Application of Extra-Trees Regression and Tree-Structured Parzen Estimators Optimization Algorithm to Predict Blast-Induced Mean Fragmentation Size in Open-Pit Mines

Abstract

Blasting is an effective technique for fragmenting rock in open-pit mining operations. Blasting operations produce either boulders or fine fragments, both of which increase costs and pose environmental risks. As a result, predicting the mean fragmentation size (MFS) distribution of rock is critical for assessing blasting operations’ quality and mitigating risks. Due to the limitations of empirical and statistical models, several researchers are turning to artificial intelligence (AI)-based techniques to predict the MFS distribution of rock. Thus, this study uses three AI tree-based algorithms—extra trees (ET), gradient boosting (GB), and random forest (RF)—to predict the MFS distribution of rock. The prediction accuracy of the models is optimized utilizing the tree-structured Parzen estimators (TPEs) algorithm, which results in three models: TPE-ET, TPE-GB, and TPE-RF. The dataset used in this study was collected from the published literature and through the data augmentation of a large-scale dataset of 3740 blast samples. Among the evaluated models, the TPE-ET model exhibits the best performance with a coefficient of determination (R²), root mean squared error (RMSE), mean absolute error (MAE), and max error of 0.93, 0.04, 0.03, and 0.25 during the testing phase. Moreover, the block size (XB, m) and modulus of elasticity (E, GPa) parameters are identified as the most influential parameters for predicting the MFS distribution of rock. Lastly, an interactive web application has been developed to assist engineers with the timely prediction of MFS. The predictive model developed in this study is a reliable intelligent model because it combines high accuracy with a strong, explainable AI tool for predicting MFS.

1. Introduction

Rock fragmentation plays a pivotal role in engineering disciplines such as mining, civil engineering, and construction [1,2,3,4]. In mining engineering, the rock fragmentation process entails the disintegration of rock mass or ore into smaller fragments using explosives, drilling apparatuses, and thermal stress. Studies have indicated that only approximately 30% of the explosive energy is effectively used for fragmentation [5,6]. The remaining energy is disseminated into other forms, which leads to adverse effects including back break, airblast, fly rock, and ground vibration [7,8,9,10,11,12,13]. The fragmentation size is the most critical factor during the blasting process, as it directly influences the efficiency of ore extraction, downstream processes, equipment management, and material handling [14,15]. Achieving an ideal fragmentation size not only enhances mines’ productivity but also improves their operational safety. Therefore, a comprehensive understanding of the fragmentation process is essential for making accurate predictions and optimizing outcomes.

Predicting the optimum fragmentation size is a challenging task due to the inherent complexity of blasting techniques [16,17]. To address this, researchers have proposed various approaches to optimize parameters and simplify the problem. For example, Kuznetsov developed a model to predict the mean fragmentation size (MFS) based on the properties of the rock mass and explosives used [18]. Cunningham subsequently refined this model, known as the Kuz–Ram model, by incorporating specific energy and additional rock mass characteristics [19,20,21]. Although these empirical models have limitations, they laid the groundwork for subsequent advancements in the field. Recognizing the need for more accurate predictions, researchers developed models that consider the full range of blasted fragment sizes, such as the distribution-free, Swebrec, and Rosin-Rammler functions [14,22,23,24]. Moreover, improvements to the Kuz–Ram model, such as the modified Kuz–Ram, Bergmann, and Riggle models, were proposed to enhance its predictive accuracy [25,26]. To further improve its practicality, researchers also developed equations for the BRW model to facilitate its real-world application.

Recently, researchers have progressively embraced artificial intelligence (AI)-based methods because of the limitations of the empirical models, especially their reliance on labor and costly field blast experiments. AI-based methods have achieved significant success in the geotechnical domain. For instance, AI-based methods have improved the precision of rock fragmentation predictions, pattern recognition, and data analysis [27,28,29,30,31]. Commonly used AI-based techniques for predicting blast-induced rock fragmentation include artificial neural networks (ANNs), fuzzy logic, genetic algorithms, and ensemble methods.

Table 1. AI-based techniques utilized over the years to predict rock fragmentation.

Source	Methods	Input Parameters	Output Parameter	Number of Datasets	Performance
[17]	GWO-v-SVR	D, H, J, S, B, ST, H/B, J/B, B/D, L/Wd, NH, L, Wd, S/B, ST/B, De, Qe, PF, UCS	X50	76	R2 = 0.8353
[32]	TPE-ET	S/B, H/B, B/D, ST/B, PF, E, XB	X50	103	R2 = 0.9463
[34]	SVM	S/B, H/B, B/D, ST/B, PF, E, XB	Xm	102	R2 = 0.962
[35]	ANN	B, S, PF, NR, D, MC, ST, H	X50	135	R2 = 0.941
[36]	ANN	BI, PF, QB	X50	100	R2 = 0.8
[37]	ANN	S/B, HL/B, B/D, ST/B, PF, XB, E	Xm	109	R2 = 0.94
[38]	BPNN	S/B, HL/B, B/D, ST/B, PF, XB, E	Xm	91	R2 = 0.941
[39]	FIS	B, S, D, Sch, DJ, PF, ST	X80	185	R2 = 0.922
[40]	MI	UCS, P, RQD, JS, ρ, q, B, ST, S/D, JPO	X80	36	R2 = 0.81
[41]	ANN	B, S, HL, SD, ST, MC, PF, GSI	X80	200	R2 = 0.94
[42]	GPR	B, S, ST, PF, MC	X80	72	R2 = 0.948
[43]	ICA	MC, B, S, ST, PF, RMR	X80	80	R2 = 0.947
[44]	PSO-ANFIS	B, S, ST, q, MC	X80	72	R2 = 0.89
[45]	FFA-ANFIS	B, S, ST, PF, MC	X80	72	R2 = 0.98
[46]	CSO	q, B, RMR, MC, ST, S	X80	75	R2 = 0.985
[45]	GA-ANFIS	B, S, ST, PF, MC, RMR	X80	88	R2 = 0.989
[47]	FFA-BGAM	PF, MC, S, ST, B, H	X100	136	R2 = 0.98
[47]	FFA-BGAM	W, P, H, T, S, B,	SDR	136	R2 = 0.98
[48]	ACO-BRT	PF, MC, S, ST, B, H	X100	136	R2 = 0.962
[49]	ANN	S/B, H/B, B/D, ST/B, PF, E, XB	X50	102	R2 = 0.87
[50]	ANN	B, S, H, D, T, PF, Is50, UCS, UTS, ρ, E, Vp, SHV, U, RQD, C, φ, XB	X50	353	R2 = 0.986
[51]	GOA-SVR	D, H, J, S, B, ST, H/B, J/B, B/D, L/Wd, NH, L, Wd, S/B, ST/B, De, Qe, PF, UCS	X50	76	R2 = 85.83
[52]	GWO-CNN	S/B, H/B, B/D, ST/B, PF, E, XB	X50	4540	R2 = 0.89772

Note: SVM—support vector machines; FIS—fuzzy inference system; MI—mutual information; GPR—Gaussian process regressor; ICA—imperialist competitive algorithm; PSO—particle swarm optimization; ANFIS—adaptive neural-fuzzy inference system; FFA—firefly algorithm; GA—genetic algorithm; CSO—cat swarm optimization; ACO—ant colony optimization; BRT—boosted regression tree; BGAM—boosted generalized additive model; GWO—grey wolf optimization; GOA—grasshopper optimization algorithm; D—hole diameter (mm); H—bench height (m); S—spacing (m); B—burden; PF—powder factor (kg/m³); ST—stemming; E—elastic modulus (GPa); SDR—size of distributed rock (m); UCS—uniaxial compressive strength (MPa); UTS—uniaxial tensile strength (MPa); RQD—rock quality designation (%); Sch—Schmidt hammer rebound number; Qe—total explosive amount (t); J—sub-grade drilling (m); L—length (m); NH—number of holes; Wd—width (m); RMR—rock mass rating; X_B—in situ block size (m); X_m—mean particle size (cm); X₈₀—80% passing size (cm); X₁₀₀—100% passing size (cm); X₅₀—50% passing size (cm). φ—friction angle; ρ—density, C—cohesion; SHV—Schmidt hardness value; JPO—joint plane orientation/bench face ratio; Vp—P wave velocity; JS—joint spacing (m); ρ—rock density (t/m³); q—specific charge (kg/m³).; Is₅₀—point load strength index (MPa); CNN—convolutional neural network; SVR—support vector regression; MC—charge per delay (kg/ms); GSI—Geological Strength Index; QB—quantity of blasted rock pile (t).

provides an overview of the AI-based techniques applied in the prediction of MFS. However, these techniques still need improvements, such as a reduction in their training time and improvement in their prediction accuracy, regardless of the quantity of the dataset. Among these techniques, tree-based algorithms have shown superior performance compared to ANNs, etc., even without extensive fine-tuning [32,33].

The existing literature reveals that most prior studies have primarily relied on small-scale blasting datasets, which limit models’ performance and increase the risk of overfitting. In contrast, AI-based models exhibit significantly improved performance when trained on large-scale datasets, as these enable the models to capture more complex patterns within the data. Moreover, large-scale datasets also improve the models’ generalization and allow them to perform better on new data.

Therefore, this study has the following objectives: (1) to employ a large-scale database for better model training; (2) to compare the predictive accuracy of the three tree-based models—extra trees (ET), gradient boosting (GB), and random forest (RF)—in predicting the MFS distribution of rock; and (3) to develop an interactive web application for easy and timely predictions. The remainder of this paper is structured as follows: Section 2 provides an in-depth description of the data source, data collection and data supplementation. Section 3 delves into the methodologies employed, including details on AI tree-based models, the Bayesian optimization algorithm, and Shapley additive explanations (SHAP), which were used for model interpretation. Section 4 presents the proposed model development and the statistical evaluation indices that were employed. Section 5 discusses the results obtained from comparative analysis of the models, hyperparameter optimization, and the SHAP model’s interpretability and provides a link to the rock fragmentation web application. Lastly, Section 6 concludes by summarizing the key findings and this study’s contributions, and outlines this study’s limitations and future research directions.

2. Data Source

2.1. Data Collection and Supplement

This study collected 187 blasting records from the published literature on in situ operations, as shown in

Table 2. Data collected from the published literature.

Data Source	Blast Samples	Input Parameters	Output Parameters
[55]	76	D, H, J, S, B, ST, L, Wd, S.B, T.B, H.B, J.B, B.D, L.W, NH, Qe, De, PF, UCS	X50 (m)
[57]	103	SB, HB, BD, TB, Pf(kg/m3), XB(m), E	X50 (m)
[58]	8	SB, HB, BD, TB, Pf(kg/m3), XB(m), E	X50 (m)
Total	187

. The MFS was represented by X₅₀. To ensure consistent input parameters, the data collected by Sharma et al. [53] required supplementation due to missing values for two parameters: the elastic modulus (E) and in situ block size (X_B). The elastic modulus was derived from the uniaxial compressive strength using Equation (1) [54], while the in situ block size was estimated by Bieniawski using the powder factor, spacing, burden, muck pile’s mean fragmentation size, and rock mass rating (RMR) with Equations (2) and (3) [36].

$$\mathrm{E}=0.3752\times \mathrm{UCS}+4.479$$

(1)

$$\mathrm{FI}=0.03\times \left(\mathrm{RMR}\times \mathrm{PF}\times \mathrm{SB}\right)+0.73$$

(2)

$${\mathrm{X}}_{\mathrm{B}}=\mathrm{FI}\times {\mathrm{X}}_{50}$$

(3)

where the units of E, UCS, and Pf are GPa, MPa, and kg/m³, respectively, RMR is Bieniawski’s rock mass rating, and FI is the Fragmentation Index. The FI is estimated using Equation (2), and the X_B is calculated using Equation (3). The RMR of the rock excavated by Sharma et al. [55] is taken as 50, which belongs to the ‘Fair’ class [56]. After supplementing the data, they are then converted to consistent input parameters with other datasets from Hudaverdi [57] and Zhu [58] as required for this study. As a result, a small-scale dataset of 187 groups of in situ blasting data with seven variables including the spacing/burden ratio (S/B), bench height/burden ratio (H/B), burden/hole diameter ratio (B/D), stemming/burden ratio (T/B), PF, X_B, and E was combined with fragmentation size data to generate a complete database.

2.2. Data Augmentation

Data from various sources may present inconsistencies, noise, incompleteness, or inadequate quantity, which can lead to challenges in data analysis. This study’s small-scale database signifies that the fragmented data may be considered incomplete. The acquisition of blasting data is time-consuming and expensive, which has led to primarily limited-scale datasets. Inadequate data diminish the validity of insights that are obtained and decrease the accuracy of models in the training process. A substantial quantity of data is essential to accurate predictions and analyses, as well as for preventing overfitting and ensuring robust models. For tree-based algorithms, the prediction accuracy depends not only on hyperparameter optimization but also on the quantity of training data. Larger training datasets enable ML algorithms to better capture patterns and relationships, increasing their likelihood of making accurate predictions on unseen data. Moreover, acquiring blasting data is both time-consuming and costly [52]. Consequently, data augmentation offers a practical and recommended approach to generating large-scale datasets and thereby enhances the performance and generalizability of ML models.

Data augmentation methods vary depending on the type and dimensionality of the data. For instance, techniques such as rotating, cropping, and filtering are commonly used for image data, while jittering, slicing, and permutation are employed for curve data [59,60]. Among these, only filtering and jittering are noise-based methods. Noise-based methods are widely adopted for data augmentation because they are versatile and applicable across different data types [61]. This study implemented noise-based data augmentation by adding noise that was equivalent to 3% of the parameter values and that followed a normal distribution. This level of noise is designed to mimic uncontrollable human errors typically associated with the measurement and collection of in situ blasting data [52]. Additionally, the 3% noise does not present significant differences between the original and the augmented dataset. The dataset is organized as a 187 × 8 matrix that contains 187 blasting samples represented as rows, with seven input parameters and one output parameter represented as columns. To expand the dataset, each data point was used to generate 20 additional points through noise-based augmentation. The newly generated data points follow a normal distribution, with the mean set being equal to the original value and the standard deviation being adjusted so that 99.7% of the generated points fall within three standard deviations of the mean. This process was applied to every value in the matrix, which resulted in a significantly larger dataset of 3740 data groups, with each retaining the same seven input and one output parameter structure. A statistical summary of the augmented large-scale dataset is provided in

Table 3. Large-scale data summary statistics.

Parameters	Min. Value	Max. Value	Mean	Standard Deviation
S/B	0.9267	1.7921	1.1788	0.1093
H/B	1.2498	6.8683	3.2153	1.3955
B/D	17.9408	52.2242	29.3455	4.7601
T/B	0.4353	4.7513	1.0477	0.5782
PF (kg/m3)	0.1625	2.5717	1.0259	0.6302
XB (m)	0.0307	2.8724	1.2029	0.4764
E (GPa)	8.8334	60.0957	23.7790	16.2551
X50 (m)	0.0184	0.9930	0.3175	0.1574

.

Figure 1 presents a correlation heatmap of the features in the dataset. The correlation was computed using the mutual information (MI) regression technique, a measure of the shared information shared between two variables. MI regression quantifies how much knowing one variable reduces the uncertainty of another variable. The MI value is equal to zero if two variables are independent, and higher values indicate stronger dependency. MI is computed using Equation (4). Unlike the Pearson correlation, the use of the MI approach can capture non-linear dependencies without assuming any specific data distribution. Given the non-linear nature of the dataset, MI regression is suitable for computing the correlation between variables. As shown in Figure 1, four features exhibit a strong relationship with X₅₀, i.e., BD, X_B, HB, and E.

$$I\left(X;Y\right)={\displaystyle \iint p\left(x,y\right)\log \left(\frac{p\left(x,y\right)}{p\left(x\right)p\left(y\right)}\right)dxdy}$$

(4)

where $I(X;Y)$ denotes the mutual information between X and Y. $p(x,y)$ denotes the joint probability density function (pdf) of X and Y. $p(x),p(y)$ denotes marginal pdfs of X and Y.

3. Methodology

3.1. Random Forest Algorithm

RF is a supervised machine-learning algorithm that leverages multiple decision trees to deliver more accurate prediction. As an ensemble learning method, RF combines the predictions of multiple decision tree models and outputs the mean prediction for regression tasks [62,63,64]. The RF regressor operates an estimator that fits a collection of N randomized decision tree regressors independently and identically distributes random vectors [53]. Each decision tree contributes a unit vote for the most popular class or value at a given input x, which ensures diversity in the training process. Figure 2 shows the framework of the RF algorithm.

In RF training, the first step is to create random subsets of a given training dataset of independently and identically distributed ${\left[0,1\right]}^{\mathrm{d}}\times \mathrm{R}$-valued random variables (d ≥ 2) with a similar distribution as independent generic pairs (x, y) by randomly resampling a certain percentage of the total dataset $t_n=\left\{\left(x_1,y_1\right),\dots ,\left(x_n,y_n\right)\right\}$. This process is known as the bagging approach, and it uses multiple decision trees (DTs) without pruning [65,66]. The second step is randomly choosing a K input for lead node splitting while training each DT. The output prediction of the n-th decision tree is denoted as $g(\mathit{x},{\mathit{t}}_N^n)$, and the average of the combined trees, taken to form a finite forest predictor, is defined in Equation (5):

$$\widehat{y}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^{n}g\left(x, t_m^n\right)}$$

(5)

where $\widehat{y}$ is the RF’s final prediction.

3.2. Extra Trees Algorithm

The extra trees (ET) algorithm proposed by Geurts et al. [67] is an example of an ensemble-based algorithm derived from the RF algorithm. Two key differences between ET and RF are as follows: (1) the ET builds trees using a random subset of the data without replacement, which helps minimize bias, and (2) the randomness in the ET comes from random splits of data rather than bootstrap aggregating [67]. In the ET method, the tree splitting process is regulated by two parameters: k is the number of nodes randomly chosen in the node, and n_min is the minimal size of the sample required to separate the node. These enhance model accuracy and help prevent overfitting in the ET algorithm [68,69].

Similar to the RF algorithm, the ET algorithm performs regression in two stages: bootstrapping and bagging. During the bootstrapping phase, a random sample of the training dataset is selected to build the decision trees. In the bagging phase, the nodes of the decision are divided using the random subsets of training data. The final decision is made by randomly selecting a subset and determining its corresponding value. Figure 3 illustrates the framework of the ET algorithm. Mathematically, the ET regression process is expressed by the equation provided by Breiman [70].

$$p\left(x,{\beta }_1,\dots ,{\beta }_n\right)={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^{m}p\left(x,{\beta }_n\right)}$$

(6)

where p is the p-th prediction tree. β is a distributed vector.

3.3. Gradient Boosting Algorithm

In contrast to bagging, the boosting approach generates base models sequentially. This method improves prediction accuracy by producing models in succession, with each new model focusing on the training samples that are more difficult to predict. During the boosting process, the training data for each successive model emphasize the samples that were misclassified by previous base models, rather than those that were correctly predicted. Each new base model corrects the errors made by its predecessors [71,72].

The boosting approach resamples the training data to provide optimal information for each model. In each training step, the data distribution is adjusted based on the errors made by previous models. Emphasis is placed on samples that were inaccurately predicted, which gives them greater weight in subsequent models [73]. Boosting involves fitting additional models that minimize a loss function, which is averaged over the training data. This loss function quantifies the deviation between the predicted and true values. The process uses a forward stage-wise modeling approach, where new base models are added sequentially without the parameters or coefficients of the previously fitted models being altered [74]. The stage-wise steps for the gradient-boosting regression tree method are as follows:

Firstly, the constant value is initialized using Equation (7).

$$F_0\left(x\right)=\underset{\alpha }{\mathrm{argmin}}{\displaystyle {\sum }_{i=1}^{m}L(y_i,\alpha )}$$

(7)

where $L(y_i,\alpha )$ is the loss function.

Secondly, the residuals can be calculated using Equation (8) as follows:

$$r_{ip}=-{\left[{\displaystyle \frac{\partial L\left(y_i,F\left(x_i\right)\right)}{\partial F\left(x_i\right)}}\right]}_{F\left(x\right)=F_{p-1(x)}}for i=1,\dots ,n$$

(8)

where r_ip is each sample’s residuals.

Thirdly, a terminal node is created, and a mean function that minimizes the loss function is found using the following equation:

$${\alpha }_{jp}=\underset{\alpha }{\mathrm{argmin}}{\displaystyle \sum _{x_i\in R_{jp}}L\left(y_i,F_{m-1}\left(x_i\right)+\alpha \right) for j=1,\dots ,J_p}$$

(9)

Finally, the model is updated using Equation (10):

$$F_p\left(x\right)=F_{p-1}\left(x\right)+v{\displaystyle \sum _{j=1}^{J_p}{\alpha }_{jp}1\left(x\in R_{jp}\right)}$$

(10)

3.4. Optimization Algorithm: Bayesian Optimization

The TPE is a Bayesian optimization approach proposed by Bergstra et al. [75]. TPEs are better than other optimization algorithms because (1) they handle high-dimensional hyperparameter search spaces; (2) they converge faster than most optimization algorithms such as meta-heuristic algorithms, grid search, and random search; (3) they are efficient for training computationally expensive machine learning algorithms; and (4) they are easy to implement with different libraries in python. To that end, three predictive models are constructed and cross-validated five times during training to assess the different models’ generalization capabilities. The TPE utilizes the Gaussian mixture model to find the best hyperparameter impacting the models [76,77]. The TPE runs on two main ideas: (1) using Parzen estimators to model the best hyperparameters for the model and (2) using a tree-like data structure known as the posterior-inference graph to optimize the algorithm runtime. Let X be a tree-structured search space and $f:X\to ℝ$ be an objective function, then the minimization problem is defined as ${\gamma }^{\ast }\in \underset{\gamma \in X}{\mathrm{argmin}}f\left(\gamma \right)$. Suppose a set of observations is $T=\left\{\left({\gamma }^{(1)},y^{(1)}\right),\dots ,\left({\gamma }^{\left(n\right)},y^{\left(n\right)}\right)\right\}$, then the TPE replicates $p\left(\gamma \left|y\right.\right)$ for every parameter $\gamma \left(\in X_i\right)$ using two probability density functions as follows:

$$p\left(\gamma \left|y\right.\right)=\left\{\begin{array}{cc}l(\gamma )& if y< y^{\ast }\\ g\left(\gamma \right)& if y^{\ast }\le y\end{array}\right.$$

(11)

where $p\left(\gamma \left|y\right.\right)$ is the conditional probability of the hyperparameter $\gamma$ and the model loss y; $l\left(\gamma \right)$ establishes the subset of the observed probability density values that are less than $y^{*}$, and $g(\gamma )$ establishes the subset of the remaining observed probability density values; the value $y^{\ast }$ is chosen to be a quantile $\lambda \in \left(0,1\right)$ of the observed $y$ values that satisfy $p(y<y^{\ast })=\lambda$.

To choose a particular candidate for evaluation, the TPE employs the expected improvement (EI) for $y^{\ast }$ and the parameter $\gamma$ as the acquisition function.

$$\begin{array}{l}E{I}_{y^{\ast }}(\gamma )={\displaystyle {\int }_{-\infty }^{y^{\ast }}\max \left(y^{\ast }-y,0\right)p\left(y\left|\gamma \right.\right)dy}\\ ={\displaystyle {\int }_{-\infty }^{y^{\ast }}\left(y^{\ast }-y\right)p\left(y\left|\gamma \right.\right)dy}\\ \alpha {\left(\lambda +{\displaystyle \frac{g\left(\gamma \right)}{l\left(\gamma \right)}}\left(1-\lambda \right)\right)}^{-1}\end{array}$$

(12)

To gain new information with the maximum EI, $g(\gamma )/l(\gamma )$ is minimized. To keep minimizing $g(\gamma )/l(\gamma )$, the process is repeated where ${\gamma }^{\ast }$ is returned in the function and fitted again until the hyperparameter optimization is complete. It can be concluded that the probability of $l(\gamma )$ is as high as possible, while the probability of $g(\gamma )$ is as low as possible when the improvement is maximized [78].

3.5. Shapley Additive Explanations for Model Interpretation

The AI-based methods are often considered unreliable in many fields due to their opacity and insufficient explanation for the predictions rendered. The internal workings of these models are complex and difficult to comprehend using human logic, which has earned them the label of ‘black box’ models [79]. To address this issue and enhance transparency, explainable AI (XAI) tools are employed to interpret these models and provide insights into the complex relationships between input features and output predictions [80]. These XAI tools include SHAP and local interpretable model-agnostic explanations (LIME). In this study, SHAP is used to interpret the models because of the following reasons: (1) it provides a global understanding of the dataset, providing insights into the overall contribution of each feature [81]; (2) it has great visualizations that make it easier to understand the model’s behavior; and (3) it handles feature interactions, showing the impact of two or more features when combined [82].

SHAP is a cumulative feature attribution algorithm that assigns a relevance score to each input feature. It leverages concepts from game theory to calculate feature importance and satisfies key properties of feature attribution, including accuracy, missingness, and consistency [83]. Given a dataset D, the Shapley value estimation of an m-th feature with i combinations of features, target feature y, and predictive model f is computed using Equation (13).

$${\varphi }_{im}=\widehat{f}(y+m)-\widehat{f}(y-m)$$

(13)

where ${\varphi }_{im}$ is the mean Shapley value of the m-th feature with the i-th feature. $\widehat{f}(y+m)$ is the prediction for target feature y with a random number of feature values, which also includes the m-th feature. $\widehat{f}=(y-m)$ is the prediction without the m-th feature. Generally, Equation (14) is used to compute the Shapley value of the m-th feature.

$${\varphi }_m(y)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left({\varphi }_{im}\right)}$$

(14)

4. Model Development and Evaluation Indices

This study employed three tree-based ensemble models to predict the MFS distribution of rock: RF, ET, and GB. The accuracy of these models depends on the optimal combination of hyperparameters. To identify this optimal combination, tree-structured Parzen estimators (TPEs) were used for hyperparameter tuning. Consequently, three hybrid models were developed: TPE-RF, TPE-ET, and TPE-GB, as illustrated in Figure 4.

The optimization process for all three models follows a similar and consistent procedure:

(1)

Data preprocessing: The dataset is divided into training and test sets with 75% (2805 samples) being allocated for training and 25% (935 samples) being reserved for testing. This 75/25 data split was chosen for several reasons: (1) it provides enough samples for the models to learn patterns, which prevents underfitting; (2) smaller test sets tend to be biased while larger test sets offer a more representative sample; and (3) this split ratio generalizes better than a 70/30 split or 80/20 split. Figure 5 show the data distribution between the training and test datasets. It can be observed that the training and test sets are normally distributed; therefore, there is no need for further pre-processing. Additionally, the training and test sets are distributed the same;

(2)

Optimization process: Many influencing parameters can influence the performance of the algorithms, but only a limited number of hyperparameters were selected to balance performance and computational cost. Five hyperparameters were optimized for the RF and ET algorithms, while six were used for the GB algorithm, as shown in Table 4. These hyperparameters are critical for improving the accuracy of the models. The performance of the Bayesian optimization algorithm is influenced by the number of trials or iterations used to search for the optimal hyperparameter combinations. More trials result in longer training times and higher costs, whereas fewer trials may lead to underfitting. To maintain consistency across models, the number of trials was set to 150 for all models;

(3)

Model evaluation: In this study, four indices, including the root mean squared error (RMSE), mean absolute error (MAE), coefficient of determination (R²), and max error were employed to evaluate the performance of the three models. These metrics are described using Equations (15)–(18):

$$\mathrm{RMSE}=\sqrt{{\displaystyle \sum _{i=1}^n\frac{{\left({\widehat{y}}_i-y_i\right)}^2}{n}}}$$

(15)

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|{\widehat{y}}_i-y_i\right|}$$

(16)

$${\mathrm{R}}^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}{{\displaystyle {\sum }_{i=1}^n{\left(y_i-{\overline{y}}_i\right)}^2}}}$$

(17)

$$\mathrm{Max} \mathrm{Error}(y,\widehat{y})=\max (\left|y_i-{\widehat{y}}_i\right|)$$

(18)

where $y_i$ represents the actual value, ${\overline{y}}_i$ represents the mean value of the predicted target, and ${\widehat{y}}_i$ represents the predicted value.

Figure 5. Data distribution between training and test datasets: (A) data distribution of SB; (B) data distribution of HB; (C) data distribution of BD; (D) data distribution of TB; (E) data distribution of Pf (kg/m³); (F) data distribution of XB (m), and (G) data distribution of E (GPa).

Figure 5. Data distribution between training and test datasets: (A) data distribution of SB; (B) data distribution of HB; (C) data distribution of BD; (D) data distribution of TB; (E) data distribution of Pf (kg/m³); (F) data distribution of XB (m), and (G) data distribution of E (GPa).

Table 4. Hyperparameters and their search space.

Hyperparameters	Regression Algorithms Hyperparameter Search Spaces			Data Type	Description
	TPE-ET	TPE-RF	TPE-GB
n estimators	[10, 3000]	[10, 3000]	[10, 3000]	Integer	Number of trees in the forest
max depth	[2, 40]	[2, 40]	[2, 40]	Integer	Maximum depth of each tree
min samples split	[2, 35]	[2, 35]	[2, 35]	Integer	The minimum number of samples required to split an internal code
criterion	‘squared_error’, ‘absolute_error’, ‘friedman_mse’, ‘poisson’	‘squared_error’, ‘absolute_error’, ‘friedman_mse’, ‘poisson’	‘squared_error’, ‘friedman_mse’,	Categorical	These functions measure the quality of a split.
min impurity decrease	[0.00001, 0.9]	[0.00001, 0.9]	[0.00001, 0.9]	Float	It splits the node if the split induces a decrease of the impurity greater than or equal to this value.
learning rate			[0.00001, 0.9]	Float	It shrinks the contribution of each tree by the value of learning_rate.

Table 4. Hyperparameters and their search space.

Hyperparameters	Regression Algorithms Hyperparameter Search Spaces			Data Type	Description
	TPE-ET	TPE-RF	TPE-GB
n estimators	[10, 3000]	[10, 3000]	[10, 3000]	Integer	Number of trees in the forest
max depth	[2, 40]	[2, 40]	[2, 40]	Integer	Maximum depth of each tree
min samples split	[2, 35]	[2, 35]	[2, 35]	Integer	The minimum number of samples required to split an internal code
criterion	‘squared_error’, ‘absolute_error’, ‘friedman_mse’, ‘poisson’	‘squared_error’, ‘absolute_error’, ‘friedman_mse’, ‘poisson’	‘squared_error’, ‘friedman_mse’,	Categorical	These functions measure the quality of a split.
min impurity decrease	[0.00001, 0.9]	[0.00001, 0.9]	[0.00001, 0.9]	Float	It splits the node if the split induces a decrease of the impurity greater than or equal to this value.
learning rate			[0.00001, 0.9]	Float	It shrinks the contribution of each tree by the value of learning_rate.

5. Results and Discussion

5.1. Performance Comparison of Models for MFS Prediction

The three optimized models were compared to determine the best model for predicting the MFS distribution of rock. As outlined in Section 4, aside from the TPE-GB model, the number of trials and hyperparameters for each model was kept consistent to ensure fairness in the results. Figure 6 presents a bar graph that shows the importance values of different hyperparameters during the optimization process. Figure 6 illustrates that all models prioritized the ‘criterion’ hyperparameter, which is rational, as it assesses potential splits at each node to optimize information gain. In the TPE-RF model, the ‘criterion’ hyperparameter significantly influenced the optimization process, whereas the other hyperparameters had a minimal impact. Figure 7 displays the performance of the three models during the optimization process involving 150 trials. The results indicate that the performance differences among the models were not substantial. However, the TPE-ET model exhibits better performance than the other two models. Additionally, it was observed that some hyperparameter combinations during the optimization process resulted in negative values of the objective function, as shown in the subfigure in Figure 7.

According to

Table 5. Evaluation indices of models during the training phase.

Models	R2	RMSE	MAE	Max Error	Scores
TPE-ET	0.97	0.03	0.02	0.14
Rank	1	1	1	1	4
TPE-GB	0.97	0.03	0.02	0.11
Rank	1	1	1	3	6
TPE-RF	0.97	0.03	0.02	0.12
Rank	1	1	1	2	5

, the TPE-GB model outperformed the other two models during the training phase, achieving R², RMSE, MAE, and max error values of 0.97, 0.03, 0.02, and 0.14, respectively. However, as shown in

Table 6. Evaluation indices of models during the testing phase.

Models	R2	RMSE	MAE	Max Error	Scores
TPE-ET	0.93	0.04	0.03	0.25
Rank	1	1	1	3	6
TPE-GB	0.92	0.04	0.03	0.28
Rank	2	1	1	1	5
TPE-RF	0.92	0.04	0.03	0.26
Rank	2	1	1	2	6

, the TPE-GB model underperformed compared to the other models during the testing phase. In contrast, the TPE-ET model demonstrated superior performance during the testing phase, with R², RMSE, MAE, and max error values of 0.93, 0.04, 0.03, and 0.25, respectively. However, the evaluation metrics across the models are closely matched, with minimal differences, which makes it challenging to definitively identify the best-performing model. The lower max error values suggest that TPE-GB and TPE-ET were the most reliable during the training and testing phases, respectively. To further analyze and compare the model performances, score rankings were visualized using stacked bar charts. As illustrated in Figure 8a,b, the TPE-GB model achieved the highest score during the training phase but ranked lower during the testing phase. Meanwhile, the TPE-ET and TPE-RF models exhibited comparable scores during the testing phase.

To further understand the models’ performance, regression plots are utilized to explore the relationship between the predicted X₅₀ values and actual X₅₀ values. Figure 9 and Figure 10 depict scatter plots of the three models, where the predicted X₅₀ is on the y-axis and the actual X₅₀ is on the x-axis. The red dashed line represents the best-fit line, indicating a perfect relationship between the predicted and the actual values. As shown in Figure 9a–c, all models demonstrate a strong relationship between the predicted and actual values during the training phase. Among them, the TPE-GB model exhibits a slightly better relationship with the best-fit line than the other models and achieves the highest R² value. Similarly, Figure 10a–c depict the relationship between the predicted and the actual values during the testing phase. All models exhibit a strong correlation between the predicted and actual values, although several outliers significantly deviate from the best-fit line, as indicated by the green circles in the figures.

5.2. Model Interpretation

Figure 11 presents a bee swarm plot that illustrates the distribution of feature contributions alongside the SHAP interaction matrix. The matrix’s diagonal elements indicate the independent contribution of each feature to the model’s predictions, whereas the off-diagonal elements reflect the interaction effects between pairs of features. The SHAP value is derived by summing the independent contributions of individual features and their interaction contributions. The figure indicates that XB(m) significantly influences the model’s predictions, with a notable interaction between XB(m) and E(GPa) enhancing this predictive effect. Conversely, SB demonstrates minimal independent contribution to the model’s predictions, and the interaction between SB and BD exhibits a negligible effect on the model’s output.

Figure 12 integrates a local bar plot with a bee swarm plot to provide a comprehensive visualization of feature contributions to the model’s predictions. The bars in the local bar plot represent the SHAP values for each feature, illustrating the individual contributions of each. The top x-axis corresponds to the mean SHAP values from the bar plot, while the bottom x-axis represents the SHAP value contributions in the bee swarm plot. In the bee swarm plot, the color coding indicates feature value ranges, with blue signifying low values and red signifying high values. It can be observed that XB plays a significant role in the model’s prediction. High values of XB positively influence the predictions, whereas low values have a negative impact. In contrast, SB has minimal influence on the model, with high values of SB contributing slightly positively to the predictions.

Figure 13 employs a scatter plot with a LOWESS fit curve to provide an intuitive understanding of how the range of feature values influences the model’s predictions. The LOWESS curve (the red curved line) represents a weighted regression curve that smooths out non-linear trends in the data. Figure 13 shows the fitted curves for all the features in the dataset. The horizontal dotted line displays SHAP values at zero (y = 0), whereas the vertical blue dotted line shows where the feature value intersects it. This intersection signifies the threshold at which the feature’s contribution to the prediction shifts from negative to positive or vice versa. For instance, for XB, values less than 1.19 negatively impact the model’s predictions, whereas values greater than 1.19 positively influence the predictions. There are two intersections in the case of E, at 10.29 and 41.66. The predictions are negatively affected by values less than and between 10.29 and 41.66, and positively affected by values above this range. However, the exact intersection values are difficult to discern in Figure 13, which prompted the use of Figure 14, which provides a clearer depiction of these critical points. For the SB, HB, TB, and Pf features, the intersections are at 1.11, 2.47, 1.29, and 1.28, respectively. This study developed a novel rock fragmentation prediction model and subsequently a user-friendly web interface through cloud deployment. The web interface overcomes the limitations of requiring software downloads on computers or mobile devices, offering seamless accessibility for rock fragmentation prediction applications. The following is the link to the web application accessed on 15 October 2024: https://rockfragmentation.streamlit.app/Data_statistics.

6. Conclusions

The prediction of MFS using AI-based techniques has practical implications in the mining industry, since it improves environmental and miners’ safety while lowering costs. To model MFS prediction, this study employed three AI tree-based techniques coupled with the TPE optimization algorithm. Furthermore, this study utilized the SHAP technique to systematically evaluate the stability, robustness, and interpretability of the model. Lastly, an interactive web application was developed to facilitate the MFS predictions. The main conclusions of this study are as follows:

(1)

Adding 3% noise to augment the dataset did not significantly distort the original dataset. Moreover, the large-scale database of 3740 samples provided deeper insights into the input and output parameters and thus enhanced the model’s predictive capabilities;

(2)

The model evaluation results demonstrated that the TPE-ET model performed better than the other models in predicting MFS, achieving R², RMSE, MAE, and max error optimal values of 0.93, 0.04, 0.03, and 0.25 on the testing set;

(3)

The model interpretability results illustrated that rock parameters and geological conditions were the most significant parameters in predicting MFS. In this study, XB (m) and E (GPa) had the most significant impact and a positive contribution to the models’ predictions.

The tree-based algorithms utilized in this study produced outstanding predictions. However, there are still limitations, such as (1) the fact that this study used a small number of input parameters, which can limit the true precision of the model’s prediction considering that there are many factors that influence MFS; (2) the fact that other powerful tree-based algorithms such as the histogram-based gradient boosting algorithm, extreme gradient boosting algorithm, and gradient boosting algorithm were not explored in the current study; and (3) the fact that noise addition during data augmentation reduced the data quality. Some of the recommendations for future research include the following: (1) employing ensemble methods may enhance the generalization capacities of the models, for example integrating tree-based models with other methods such as SVMs or ANNs; (2) integrating physics-informed models with machine learning models would enhance the accuracy and reliability of this method, as machine learning models do not adhere to the physical laws governing the time-dependent dynamics of fragmentation systems; and (3) utilizing sensors to gather real-time fragmentation data and employing machine learning models for instantaneous predictions.