number in the range of [0,1].

Because the traps laid by the ant lion affect the random movement of the ants, the algorithm restricts the movement of the ants is restricted in the range of a certain sphere of ant lion. To better simulate the survival of the fittest in nature, the roulette strategy was used to give more adaptable ant lions chance to catch ants. Based on this strategy, the ant lions can build traps that match adaptive scale. When ants fall into the trap, their movement range is adaptively reduced to constantly approach the ant lion’s position. After a successful predation, the ant lion updated its position to the prey position to improve the success of subsequent hunting. Its location is updated as follows:

$$A{L}_i^l=A_j^l , if f(A_j^l)>A{L}_i^l$$

(3)

where A^l_j represents the position of the j-th ant at the l-th iteration and AL^l_i represents the updated position of the i-th ant lion at the l-th iteration.

Eventually, the optimal ant lion in each iteration is retained as an elite that can influence the random movement of all ants in the process.

2.3. Multi-Verse Optimizer (MVO)

Multi-Verse Optimization (MVO) is a class of nature-inspired algorithms proposed by Mirjalili [84]. Physicists believe that the Big Bang can be thought of as a white hole and an integral part of the universe; the black hole absorbs all matter, including light; and the wormhole is a tunnel that connects parts of the universe to each other. Therefore, the algorithm is inspired by a cyclic multiverse model [85]: the stability among multiple universes is controlled by an interaction of black holes, white holes, and wormholes.

Moreover, the objects with high expansion rate in the universe (white holes in the algorithm) always tend to move toward the objects with low expansion rate (black holes in the algorithm), and this movement is made possible by wormholes. In the MVO algorithm, wormholes are used as the channel medium to connect white holes and black holes for detecting the search space, as shown in Figure 1.

The algorithm contains two important parameters: traveling distance rate (TDR) and wormhole existence probability (WEP). The TDR represents the distance an object can travel through a wormhole near the optimal universe and can be expressed as follows:

$$\mathit{TDR}=1-\frac{h^{1/q}}{H^{1/q}}$$

(4)

where q represents the exploration accuracy during the iteration, h represents the current iteration, and H represents the maximum number of iterations.

The WEP expresses the probability of wormhole occurrence and can be expressed by the following equation:

$$\mathit{WEP}=\mathit{MIN}+n\times \left(\frac{\mathit{MAX}-\mathit{MIN}}{N}\right)$$

(5)

where MAX represents the maximum value (1 in this paper), and MIN represents the minimum value (0.2 in this paper), n is the current iteration, N shows the maximum iterations.

2.4. Grasshopper Optimization Algorithm (GOA)

Saremi et al. [86] proposed the grasshopper optimization algorithm (GOA) to solve optimization problems. This is a class of bionic optimization algorithms that simulates the group behavior of grasshoppers. The algorithm divides the behavioral trends of grasshopper swarms into exploration and exploitation. In the exploration phase, grasshoppers normally move abruptly, while they prefer to move more regionally in the exploitation phase. These two patterns of grasshoppers behavior are modeled mathematically.

The position of the i-th grasshopper is denoted by x_i in the algorithm, and the group behavior of the grasshoppers can be expressed by the following equation:

$$X_i^d=k\left({\displaystyle \underset{\begin{array}{c}j=1\\ j\ne i\end{array}}{\overset{N}{\mathsf{-}}}k\frac{u{b}_d+l{b}_d}{2}}g\left(\left|x_j^d-x_i^d\right|\right)\frac{x_j-x_i}{d_{ij}}\right)+\stackrel{\wedge }{F_d}$$

(6)

The social definition function g(r) can be expressed as follows:

$$g(r)=f{e}^{\frac{-r}{c}}-e^{-r}$$

(7)

where ub_d and lb_d are the upper and lower bounds of the d-dimensional space, respectively, and $d_{ij}=\left|x_j^d-x_i^d\right|$ is the distance between the i-th grasshopper and the j-th grasshopper; $\stackrel{\wedge }{F_d}$ is the location vector of the grasshopper target food; f denotes the strength of attraction; c shows the attractive length scale. The reduction of the grasshopper’s comfort zone is captured by the coefficient k:

$$k=k\max -t\frac{k\max -k\min }{T}$$

(8)

where kmax represents the maximum (1 in this work) and kmin represents the minimum values (0.00001 in this work), respectively; t is the current iteration, while T represents the maximum iteration.

3. Database Preparation

In this study, the data were collected from the Singapore Circle Line rail traffic project [61,87,88,89]. The projects were constructed with EPB machines and some basic parameters are shown in

Table 3. Tunnel project construction parameters.

Contract	C705	C823	C825
TBM manufacture	Hitachi-Zosen	Hitachi-Zosen	Herrenknecht
Drive length (km)	1.3	3.3	1.5
Tunnel drive (no.)	2	2	2
Outside diameter(m)	6.44	6.63	6.58
Internal diameter (m)	5.8	5.8	5.8
Stations	Boon Keng and PotongPasir	Mountbatten, Dakota and PayaLebar	DhobyGhaut, Bras Basah, Esplanade and Promenade
Geology (general description)	Mostly Old Alluvium with Marine clay	Fill overlying Kallang formation and Old Alluvium	Soft marine clay, Old Alluvium, Fort Canning Boulder bed, Jurong formation

. The tunnel was lined with a precast concrete tunnel liner with an internal diameter of 5.8 m. The spacing of the measurement points for ground settlement was about 25 m. After the excavation of each section of the tunnel project was completed, the settlement of that section was measured.

The construction section contains mainly four different types of soil conditions as shown in Figure 2. The stratigraphic characteristics are systematically described by Goh et al. [61], Hulme and Burchell [87] and Shirlaw et al. [89,90].

3.1. Data Structure

The database contains 148 ground settlement data and other related parameters are represented and described as shown in Figure 3. These parameters affecting ground settlement can be divided into three categories [53,61]: geological conditions, tunnel geometry conditions, and operational parameters.

The geological parameters included in the database of this study are the mean SPT N (standard penetration test) value from the soil crown to the surface (Sm), the mean SPT N value in the bottom, middle, and crown layers of the soil (Sa), the mean modulus of elasticity of the soil layer (E) and the mean water content of the soil layer (MC). Since the groundwater levels of the three projects are similar, they are not considered to be input variables. The geometric parameters of the tunnels are mainly considered to be the inner diameter and the depth to the surface (H) of the tunnel. Since the inner diameter of the tunnel was the same in all three projects, it was not used as an input variable in this study. The operational parameters of the tunnel that can be used as input parameters for this study are the tunnel advance rate (AR), the EPB earth pressure (EP) and the grout pressure used for injecting grout into the tail void (GP). In addition, the ground settlement (GS) is output parameter in this study.

3.2. Sensitivity Analysis

Parameters sensitivity analysis can help reduce redundancy in the data and improve prediction efficiency. Therefore, the correlation between all parameters of used database is examined using Pearson correlation analysis, and the results are shown in Figure 4.

As illustrated in Figure 4, it is not difficult to find that the correlation coefficient between E and Sa is high of 0.99. The result indicated that there is an extremely strong correlation between these two parameters, and further analysis reveals that the correlation between the two types of characteristic parameters and the others is similar. Therefore, Sa is not considered to be an input parameter. The correlations between the remaining parameters and GS meets the requirements of the study. Therefore, the input parameters used in this study are H, AR, EP, Sm, MC, E, and GP, and the output parameter is GS.

3.3. Evaluation Indicators

The performance evaluation is a key part for the model construction, which is related to the optimal model selection after finishing prediction [78,91,92]. In this study, four statistical indices named the coefficient of determination (R²), the mean absolute error (MAE), the mean absolute percentage error (MAPE), and the root mean square error (RMSE) were used to evaluate the model performance. The definition and calculation equation of each evaluation index is shown in

Table 4. Definition of evaluation indicators and their calculation.

Metric	Definition	Computational Formula
MAE (Mean Absolute Error)	The difference between each actual and predicted value is summed and finally divided by the number of observations	$\mathrm{MAE}=\frac{1}{m}{\displaystyle \underset{i=1}{\overset{m}{\mathsf{-}}}\left|{\stackrel{\wedge }{y}}_i-y_i\right|}$
MAPE (Mean Absolute Percentage Error)	The error of each point is normalized	$\mathrm{MAPE}=\frac{100\%}{m}{\displaystyle \underset{i=1}{\overset{m}{\mathsf{-}}}\left|\frac{{\stackrel{\wedge }{y}}_i-y_i}{y_i}\right|}$
R2 (Coefficient of determination)	Reflects the extent to which independent variables explain changes in dependent variables	${\mathrm{R}}^2=1-\frac{{\displaystyle \underset{i=1}{\overset{m}{\mathsf{-}}}{({\stackrel{\wedge }{y}}_i-y_i)}^2}}{{\displaystyle \underset{i=1}{\overset{m}{\mathsf{-}}}{({\stackrel{\mathsf{-}}{y}}_i-y_i)}^2}}$
RMSE (Root meansquare error)	Mean square error squared, commonly used as a measure of machine learning model prediction results	$\mathrm{RMSE}=\sqrt{\frac{1}{m}{\displaystyle \underset{i=1}{\overset{m}{\mathsf{-}}}{({\stackrel{\wedge }{y}}_i-y_i)}^2}}$

Note: ${\stackrel{\wedge }{y}}_i$ represents the predicted value $y_i$ represents the true value, and ${\stackrel{-}{y}}_i$ represents the mean of the true values.

.

3.4. Hybrid Models

Hyperparameters have a significant impact on model performance, thus three metaheuristics (ALO, WVO, and GOA) were selected to optimize hyperparameters of RF model in this study. The model constructing process of ALO-RF, WVO-RF and GOA-RF models is shown in Figure 5. The number of populations was set to 10, 20, 50, 100, and the iteration was set to 500 to ensure that all models were fully optimized. One-hundred forty-eight data sets were divided into a training set and a test set in a 3:1 ratio. Furthermore, a five-fold CV approach was used to choose the best optimization parameters for the training set data in order to reduce model overfitting [93,94]. Then, three metaheuristic approaches: ALO, WVO, and GOA, were used to optimize the RF hyperparameters and enhance the accuracy of the model. The MAE, MAPE, RMSE, and R² were used as evaluation indices to evaluate the predictive capability of ALO-RF, WVO-RF, and GOA-RF models for estimating GS and to determine the optimal model. Finally, the importance of each input parameter was analyzed to grasp the influence of them on GS.

4. Discussion

4.1. Model Efficiency Evaluation

With the help of the three-class metaheuristic technique, the predictive ability of the RF model is enhanced, and the ALO-RF, WVO-RF, and GOA-RF hybrid models are obtained. The RMSE was chosen to calculate the fitness values of each developed model. As illustrated in Figure 6, the fitness values of all developed models with different populations had been minimized and stabilized after reaching the maximum number of iterations.

As can be seen in this picture, it is easy to find that the convergence time of GOA-RF hybrid models with different populations is slightly faster than ALO-RF and WVO -RF models. In conclusion, the variation of the fitness function confirms the effectiveness of the three types of optimization algorithms and visualizes the optimization efficiency of the different algorithms. The predictive performance of the three types of hybrid models for GS is evaluated more fully in Section 4.2.

4.2. Model Prediction Performance Evaluation

To better assess the optimization capability of ALO, WVO and GOA, the unoptimized RF model was introduced to compare predictive performance with three hybrid RF models.

Figure 7 illustrates the total scores of four evaluation indices for each hybrid model. The larger sector area represents a higher score obtained by a model. In other words, the hybrid model has a superior performance than other models for the ground settlement prediction. Figure 7a displays the cumulative scores of ALO-RF models with population sizes of 10, 20, 50, and 100. It can be obviously observed that the ALO-RF model with 10 population obtains the highest cumulative score. This result indicated that the ALO-RF model with 10 is the best model for the GS prediction than other similar models. Similarly, in Figure 7b, the WVO-RF model with a population size of 100 receives the highest score for each evaluated indices, who demonstrates a superior capacity to predict GS. The cumulative score of GOA-RF with a population size of 10 was then greater than other models considering other population sizes, thus it is the best prediction model for forecasting the GS.

To better visualize the performance of each ALO-RF, WVO-RF, and GOA-RF models, the relationships between predicted and measured GS values were depicted in Figure 8. Each sphere in this figure represents the distribution of a set of measured and predicted values. If the sphere lies on the diagonal line, the measured value is equal to the predicted value. Moreover, the histogram of differences located in the upper right corner of the figure represents the distribution of differences between predicted and measured values. The proximity of measured values and predicted values by ALO-RF, WVO-RF, and GOA-RF models to the diagonal dashed line indicates that all hybrid models are effective for predicting the GS. By comparison of details, the distribution of GOA-RF (Pop = 10) is more concentrated near the diagonal dashed line than ALO-RF (Pop = 10) and WVO-RF (Pop = 100) models. In the training set, the MAE of GOA-RF model is significantly lower than that of ALO-RF and WVO-RF models (ALO-RF: 2.9516, WVO-RF: 2.865, GOA-RF: 2.8224); the RMSE of GOA-RF model is also the lowest among three models (ALO-RF: 5.4133, WVO-RF: 5.3441, GOA-RF: 4.93); and the R² value of GOA-RF model is higher than ALO-RF and WVO-RF models (ALO-RF: 0.9338, WVO-RF: 0.9409, GOA-RF: 0.9487); the MAPE of GOA-RF model is similar to the WVO-RF model, which are both at a low level (ALO-RF: 50.5681, WVO-RF: 40.3814, GOA-RF: 40.5629). In the test set, the MAE of GOA-RF model is the lowest among three models (ALO-RF: 2.819, WVO-RF: 2.5786, GOA-RF: 2.3507); similarly, the RMSE of GOA-RF model is also the lowest value (ALO-RF: 3.5978, WVO-RF: 3.2843, GOA-RF: 3.1576); and the R² of GOA-RF presents is higher than ALO-RF and WVO-RF models (ALO-RF: 0.9055, WVO-RF: 0.9202, GOA-RF: 0.9282); the MAPE of GOA-RF is at a low value as the same as with WVO-RF (ALO-RF: 48.5579, WVO-RF: 38.7341, GOA-RF: 38.7537). This result demonstrated that GOA-RF is the best hybrid model for GS prediction. In the training phase, the error range of all three hybrid models is primarily concentrated within 10 mm, whereas it is concentrated within 8 mm in the testing phase. It is evident that all hybrid models have an effective error control.

After determining the optimal population size for each hybrid model (ALO-RF, WVO-RF, and GOA-RF), further comparisons were made with the unoptimized RF model. Figure 9 depicts Taylor diagrams for four models and lists the corresponding evaluation indices. Taylor diagrams can demonstrate the model fitting performance for solving regression problems. The picture depicts a negative correlation between the azimuth of the points and the correlation coefficient, while the distance between the point and the reference point represents the RMSE. This tool is commonly used to compare the observed results and predicted results by prediction model [95]. The positions of ALO-RF (Pop = 10), WVO-RF (Pop = 100), and GOA-RF (Pop = 10) models are closer to the observed points in the Taylor diagram than the unoptimized RF model, indicating that the three models have a lower RMSE than the unoptimized RF model. In addition, the smaller azimuth showed that ALO-RF, WVO-RF, and GOA-RF have greater correlation coefficients than unoptimized RF sites, which demonstrated that the three optimized algorithms play a significant role in improving the accuracy of GS prediction. Based on the calculated evaluation indices of all models in the training and testing phases, the R² values of ALO-RF (Pop = 10) (Train: R² = 0.9338, Test: R² = 0.9055), WVO-RF (Pop = 100) (Train: R² = 0.9409, Test: R² = 0.9202), and GOA-RF (Pop = 10) (Train: R² = 0.9487, Test: R² = 0.9282) are significantly greater than the unoptimized RF (Train: R² = 0.7912, Test: R² = 0.7195), indicating that the prediction accuracy of the three hybrid RF models are more accurate than the unoptimized RF model. Furthermore, all hybrid RF models provided significantly lower RMSE, MAE and MAPE values than the unoptimized RF model for the GS prediction.

Figure 7d ranks the evaluation metrics for ALO-RF (Pop = 10), WVO-RF (Pop = 100), GOA-RF (Pop = 10), and RF as shown in Figure 10. The Taylor diagrams in Figure 9 show that the point represented by the GOA-RF model is the closest to the observation point (indicating its smallest RMSE) and its azimuth is the smallest (indicating its largest correlation coefficient), both in the training and testing phases. In conjunction with the comparison and the distribution of Taylor diagrams points shown in Figure 9b, the GOA-RF (Pop = 10) model achieves the optimal GS prediction performance among all hybrid models.

Table 5. Comparison of the prediction effect of GOA-RF (Pop = 10) and previous studies.

Method	R2
Training set
GOA-RF(Pop = 10)	0.9487
RF	0.7912
MARS Methodology [61]	0.906
Test set
GOA-RF(Pop = 10)	0.9282
RF	0.7195
MARS Methodology [61]	0.721

displays the results of the predictive performance comparison between the proposed GOA-RF model, the unoptimized RF model, and previous studies. The results indicated that the GOA-RF model has significantly improved the GS prediction accuracy compared to previous published results. On the other hand, the GOA algorithm has a significant advantage in optimizing the hyperparameters of RF model for the GS prediction. In addition, there is no significant over- or under-fitting occurred in the GOA-RF (Pop = 10) model using the train and test sets. Therefore, it is recommended to use this model to predict the GS.

4.3. Parameter Importance Analysis

In this study, GOA-RF (Pop = 10) is the best model for the GS prediction. Typically, the out-of-bag (OOB) data error rate is used as the basis for assessing correlation in the RF model, which also represents the contribution degree of variables to the results [96].

Figure 10 illustrates the importance of each relevant parameter for the GS prediction based on the GOA-RF (Pop = 10) model. The results clearly illustrate that MC, AR, Sm, and GP have achieved higher importance scores (2.586, 2.277, 2.078, and 1.875) than E, EP, and H with relatively lower importance scores of 1.55, 1.078, and 1.052, respectively. Therefore, MC, AR, Sm, and GP are more important for the GS prediction than E, EP, and H. Although these seven parameters are different from the GS, the difference in importance scores indicates that neither parameter can be discarded for predicting GS. In future research, additional geological, design, and operational parameters should be considered to improve the model efficiency and precision.

5. Conclusions

In this investigation, three RF models optimized by ALO, WVO, and GOA were developed to predict GS. To avoid data redundancy and improve the scientific rigor of the data, the Pearson correlation analysis was used to exclude feature parameters with strong correlation prior to database construction. The database contains seven input parameters, namely H, AR, EP, Sm, MC, E, and GP.

After completing model construction, the ALO-RF, WVO-RF, and GOA-RF models were evaluated using the evaluation indices MAE, MAPE, R², and RMSE. The GOA-RF (Pop = 10) model exhibited the best performance in the overall GS prediction, with MAE (Train set: 2.8224; Test set: 2.3507), RMSE (Train set: 4.93; Test set: 3.1576), R² (Train set: 0.9487; Test set: 0.9282), and MAPE (Train set: 40.5629; Test set: 38.5637). Finally, the importance of each input parameter is calculated using the best prediction model (GOA-RF), and the results indicated that MC, AR, Sm, and GP play a crucial role in the GS prediction.

In future research, more geological, design, and operational parameters should be taken into account to enrich the database and make the prediction of ground settlement more accurate and efficient, therefore providing a more solid scientific foundation for the prevention and control of disasters related to ground settlement. In addition, the proposed GOA-RF model can inspire the development of prediction models in other disciplines.