Performance evaluation of rock fragmentation prediction based on RF-BOA, AdaBoost-BOA, GBoost-BOA, and ERT-BOA hybrid models-深地科学

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Performance evaluation of rock fragmentation prediction based on RF-BOA, AdaBoost-BOA, GBoost-BOA, and ERT-BOA hybrid models

Abstract

Rock fragmentation is an important indicator for assessing the quality of blasting operations. However, accurate prediction of rock fragmentation after blasting is challenging due to the complicated blasting parameters and rock properties. For this reason, optimized by the Bayesian optimization algorithm (BOA), four hybrid machine learning models, including random forest, adaptive boosting, gradient boosting, and extremely randomized trees, were developed in this study. A total of 102 data sets with seven input parameters (spacing-to-burden ratio, hole depth-to-burden ratio, burden-to-hole diameter ratio, stemming length-to-burden ratio, powder factor, in situ block size, and elastic modulus) and one output parameter (rock fragment mean size, X50) were adopted to train and validate the predictive models. The root mean square error (RMSE), the mean absolute error (MAE), and the coefficient of determination ( $𝑅^{2}$ ) were used as the evaluation metrics. The evaluation results demonstrated that the hybrid models showed superior performance than the standalone models. The hybrid model consisting of gradient boosting and BOA (GBoost-BOA) achieved the best prediction results compared with the other hybrid models, with the highest R2 value of 0.96 and the smallest values of RMSE and MAE of 0.03 and 0.02, respectively. Furthermore, sensitivity analysis was carried out to study the effects of input variables on rock fragmentation. In situ block size (XB), elastic modulus (E), and stemming length-to-burden ratio (T/B) were set as the main influencing factors. The proposed hybrid model provided a reliable prediction result and thus could be considered an alternative approach for rock fragment prediction in mining engineering.

Highlights

A large data set that consists of 102 data was used in this study.
Rock fragmentation prediction models were developed by the combination of four machine learning methods and a Bayesian optimizer.
R2, the root mean square error, and the mean absolute error were used to evaluate the prediction capabilities of the hybrid prediction models.
The factors influencing rock fragmentation were determined through sensitivity analysis.

1 INTRODUCTION

Blasting, as a common method for tunnel excavation and rock breakage, is widely used in mining and tunneling engineering. The distribution of rock fragmentation after blasting directly affects downstream operations like shoveling, loading, and milling (Amoako et al., 2022; Enayatollahi et al., 2014; Inanloo Arabi Shad et al., 2018; Kinyua et al., 2022; Shehu et al., 2022). Oversized fragments require secondary crushing to improve the efficiency of shovels and loaders, resulting in higher explosive costs and safety issues. Finer particles, on the contrary, are more susceptible to water and easily cemented, increasing the risk of chute blockage. In addition, blasting-induced rock fracture behavior characterization cannot be overlooked in engineering. To address this problem, several experiments and theoretical analyses of rock fracture mechanisms under different conditions have been conducted (Han, Li, Li, et al., 2023; Han, Li, Wang, et al., 2023). Since the above phenomenon occurs in most cases, it is crucial for engineers to quickly obtain rock fragment size after the blasting operation.

Rock fragment size can be measured by direct and indirect methods. The most well-known direct method is the sieving method. Although the sieving method can render more precise results, it is time-consuming, costly, and only suitable for small-scale blasting tests (Amoako et al., 2022; Bamford et al., 2021; Kinyua et al., 2022; Qiao et al., 2021). As a result, empirical models and indirect methods have been widely applied in engineering. The Kuz–Ram model (Cunningham, 1987), the extended Kuz–Ram model (Hekmat et al., 2019; Lawal, 2021; Morin & Ficarazzo, 2006), the KCO model (Ouchterlony, 2005), the SveDeFo model (Hjelmberg, 1983), and the Kou–Rustan model (Kou & Rustan, 1993) are the most commonly used empirical models. However, the empirical models also have their limitations: (1) they are established based on a single mathematical equation, thus failing to capture the complex relationships between the influential factors and rock fragment size (Amoako et al., 2022; Li, Yang, et al., 2021), and (2) the key influential factors that affect rock fragmentation, such as rock properties, charge structure, and detonation method, cannot be explained well quantitatively.

Artificial intelligence (AI)-based methods have displayed excellent advantages in terms of complex nonlinear problems in recent years, and are thus being extensively used in engineering fields, such as rock physical and mechanical parameter prediction (Ceryan et al., 2013; He et al., 2021; Li et al., 2020; Momeni et al., 2015), rock lithology classification (Li et al., 2022a, 2022b), and rock burst prediction (Li et al., 2022a, 2022b; Zhou et al., 2020). The remarkable performance of AI-based methods has attracted much attention from scholars engaged in rock fragmentation prediction. For instance, Kulatilake et al. (2010) trained a shallow artificial neural network (ANN) to predict X50. The Levenberg–Marquardt algorithm was adopted to optimize the hidden layer units, and the trained model was proven to be practical in mines. Enayatollahi et al. (2014) developed a two-hidden layer ANN model, and the experimental results demonstrated that the ANN model outperformed the regression models and the empirical Kuz–Ram model. Dimitraki et al. (2019) introduced a simple ANN model for rock fragment size prediction. The model required only three input parameters (blastability index, powder factor, and quantity of blasted rock pile) and produced better results with a coefficient of determination ( $𝑅^{2}$ ) value of 0.8, which was higher than that of the regression model ( $𝑅^{2}$ value of 0.7). Hasanipanah et al. (2018) integrated an adaptive neuro-fuzzy inference system model and an optimization algorithm for the prediction of X50. The model predictive performance outperformed the multiple regression equation (MR) method. Moreover, Asl et al. (2018) developed a hybrid model using ANN and the firefly optimization algorithm. The values of $𝑅^{2}$ and the root mean square error (RMSE) were 0.94 and 0.10, respectively, and the optimized model reduced rock fragmentation size by 32.9%. Li, Koopialipoor, et al. (2021) proposed an optimized support vector regression (SVR) model by combining five optimization algorithms, and the optimized model achieved satisfactory performance with 19 input parameters.

Although ANN methods have achieved reliable results in rock fragmentation prediction, the performance of ensemble learning algorithms deserves in-depth study. Ensemble learning algorithms such as random forest (RF), adaptive boosting (AdaBoost), gradient boosting (GBoost), and extremely randomized trees (ERT) are simple and practical. Accordingly, they have received considerable attention in recent years. Zhang et al. (2021) and Zhou et al. (2019) used an RF-based method for material strength prediction, and the trained model showed good prediction capabilities. Zhou et al. (2020) developed an RF model to estimate the ground vibration by using the Bayesian optimization algorithm (BOA). The model obtained 92.95% and 90.32% accuracy on the training and testing sets, respectively. Dai et al. (2022) combined an RF model and a particle swarm optimization algorithm to estimate the backbreak after blasting, and the $𝑅^{2} obtained by the optimized model$ reached 0.9961 on the testing data set. AdaBoost, another ensemble learning algorithm, has also been used in a wide range of applications, including rock mass classification (Liu et al., 2020), rockburst prediction (Wang et al., 2021), rock strength estimation (Liu et al., 2022), tunnel boring machine (TBM) performance prediction (Zhang et al., 2020), and engineering geology (Wu et al., 2020; Yang et al., 2019). According to Geurts and Louppe (2011), the ERT model had a higher prediction accuracy than the RF model. Overall, the studies mentioned above have verified the excellent abilities of ensemble learning algorithms. This paper aims to use four ensemble learning algorithms to predict rock fragmentation. In addition, the BOA, proposed by the American scholar Pelikan et al. (1999), is utilized to obtain the optimal hyperparameters of the ensemble learning models. BOA boasts of advantages of fewer iterations and higher processing speed. Accordingly, it has been extensively utilized to fine-tune the parameters of machine learning (ML) models (Canayaz et al., 2022; Egberts et al., 2022; Snoek et al., 2012). In all, four hybrid models, namely, RF-BOA, AdaBoost-BOA, GBoost-BOA, and ERT-BOA, are established to predict rock fragmentation.

This paper is organized as follows: In Section 2, the background of the four ensemble learning algorithms and the BOA is introduced. In Section 8, the process of hybrid model development is described. In Section 13, the prediction capabilities of the four hybrid models (RF-BOA, AdaBoost-BOA, GBoost-BOA, and ERT-BOA) are analyzed and discussed, and the optimal hybrid model for rock fragmentation prediction is confirmed. Finally, sensitivity analysis is implemented to investigate the effects between the input variables and the output X50.

2 METHODOLOGY

In this paper, four ensemble learning algorithms, namely, RF, adaptive boosting, gradient boosting, and ERT, are used to predict rock fragmentation, and the structures are shown in Figure 1. Furthermore, the BOA is adopted to determine the best parameters of all models to improve model prediction accuracy.

Details are in the caption following the image — Figure 1
Open in figure viewer PowerPoint

Flowchart of the ensemble learning algorithms. (a) Random forest method; (b) adaptive boosting method; (c) gradient boosting method (a–c are modified from Chen et al. [ 2022]); and (d) extremely randomized trees method (modified from Yang et al. [ 2021]).

2.1 RF

RF is a commonly used method for addressing classification and regression issues. The RF model has strong generalization capabilities, and thus can achieve better results even with noise data sets (Zhang et al., 2022). Typically, only two parameters need to be fine-tuned: the maximum depth of the tree and the number of estimators, which makes it more user-friendly. For regression prediction, the RF model generates N regression trees and renders the final result by integrating the output of all the independent regression trees, as shown in Equation ( 1). The structure is shown in Figure 1a.

𝑓 (𝑥) = \frac{1}{𝐾} \sum_{𝑘 = 1}^{𝐾} 𝑡_{𝑖} (𝑥),

(1)

where

𝑓 (𝑥)

indicates the final result;

𝑥

represents the input vector;

𝑡_{𝑖}

is the result of each decision tree; and

𝐾

is the number of regression trees.

2.2 Adaptive boosting

AdaBoost is a typical ensemble boosting method that consolidates weak learners with strong learners. It gives full consideration to the weights of all learners so as to obtain a more accurate and less overfitted model. Its framework is shown in Figure 1b and the procedure of the algorithms is shown as follows:

(1)
Determine the regression error rate of the weak learner.

Obtain the maximum error;

$𝐸_{𝑡} = \max | 𝑦_{𝑖} - ℎ_{𝑡} (𝑥_{𝑖}) | (𝑖 = 1, 2, \dots, 𝑚),$ (2)
where
$𝑥_{𝑖}, and 𝑦_{𝑖}$
indicate the input data; m is the number of samples; and
$ℎ_{𝑡} (𝑥)$
represents the weak learner of the t-th iteration.
Estimate the relative error for each sample by using the linearity loss function;

$𝜖_{ti} = \frac{| 𝑦_{𝑖} - ℎ_{𝑡} (𝑥_{𝑖}) |}{𝐸_{𝑡}} .$ (3)
Determine the regression error rate;

$𝜖_{𝑡} = \sum_{𝑖 = 1}^{𝑛} 𝑤_{𝑡 𝑖} 𝜖_{𝑡 𝑖},$ (4)
where
$𝑤_{ti}$
denotes the weights of the sample
$𝑥_{𝑖}$
.

(2)
Determine the weight coefficient $𝛼$ of the weak learner.

$𝛼_{𝑡} = \frac{𝜖_{𝑡}}{1 - 𝜖_{𝑡}} .$ (5)
(3)
Sample weight updating for the $𝑡 + 1$ iteration round.

$𝑊_{𝑡 + 1} (𝑥_{𝑖}) = \frac{𝑊_{𝑡} (𝑥_{𝑖})}{𝑍_{𝑡}} {𝛼_{𝑡}}^{1 - 𝜖_{ti}},$ (6)

$\begin{array}{l} 𝑍_{𝑡} = \sum_{𝑖 = 1}^{𝑚} 𝑊 (𝑥_{𝑡 𝑖}) {𝛼_{𝑡}}^{1 - 𝜖_{𝑡 𝑖}}, \end{array}$ (7)
where Zt denotes the normalization factor.
(4)
Build the ultimate learner.

$𝐻 (𝑥) = \sum_{𝑡 = 1}^{𝑇} \ln (\frac{1}{𝛼_{𝑡}}) 𝑓 (𝑥),$ (8)
f (x) is the median value of αt ht (x)(t = 1, 2, …, T).

2.3 Gradient boosting

GBoost is another empirical ensemble algorithm derived from a gradient descent, whose flowchart is shown in Figure 1c. Given a set of training data sets

𝐷 = {(𝑥_{1}, 𝑦_{1}), (𝑥_{2}, 𝑦_{2}), \dots, (𝑥_{𝑚}, 𝑦_{𝑚})}

, the maximum iterations number

𝑇

, and the squared error loss function

𝐿 (𝑦, 𝐻 (𝑥))

, a strong learner could be obtained as follows:

(1)
Initialize the base learner.

$𝐻_{0} (𝑥) = arg \min_{𝑐} \sum_{𝑖 = 1}^{𝑚} 𝐿 (𝑦_{𝑖}, 𝑐),$ (9)

$𝑐 = \frac{\sum_{𝑖 = 1}^{𝑚} 𝑦_{𝑖}}{𝑚},$ (10)
where
$𝐿 (𝑥)$
indicates the squared error loss function;
$𝑐$
is a constant; and
$𝑦_{𝑖}$
is the target value.
(2)
Calculate the negative gradient.
Then, the leaf node region $𝑅_{tj}$ of the t-th regression tree ( $𝐽$ is the number of leaf nodes of a regression tree $𝑡$ ) could be obtained according to $(𝑥_{𝑖}, 𝑟_{ti})$ .

$𝑟_{ti} = - \frac{\partial 𝐿 (𝑦_{𝑖}, ℎ_{𝑡 - 1} (𝑥_{𝑖}))}{\partial ℎ_{𝑡 - 1} (𝑥_{𝑖})} (𝑖 = 1, 2, \dots, 𝑚) .$ (11)
(3)
Obtain the optimal-fitted value $𝑐_{tj}$ for the leaf region j = 1, 2, …, J, and update the strong learner.

$𝑐_{tj} = arg \min_{𝑐} \sum_{_{𝑥_{𝑖} \in 𝑅_{𝑡 𝑗}}} 𝐿 (𝑦_{𝑖}, ℎ_{𝑡 - 1} (𝑥_{𝑖}) + 𝑐),$ (12)

$ℎ_{𝑡} (𝑥) = ℎ_{𝑡 - 1} (𝑥) + \sum_{𝑗 = 1}^{𝐽} 𝑐_{tj} 𝐼 (𝑥 \in 𝑅_{tj}),$ (13)
where I is the identity matrix.
(4)
Finally, a strong learner can be obtained.

𝑓 (𝑥) = 𝑓_{𝑇} (𝑥) = \sum_{𝑡 = 1}^{𝑇} \sum_{𝑗 = 1}^{𝐽} 𝑐_{tj} 𝐼 (𝑥 \in 𝑅_{tj}) .

(14)

The advantage of GBoost consists of its flexibility in choosing the loss function, which makes it possible to use any continuously derivable loss function. Also for this reason, it is feasible to adopt some robust loss functions to make the model more robust. Therefore, the GBoost method has achieved many successful applications and more modified versions have been developed (Chen & Guestrin, 2016; Ke et al., 2017; Prokhorenkova et al., 2018).

2.4 ERT

Figure 1d shows the architecture of the ERT model, which is similar to that of the RF model. The ERT model sets K features randomly from the entire sample. Each of these K features yields K classification nodes, and the node with the highest score is considered the final split node. According to Geurts and Louppe (2011), ERT showed better prediction ability than the RF model. Hameed et al. (2021) applied ERT to obtain the discharge coefficient for side weirs, and the model prediction results were more reliable than the field engineer's analysis results. Djarum et al. (2021) explored ML algorithms for river water quality prediction using linear discriminant analysis. The results revealed that the ERT-based model outperformed other ensemble models. In addition, research has been carried out to study the application of the ERT model to predict stock prices (Pasupulety et al., 2019; Polamuri et al., 2019).

2.5 BOA

Compared with the commonly used grid and random search algorithms, BOA takes full advantage of prior information to determine the parameters that maximize the target function globally. The algorithm is composed of two parts: (1) Gaussian process regression, which aims to determine the values of the mean and variance of the function at each point, and (2) the acquisition function, which is used to obtain the search position of the next iteration, as shown in Figure 2. BOA boasts of advantages of fewer iterations and higher processing speed, because of which it has been applied in several fields. Li, Zhao, and Ma (2022) proposed an intelligent model for rockburst prediction using the BOA for hyperparameter optimization. Lahmiri et al. (2023) used the BOA to obtain the optimal parameters of models for house price prediction. Bo et al. (2023) developed an ensemble classifier model to assess tunnel squeezing hazards using the BOA method, and the optimal values of the 17 parameters were obtained.

3 IMPLEMENTATION OF HYBRID MODELS

Figure 3 depicts the overall framework for rock fragmentation prediction, which primarily consists of three parts: (1) statistical analysis, data transformation, and outliers' elimination; (2) training and optimization of ensemble learning models; and (3) sensitivity analysis to study the relationships between the input variables and the output X50. The details of each part are described in the subsequent subsections.

3.1 Data analysis and preprocessing

In this paper, 102 blasting events collected from the previous study of Hudaverdi et al. (2011) were used to develop the predictive model. The blasts were conducted at various mines around the world, including Akdaglar quarry, the Dongri–Buzurg open-pit manganese mine, the Enusa mine, the Murgul copper mine, Mrica quarry, the Reocin underground mine, the Soma basin mine, and Ozmert quarry. The data set includes seven input parameters (i.e., spacing-to-burden ratio [S/B], hole depth-to-burden ratio [H/B], burden-to-hole diameter ratio [B/D], stemming length-to-burden ratio [T/B], powder factor [fp], in situ block size [XB], and elastic modulus [E]) and one output parameter (rock mean fragment size X50). S/B, H/B, B/D, and T/B are the blast design parameters; fp is the explosive parameter; and XB and E are the rock mass properties. Figure 4 shows the detailed distribution of all variables. The abscissa indicates the value range of each parameter, and the ordinate represents the frequency corresponding to each parameter.

3.1.1 Exploratory data analysis (EDA)

A standard or approximately normal distribution data set is usually more convenient for statistical analysis. However, it is not always possible for real-world data. In this case, EDA is needed. In this paper, the statistical index description of the data set and the corresponding Pearson correlation matrix are presented first, as shown in Table 1 and Figure 5. In summary, all variables have a strong nonlinear relationship with the rock fragment mean size X50.

Table 1. Statistical description of all variables.

Statistical indexes	S/B	H/B	B/D	T/B	fp (kg/m3)	XB (m)	E (GPa)
Mean value	1.20	3.46	27.22	1.27	0.53	1.18	30.59
Standard deviation	0.11	1.64	4.80	0.69	0.24	0.48	17.76
Min value	1.00	1.33	17.98	0.50	0.22	0.29	9.57
25th percentiles	1.16	2.40	24.72	0.83	0.35	0.83	15.48
50th percentiles	1.20	3.19	27.27	1.13	0.48	1.08	24.45
75th percentiles	1.25	4.80	30.30	1.39	0.65	1.56	45.00
Max value	1.75	6.82	39.47	4.67	1.26	2.35	60.00

3.1.2 Data preprocessing

The skewness values of all variables are calculated and shown in Table 2. As can be seen, the values of S/B, T/B, and fp are all greater than one, thus being highly skewed. The other three parameters, H/B, B/D, and XB, have smaller values and are closer to the normal distribution.

Table 2. Skewness values of all variables.

Parameters	S/B	H/B	B/D	T/B	fp	XB
Skewness	1.343	0.393	0.111	2.722	1.422	0.281

The two most commonly used data transformation and normalization approaches are square root transformation and logarithmic transformation (Nishida, 2010). Logarithmic transformation is more suitable for highly skewed data and is thus used in this paper. As shown in Table 3, the skewness values of S/B, T/B, and fp are all reduced after logarithmic transformation.

Table 3. Comparison results of the skewness values after transformation.

Skewness values	S/B	H/B	B/D	T/B	fp	XB
Before transformation	1.343	0.393	0.111	2.722	1.422	0.281
After transformation	0.581	0.393	0.111	0.948	0.495	0.281

Last but not least, data cleaning is also an extremely significant step. The distribution results of the three transformed parameters are shown in Figure 6, and it can be seen that there are some outliers of parameters S/B and T/B. The outlier is defined as a value less than Q1-1.5IQR or greater than Q3 + 1.5IQR. Q1 is the 25th percentile of the data; Q3 refers to the 75th percentile of the data; and IQR (the abbreviation of interquartile range) equals Q3–Q1. According to this principle, the outliers of parameters S/B and T/B are all removed, as shown in Figure 7, and the result is closer to the normal distribution.

3.2 Model training

The training codes are written in Python programming language, and all the aforementioned algorithms are performed based on the framework of the Scikit-learn library (Pedregosa et al., 2011).

4 RESULTS, ANALYSIS, AND DISCUSSION

4.1 Evaluation indexes

Three evaluation metrics, namely, RMSE, MAE, and

𝑅^{2}

, are used to evaluate the predictive performance of all models. The lower the RMSE and MAE values, the better the model performs. This indicates that the prediction results are closer to the measured values. Conversely, the greater the value of

𝑅^{2}

, the more robust the model will be, and the maximum value is equal to 1.

RMSE = \sqrt{\frac{1}{𝑚} \sum_{𝑖 = 1}^{𝑚} {(𝑦_{𝑖} - 𝑦_{𝑖}^{'})}^{2}},

(15)

MAE = \frac{1}{𝑚} \sum_{𝑖 = 1}^{𝑚} | 𝑦_{𝑖} - 𝑦_{𝑖}^{'} |,

(16)

𝑅^{2} = 1 - \frac{\sum_{𝑖 = 1}^{𝑚} {(𝑦_{𝑖} - 𝑦_{𝑖}^{'})}^{2}}{\sum_{𝑖 = 1}^{𝑚} {(𝑦_{𝑖} - \bar{𝑦})}^{2}},

(17)

where

𝑦_{𝑖}

𝑦_{𝑖}^{'}

, and

\overset{̅}{𝑦}

represent the target value, the prediction result, and the average of all the target values, respectively.

The prediction capabilities of the four standalone ensemble models, the hybrid models (RF-BOA, AdaBoost-BOA, GBoost-BOA, and ERT-BOA), and the previous studies (Amoako et al., 2022; Hudaverdi et al., 2011) are compared in this section. Table 4 shows the testing data set, which is completely consistent with the previous studies.

Table 4. Testing data set (Hudaverdi et al., 2011; Kulatilake et al., 2010).

ID	S/B	H/B	B/D	T/B	fp (kg/m3)	XB (m)	E (GPa)	X50 (m)
Ad23	1.11	4.44	18.95	1.67	1.25	1.63	16.90	0.21
Ad24	1.28	3.61	18.95	1.67	0.89	0.61	16.90	0.20
Db10	1.15	4.35	20.00	1.75	0.89	1.00	9.57	0.35
En13	1.24	1.33	27.27	0.78	0.48	1.11	60.00	0.47
Mg8	1.10	2.40	30.30	0.80	0.55	1.23	50.00	0.44
Mg9	1.00	2.67	27.27	0.89	0.75	0.77	50.00	0.25
Mr12	1.25	6.25	31.58	0.63	0.48	1.03	32.00	0.20
Ru7	1.13	5.00	39.47	3.11	0.31	2.00	45.00	0.64
Sm8	1.25	2.50	28.57	0.83	0.42	0.50	13.25	0.18
Oz8	1.20	2.40	28.09	1.00	0.53	0.82	15.00	0.23
Oz9	1.11	3.33	30.34	1.11	0.47	0.54	15.00	0.17

Where the symbols Ad, Db, En, Mg, Mr, Ru, Sm, and Oz indicate the abbreviation of Akdaglar quarry, the Dongri–Buzurg open-pit manganese mine, the Enusa mine, the Murgul copper mine, Mrica quarry, the Reocin underground mine, the Soma basin mine, and Ozmert quarry, respectively.

4.2 Optimization of the hybrid models

The hyperparameters for the RF model that need to be optimized include the maximum number of estimators, maximum features, and maximum decision tree depth. However, the stability of the AdaBoost algorithm is dependent on the parameters' learning rate, the maximum decision tree depth, and the loss function. The hyperparameters of the GBoost algorithm that need to be optimized are maximum number of estimators, maximum decision tree depth, learning rate, and maximum features. The main optimization parameters of the ERT method include maximum features, maximum decision tree depth, minimum number of samples required to split an internal node, and minimum number of samples required to be at a leaf node. To optimize the corresponding parameters of all the ensemble learning models, BOA is used to obtain the optimal results with 500 iterations.

In this section, the prediction performance of RF-BOA, AdaBoost-BOA, GBoost-BOA, and ERT-BOA hybrid models is systematically evaluated and compared. Figure 8a depicts the loss convergence of the four hybrid models during the training process. The loss values of hybrid models RF-BOA and ERT-BOA converge to the same level and yield comparable prediction performance, as shown in Figure 8a,b. The values of $𝑅^{2}$ , RMSE, and MAE of RF-BOA are 0.88, 0.05, 0.05, and those of ERT-BOA are 0.89, 0.05, and 0.04, respectively. Although the minimum loss value of the AdaBoost model is smaller than that of the RF-BOA and ERT-BOA models, the entire loss curve is more oscillating. Obviously, the hybrid model GBoost-BOA is the best-performing model among the four hybrid models. The loss value, on the one hand, converges to the lowest level; the value of $𝑅^{2}$ is the highest ( $𝑅^{2} = 0.96$ ); and the values of RMSE and MAE are 0.03 and 0.02, which are also the best. The optimal hyperparameters of the four hybrid models are obtained by using the BOA, as shown in Table 5.

Table 5. Optimized values of the hyperparameters for different machine learning models.

Models	Hyperparameters	Optimal value	Search space
RFR	Maximum number of estimators	65	1–200
	Maximum features	5	1–10
	Maximum depth of the tree	8	1–50
ABR	Learning rate	0.1	0–1
	Maximum number of estimators	96	1–200
	The loss function	Square	Linear, square, etc.
GBR	Maximum number of estimators	125	1–200
	Maximum depth of the tree	6	1–50
	Learning rate	0.15	0–1
	Maximum features	6	1–10
ERT	Maximum features	4	1–10
	Maximum depth of the tree	5	1–50
	Minimum number of samples required to split an internal node	2	2–30
	Minimum number of samples required to be at a leaf node	1	1–30

Abbreviations: ABR, adaptive boosting regression; ERT, extremely randomized trees; GBR, gradient boosting regression; RFR, random forest regression.

Besides, Figure 9a,b show the rock fragmentation prediction results of the four hybrid models. The abscissa axis indicates the measured values; the ordinate axis denotes the model prediction results; and the green dotted line $𝑦 = 𝑥$ represents the reference line. The closer the prediction result is to the reference line, the better the model performs. As can be seen, the prediction results of the GBoost-BOA hybrid model almost match the reference line, whereas the other three hybrid models perform slightly worse. Accordingly, the model prediction capability is ranked in descending order as follows: GBoost-BOA, AdaBoost-BOA, ERT-BOA, and RF-BOA.

4.3 Comparison between hybrid models and standalone models

Figure 10 summarizes the prediction results of all standalone ensemble learning models and the corresponding hybrid models on the testing data set. The red pentagrams indicate the prediction results of the standalone ensemble learning model; the solid balls denote the prediction results of the hybrid models; and the green dotted line represents the reference line. As shown in Figure 10, compared with solid balls, the red pentagrams are further away from the reference line, indicating that the hybrid model is more robust than standalone models and that the optimization algorithm is useful for the improvement of the model prediction capability. In addition, a box plot is applied to show the detailed errors between all the predictive model results and the measured values, as shown in Figure 11. Obviously, the prediction performance of the four hybrid models is better than that of the corresponding standalone models.

Furthermore, as shown in Figure 12, the evaluation indices $𝑅^{2}$ , RMSE, and MAE are calculated to quantitatively compare the prediction capabilities of the standalone models and the hybrid models. All hybrid models have higher $𝑅^{2}$ values than the standalone models, with an improvement rate greater than 15%. The largest amplitude of improvement appears between the AdaBoost–BOA hybrid model and the corresponding standalone model, reaching up to 37.44%. The values of RMSE and MAE of all the hybrid models, on the contrary, are lower than those of the standalone models, and the reduction rates of RMSE and MAE are more than 30% and 25%. According to the evaluation metrics, all hybrid models outperform the corresponding standalone models.

4.4 Comparison results between the hybrid model and previous literature methods

Furthermore, several studies on rock fragmentation prediction are proposed using this data set. Table 6 shows a comparison of performance between the previous studies and the GBoost–BOA hybrid model.

Table 6. Comparison results of X 50 predicted by different methods. m

ID	X50	X50-GBoost-BOA	X50-K	X50-MVR	X50-ANN	X50-SVR
Ad23	0.21	0.22	0.12	0.19	0.21	0.20
Ad24	0.20	0.20	0.13	0.15	0.21	0.19
Db10	0.35	0.35	0.09	0.16	0.21	0.52
En13	0.47	0.41	0.48	0.39	0.44	0.38
Mg8	0.44	0.40	0.42	0.40	0.38	0.41
Mg9	0.25	0.29	0.33	0.24	0.25	0.25
Mr12	0.20	0.19	0.27	0.14	0.15	0.14
Ru7	0.64	0.64	0.71	0.51	0.68	0.61
Sm8	0.18	0.19	0.38	0.17	0.19	0.19
Oz8	0.23	0.24	0.22	0.17	0.17	0.18
Oz9	0.17	0.20	0.25	0.17	0.17	0.19

Note: The results of the X50-K, X50-ANN, and X50-SVR were presented by Amoako et al. (2022), and the result of the X50-MVR was reported by Hudaverdi et al. (2011). Abbreviation: MVR, multivariate regression.

Figure 13 displays the comparison of the evaluation metrics between the existing models and the GBoost–BOA model. As can be seen, the existing models are the same as those described in Table 6. First, it can be concluded that the prediction methods based on ML or ANN-based prediction methods (such as the GBoost-BOA, ANN, and SVR model) outperform traditional methods (Kuz–Ram and MVR methods). Furthermore, as illustrated in Figure 14, the median values of the box plots for the GBoost-BOA, ANN, and SVR models are lower than those of the Kuz–Ram model and the MVR method. Second, the GBoost–BOA hybrid model shows the best prediction performance compared with Kuz–Ram, MVR, ANN, and SVR.

In summary, the findings of this paper show that the hybrid model GBoost–BOA developed in this paper outperforms the models developed in the previous studies (Amoako et al., 2022; Hudaverdi et al., 2011) as well.

4.5 Sensitivity analysis

In addition, sensitivity analysis was conducted to better understand the intrinsic relationships between the seven independent variables and rock fragmentation. The relevancy factor is a commonly used method to illustrate the sensitivity scale (Bayat et al., 2021; Chen et al., 2014). Accordingly, it is applied in this paper to assess the influence of each variable on rock fragmentation. It is common knowledge that the absolute value of the RF between the independent and dependent variables is positively correlated with the influence of the variable on rock fragmentation. In some cases, it is also necessary to specify whether the correlation is positive or negative. The main form of the sensitivity relevancy factor ( SRF) is as follows:

̅ ̅

SRF = \frac{\sum_{𝑖 = 1}^{𝑛} (𝑥_{li} - \overset{̅}{𝑥_{𝑙}}) \times (𝑦_{𝑖} - \bar{𝑦})}{\sqrt{\sum_{𝑖 = 1}^{𝑛} {(𝑥_{li} - \overset{̅}{𝑥_{𝑙}})}^{2} \sum_{𝑖 = 1}^{𝑛} {(𝑦_{𝑖} - \bar{𝑦})}^{2}}},

(18)

where

\overset{̅}{𝑥_{𝑙}}

denotes the mean value of all data for variable l ( l includes S/B, H/B, B/D, T/B, f p, X B, and E);

𝑥_{li}

represents the ith value of variable l;

𝑛

indicates the number of the variable data; and

𝑦_{𝑖}

and

\overset{̅}{𝑦}

are the ith measured value of variable l and the average value of the prediction results, respectively.

Figure 15 shows the results of the sensitivity analysis. Four factors (B/D, T/B, XB, E) show a positive impact on rock fragmentation, whereas S/B and fp exert a negative impact. The RF value of H/B is close to zero and thus can be ignored. Among all variables, XB ( $SRF$ is 0.73) has the greatest effect on rock fragmentation, followed by T/B, E, and B/D. The sensitivity analysis results in this paper are highly consistent with previous studies (Mehrdanesh et al., 2018; Sayadi et al., 2013).

5 CONCLUSIONS

In this study, hybrid predictive models combined with ensemble learning algorithms and Bayesian optimization method were developed for rock fragmentation prediction. In total, 102 samples were used to evaluate the prediction performance of the predictive models. The evaluation results revealed that the hybrid models outperformed the corresponding standalone models. The hybrid model GBoost–BOA achieved far better prediction results than other hybrid models. Specifically, it obtained the highest R2 value of 0.96, and the smallest values of RMSE and MAE, 0.03 and 0.02, respectively, on the testing data set. Moreover, the proposed GBoost–BOA hybrid model showed better accuracy and generalization ability than other models developed previously.

In addition, the sensitivity analysis results indicated that four factors (XB, T/B, E, and B/D) showed positive impacts, and the sensitivity relevancy factors were 0.73, 0.63, 0.60, and 0.54, respectively. S/B and fp reflected negative influences, with the values of the sensitivity relevancy factor being −0.27 and −0.35. The contribution of H/B in the prediction of rock fragmentation was much less and thus negligible. Therefore, XB, T/B, and E were the primary influencing factors among all the variables.

6 LIMITATIONS

Further improvements can be achieved in the following two aspects. First, small-scale data sets are more likely to yield satisfactory experimental results. Accordingly, more field data should be supplemented to enhance the model generalization capabilities. Second, considering the sensitivity analysis results, it is worthwhile to quantitatively analyze the relationships between the main influencing factors and rock fragmentation, which can provide valuable technical instructions for the in situ blasting operations.

AUTHOR CONTRIBUTIONS

Junjie Zhao: Writing—original draft; coding; model training. Diyuan Li: Supervision; writing review. Jian Zhou and Danial J. Armaghani: Writing review. Aohui Zhou: Data collection and process.

ACKNOWLEDGMENTS

This work was financially supported by the National Natural Science Foundation of China (Grant No.: 52374153).

CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.

Biography

Diyuan Li is a professor and doctoral supervisor at Central South University (CSU). He received the PhD degree in geotechnical engineering in 2010 from Central South University with 2 years of overseas joint study in the Norwegian University of Science and Technology from 2007 to 2009. He was selected for the National Youth Talent Program, currently serving as the vice dean of the School of Resources and Safety Engineering at Central South University. He is a director of the Chinese Society of Rock Mechanics and Engineering and Executive Committee Member of the Rock Dynamics and Rock Mechanics Testing Committees. He is mainly engaged in teaching and research work in rock mechanics and rock engineering, and has achieved some innovative results in testing techniques and characterization of rock crack propagation characteristics, rock fracture mechanics behavior and testing, and slabbing failure mechanism and criteria of hard rocks. He has presided over four National Natural Science Foundation projects and two Hunan Provincial Natural Science Foundation projects (including the Outstanding Youth Project). He has already published over 200 academic papers, including 127 indexed by SCI with an H-index of 36. Over the past 3 years, he has been selected as one of the “Highly Cited Scholars in China” by Elsevier. He has been granted 18 national invention patents and has received eight provincial-level and industry association science and technology awards.