Bagged Ensemble of Gaussian Process Classifiers for Assessing Rockburst Damage Potential with an Imbalanced Dataset

Abstract

The evaluation of rockburst damage potential plays a significant role in managing rockburst risk and guaranteeing the safety of personnel. However, it is still a challenging problem because of its complex mechanisms and numerous influencing factors. In this study, a bagged ensemble of Gaussian process classifiers (GPCs) is proposed to assess rockburst damage potential with an imbalanced dataset. First, a rockburst dataset including seven indicators and four levels is collected. To address classification problems with an imbalanced dataset, a novel model that integrates the under-sampling technique, Gaussian process classifier (GPC) and bagging method is constructed. Afterwards, the comprehensive performance of the proposed model is evaluated using the values of accuracy, precision, recall, and F₁. Finally, the methodology is applied to assess rockburst damage potential in the Perseverance nickel mine. Results show that the performance of the proposed bagged ensemble of GPCs is acceptable, and the integration of data preprocessing, under-sampling technique, GPC, and bagging method can improve the model performance. The proposed methodology can provide an effective reference for the risk management of rockburst.

1. Introduction

With the increase in mining depth, rockbursting has become an increasingly prominent issue [1,2,3]. It is induced by the instantaneous release of elastic strain energy, and often is accompanied by ejection and collapse of massive rock [4,5,6]. Many mines have suffered rockburst disasters, causing serious economic losses and casualties. For example, a rockburst with a magnitude of 3.5 happened in Falconbridge nickel mine, resulting in four deaths [7]; a rockburst with a magnitude of 2.47 occurred in Junde coal mine, causing five deaths and the destruction of a shearer and scraper conveyor [8]; and a rockburst with a magnitude of 5.2 appeared in the Klerksdorp district of South Africa, leading to two deaths and fifty-eight injuries [9]. Due to such serious consequences, assessing rockburst damage potential is necessary and significant.

According to the differences between the locations of damage and seismic event, rockbursts can be classified into self-initiated rockburst and remotely triggered rockburst [10]. For the former, the locations of damage and seismic event are consistent. While for the latter, rockburst is triggered by remote and relatively large magnitude seismic events. For different types of rockburst, the influencing factors are different, resulting in the disparity of the rockburst damage potential evaluation. This study aims to assess the damage potential of remotely triggered rockburst. Because the location of the damage is not consistent with that of the microseismic event, it is difficult to evaluate the damage potential only based on the microseismic event. The microseismic event information, stress wave propagation paths, and rock mass conditions on the excavation face should be considered simultaneously. Due to the complex mechanisms and numerous influencing factors, the evaluation of rockburst damage potential is still a difficult issue.

Scholars have proposed some methods to assess rockburst damage potential. Kaiser et al. [11] developed a rockburst damage assessment procedure. It mainly included four steps: propose rock and support damage scales, put forward an initial condition index, calculate the scaled distance, and establish relationships among the initial condition index, scaled distance, and rock and support damage scales. Durrheim et al. [12] summarized the influencing factors of rockburst damage according to the investigations of rockbursts in South African gold mines, which was valuable for the evaluation of rockburst damage potential. Brink et al. [13] proposed an approach for seismic risk evaluation, which can be summarized in four steps: determine an evaluation indicator system, score each sub-category according to the risk rating, calculate the score of each category, and determine the risk levels. Albrecht and Sharrock [14] investigated ten indicators that affect rockburst damage, and established the relationship between them and rockburst damage based on field rockburst incidences. With the increase of rockburst cases, machine learning (ML) algorithms were used to evaluate rockburst damage potential. Heal et al. [15] proposed the concept of excavation vulnerability potential (EVP), and then adopted logistic regression to assess rockburst damage potential. Zhou et al. [16] employed a stochastic gradient boosting approach for the evaluation of rockburst damage. Li et al. [17] put forward a rockburst damage scale index using rock engineering systems and artificial neural networks to evaluate rockburst damage.

When a large number of rockburst cases accumulate, ML is a possible way to evaluate rockburst damage potential [18,19]. However, due to the fact that most rockburst damage levels are slight, while the strong or even extremely strong type is relatively rare, the distribution of sample data for each level is usually imbalanced [20,21,22]. Considering the specific characteristics of rockburst data, two key issues need to be solved. The first one is the handling of the imbalanced rockburst dataset. Generally, classical ML algorithms are conceived on the premise of balanced datasets [23]. It is difficult to handle classification problems with an imbalanced dataset, especially for discriminating the minority category cases [24]. Therefore, traditional ML algorithms should be improved to deal with imbalanced datasets. The corresponding strategy can be roughly divided into four groups: data level, algorithm level, cost-sensitive level, and ensemble level [25]. Among them, combining bagging ensemble learning with under-sampling techniques is an effective way to deal with imbalanced datasets [26].

The second one is the selection of algorithms. A large number of ML algorithms have been used to solve classification problems. Although some other statistical algorithms, such as Monte Carlo methods, can also be adopted to solve multidimensional problems and obtain probabilistic results, the probability density functions need to be determined in advance [27,28,29]. Gaussian process classifier (GPC) is a promising statistical model because it can deal with high-dimensional and nonlinear problems, tune hyperparameters directly based on training data, and obtain probabilistic outputs [30,31]. However, due to the characteristics of imbalance and strong noise in rockburst data, a single GPC is hard to have stable prediction ability. Ensemble learning can overcome this drawback by combining multiple base classifiers to some extent [32,33,34]. Combining bagging ensemble learning with Gaussian process classifiers (GPCs) may improve the generalization ability and robustness of models.

This study proposes a novel model that integrates the under-sampling technique, GPC, and bagging method to assess rockburst damage potential with an imbalanced dataset. First, the rockburst dataset is collected and preprocessed by the Yeo-Johnson transformation and standardization process. Then, the reliability of the proposed methodology is verified, and the comprehensive performance is evaluated using four metrics. Finally, the proposed bagged ensemble of GPCs is applied to assess rockburst damage potential in the Perseverance nickel mine.

2. Data Acquisition

According to the original work of Heal [35], a total of 254 rockburst cases were collected from 13 underground metal mines in Canada and Australia. These cases were obtained based on the rock mass failure conditions caused by a single microseismic event. This database contains 83 microseismic events and 254 failure locations. It indicates some failure locations are caused by the same microseismic event. Based on the damage status of rock mass and support, the degree of rockburst damage was divided into five levels: none (L₁), low (L₂), moderate (L₃), high (L₄), and strong (L₅). Among them, L₁ indicated the rock mass showed no damage or minor loss, and the support was not damaged; L₂ indicated the rock mass was slightly damaged, less than 1 ton of rock was displaced, the support system was loaded, the meshes were loose and the plates were deformed; L₃ indicated 1 ton to 10 tons of rock was displaced and some bolts were broken; L₄ indicated 10 tons to 100 tons of rock was displaced and the support system was severely damaged; L₅ indicated above 100 tons of rock was displaced and the support system was completely destroyed. As the damage locations of L₁ were not reported during investigations, the original database only contained L₂, L₃, L₄, and L₅. The sample sizes at these levels were 116, 48, 63, and 27, respectively.

The original database included nine indicators: the ratio of total maximum principal stress to uniaxial compressive strength (I₁), the energy capacity of support system (I₂), excavation span (I₃), geology factor (I₄), Richer magnitude of seismic event (I₅), distance between rockburst location and microseismic event (I₆), peak particle velocity (I₇), rock density (I₈) and support types (I₉). The specific meaning of each indicator can be referred to in literature [35]. Different indicator combinations have a significant impact on evaluation results. Heal [35] and Zhou et al. [16] selected I_1, I_2, I_3, I₄ and I₇; and Li et al. [17] chose I_1, I_2, I_3, I_4, I_5, I₇ and I₈. Considering I₇ was calculated by I₅ and I₆ based on an empirical formula and I₉ was difficult to be quantified, this study adopted I_1, I_2, I_3, I_4, I_5, I₆ and I₈ to evaluate the rockburst damage.

From this dataset, it can be seen that there were some duplicate sample data, and some samples had the same indicator values, but the corresponding rockburst damage levels were different. To improve the prediction accuracy, the duplicate samples were first removed. For the samples with the same indicator values but different levels, only the samples with the highest level were selected to ensure safety. Consequently, the number of samples in the updated dataset was 236, and the sample sizes at L₂, L₃, L₄ and L₅ were 107, 45, 57 and 27, respectively. The corresponding ratio of sample sizes at different levels was 4.0:1.7:2.1:1.0. It shows that the distribution is relatively unbalanced, which may affect the accuracy of evaluation results. The detailed dataset was listed in Appendix A.

To quantitatively analyze the correlations between these seven indicators, the heat map of the correlation coefficient was obtained, as shown in Figure 1. It can be seen that some indicators were positively correlated, such as I₁ and I₂, whereas some indicators were negatively correlated, such as I₁ and I₃. Overall, the correlations between these indicators were generally small. Although I₅ and I₆ had the largest correlation coefficient of 0.48, they were distinctly based on their physical meanings. Therefore, these indicators were relatively independent, which verified the rationality of the selected indicators.

The box plot of all indicators for each rockburst damage level was shown in Figure 2. It can be seen that all indicators had some outliers. Especially for I₂, I₃, I₆ and I₈, outliers were more obvious. Some overlapping parts existed in the range of indicator values for various levels. As a result, it was difficult to differentiate the level of rockburst damage only using one indicator. Second, there was no obvious correlation between rockburst damage level and each indicator. In addition, the distribution of indicator values was uneven. All these characteristics illustrated the complexity of rockburst damage evaluation.

3. Methodology

3.1. Gaussian Process Classifier

GPC is a statistical learning algorithm based on the Gaussian process and Bayesian theory, which has a solid mathematical foundation. By assuming the implicit function obeys the prior distribution of a Gaussian process, the posterior distribution can be obtained according to Bayesian inference [36]. Then, the probability of different classes can be determined. The main calculation steps are as follows.

Suppose the training set is:

$$D=(X,Y)=\left\{(x_i,y_i)\left|y_i=\pm 1,i=1,2,\cdots ,m\right.\right\},$$

(1)

where $x_i=\left(x_{1i},x_{2i},\cdots ,x_{Bi}\right)$ is the input; $y_i$ is the output; and $m$ is the number of samples in the training set.

To reflect the mapping relationship between $x_i$ and $y_i$, the implicit function that obeys the Gaussian process distribution is defined as:

$$f={[f(x_1),f(x_1),\cdots ,f(x_i),\cdots ,f(x_m)]}^T,$$

(2)

Suppose $f$ satisfies a Gaussian process distribution with a zero mean and covariance matrix $K$, then:

$$p(f\left|X\right.)~N(f\left|0,K\right.),$$

(3)

where $K$ can be calculated by a covariance function $k(x,x\prime )$, and is specifically defined according to the actual situation.

In general, the radial basis function is selected as the covariance function:

$$k(x,x\prime )={\theta }_1{e}^{-\frac{{‖x-x\prime ‖}^2}{{\theta }_2}},$$

(4)

where ${\theta }_1$ and ${\theta }_2$ are hyperparameters.

Based on Equations (3) and (4), the prior probability can be determined as:

$$p(f\left|X\right.)=\frac{1}{{(2\pi )}^{0.5}{\left|K\right|}^{0.5}}\mathrm{e}\mathrm{x}\mathrm{p}(-0.5f^T{K}^{-1}f).$$

(5)

Then, to obtain the probability of the predicted category, a likelihood function is used to map the output value of the implicit function to the interval $[0, 1]$. The logistic function is generally used as the likelihood function:

$$p(Y\left|f\right.)=\psi (z)=\frac{1}{1+\exp (-z)}.$$

(6)

Based on Bayes’ theorem, the posterior probability of the implicit function is:

$$p(\widehat{f}\left|X,Y\right.)=\frac{p(Y\left|f\right.)p(f\left|X\right.))}{p(Y\left|X\right.)},$$

(7)

where $p(Y\left|X\right.)$ is the marginal likelihood function, which indicates the probability distribution of a training set.

$p(Y\left|X\right.)$ can be calculated by:

$$p(Y\left|X\right.)={\displaystyle \int p(D\left|f\right.)p(f)}df.$$

(8)

Suppose the sample to be predicted is ($\tilde{x},\tilde{y}$), the probability of $\tilde{y}=+1$ can be determined by:

$$p(\tilde{y}=+1\left|X,Y,\tilde{x}\right.)={\displaystyle \int p(\tilde{y}\left|\widehat{f}\right.)p(\widehat{f}\left|X,Y,\tilde{x}\right.)}d\widehat{f},$$

(9)

where $\widehat{f}$ indicates the implicit function of $\tilde{x}$.

Since there is no analytical solution in Equations (7)–(9), Laplace approximation algorithm is often to obtain the solutions. Namely, the posterior probability distribution $p(\widehat{f}\left|X,Y\right.)$ is first obtained, then the implicit function $\widehat{f}$ can be determined.

Finally, the probability of $\tilde{y}=+1$ can be calculated by

$$p(\tilde{y}=+1\left|X,Y,\tilde{x}\right.)={\displaystyle \int \psi (\widehat{f})p(\widehat{f}\left|X,Y,\tilde{x}\right.)}d\widehat{f}.$$

(10)

If $p(\tilde{y}=+1\left|X,Y,\tilde{x}\right.)\ge 0.5$, then the prediction result is a positive class, otherwise it is a negative class.

For multi-classification issues, the binary Gaussian process classifier can be extended with a “one-vs-rest” or “one-vs-one” strategy [37]. For the “one-vs-rest” strategy, the binary Gaussian process classifier classifies one of the classes and the remaining classes respectively. In this case, the class with the highest probability is selected as the final result. For the “one-vs-one” strategy, the binary Gaussian process classifier classifies the two classes respectively. In this case, each classification is equivalent to one vote, and the class with the highest votes is selected as the final result.

3.2. Bagged Ensemble of Gaussian Process Classifiers

A bagged ensemble of GPCs is proposed to handle classification problems with imbalanced datasets, as shown in Figure 3. This model integrates the under-sampling technique, GPC and bagging method. The under-sampling technique is used to make the training samples balanced. The samples of classes except the minority class are resampled and combined with the minority class samples into a new dataset. GPC has a strong probabilistic prediction ability for unknown data by learning from the existing dataset. By integrating multiple classifiers, the bagging method can avoid over-fitting to a certain extent, and has better anti-noise ability and robustness. In addition, the defect of data loss from a single under-sampling can be overcome through multiple under-samplings with replacement. The specific steps are as follows.

First, the under-sampling technique with replacement is used to generate the balanced sample sets from the original dataset.

Second, the GPCs are independently trained based on the generated training sets.

Last, the final result is obtained by integrating the evaluation results of each GPC based on a voting classifier.

3.3. Establishment of Rockburst Damage Evaluation Model

The proposed ensemble model is used to evaluate rockburst damage. The detailed procedure is shown in Figure 4, which is described as follows.

First, the original rockburst damage dataset is preprocessed based on the Yeo-Johnson transformation and standardized processing. In many modeling scenarios, data needs to be normalized to improve predictive performance. Power transformation maps sample data from an arbitrary distribution to a Gaussian distribution as close as possible. It builds a set of monotonic functions to stabilize variance and minimize skewness. There are two transformation methods: the Yeo-Johnson and Box-Cox transformation. Since the Box-Cox transformation only works for positive data, the Yeo-Johnson transformation is adopted in this study. The calculation formula is [38]:

$$x_i^{(\lambda )}=\left\{\begin{array}{ll}\left[{(x_i+1)}^{\lambda }-1\right]\lambda & \mathrm{if}\lambda \ne 0,x_i\ge 0\\ \ln (x_i+1)& \mathrm{if}\lambda =0,x_i\ge 0\\ -\left[{(-x_i+1)}^{2-\lambda }-1\right]/(2-\lambda )& \mathrm{if}\lambda \ne 2,x_i<0\\ -\ln (-x_i+1)& \mathrm{if}\lambda =2,x_i<0\end{array}\right.,$$

(11)

where $x_i$ is the data to be transformed; and $\lambda$ is a parameter, which can be estimated by the maximum likelihood method.

In addition, there is a large gap between some indicator values. For example, $I_8$ is three orders of magnitude larger than $I_5$. In this case, it may lead to the dominance of this indicator, while the roles of other indicators are ignored. Therefore, the initial indicator values need to be firstly standardized. In this study, they are converted into a standard normal distribution with a mean of zero and a standard deviation of one. The conversion formula is:

$$x_i^{(\lambda )}=(x_i^{(\lambda )}-\mu )/\sigma ,$$

(12)

where $\mu$ is the mean value and $\sigma$ is the standard deviation of sample data.

Second, the preprocessed dataset is randomly divided into training and test sets with a ratio of 4:1. Furthermore, the ratio of sample size for different levels in these two sets is kept consistent to make the results more stable.

Third, the hyperparameter of the bagged ensemble of GPCs is optimized using five-fold cross-validation. The number of GPC is adopted as the hyperparameter to be optimized, and both the hyperparameters ${\theta }_1$ and ${\theta }_2$ in kernel function of GPC are selected as 1.0. Then, the optimal hyperparameter value is determined based on the average accuracy of five-fold cross-validation.

Fourth, the model with the optimal hyperparameter value is fitted based on the training set, and then the optimal training model is obtained.

Fifth, the comprehensive performance of the proposed methodology is evaluated based on the test set. The accuracy, precision, recall, and F₁ are chosen as the evaluation metrics, which can be calculated by a confusion matrix.

Suppose the confusion matrix is:

$$S=\left[\begin{array}{cccc}{s}_{11}& s_{12}& \cdots & s_{1q}\\ s_{21}& s_{22}& \cdots & s_{2q}\\ ⋮& ⋮& \ddots & ⋮\\ s_{q1}& s_{q2}& \cdots & s_{qq}\end{array}\right],$$

(13)

where $q$ is the number of levels.

Then, the accuracy can be calculated by

$$\mathrm{Accuracy}=\frac{1}{{\displaystyle \sum _{j=1}^q{\displaystyle \sum _{k=1}^q{s}_{jk}}}}{\displaystyle \sum _{j=1}^q{s}_{jj}};$$

(14)

The precision can be calculated by

$$\mathrm{Precision}=\frac{s_{jj}}{{\displaystyle \sum _{j=1}^q{s}_{jk}}};$$

(15)

The recall can be calculated by

$$\mathrm{Re}\mathrm{call}=\frac{s_{jj}}{{\displaystyle \sum _{k=1}^q{s}_{jk}}};$$

(16)

The F₁ can be calculated by

$$F_1=\frac{2\times \mathrm{Precision}\times \mathrm{Recall}}{\mathrm{Precision}+\mathrm{Recall}}.$$

(17)

Finally, if the prediction performance is reliable, the entire preprocessed dataset can be used as the training set to fit the model. Then, the rockburst damage level in actual engineering can be evaluated. Conversely, if the prediction performance is unreliable, improvements can be made in terms of database quality, data preprocessing, and evaluation models. Moreover, new cases can be adopted to update the original rockburst dataset, and the evaluation process in the next stage can be conducted.

4. Validity Verification

The collected rockburst dataset was used to verify the feasibility of the proposed methodology. Based on Equations (11) and (12), all indicator values were preprocessed. The distribution of indicator values before and after preprocessing was shown in Figure 5. The values of all indicators after preprocessing followed the standard normal distribution with a mean of zero and a standard deviation of one.

To make the model performance more reliable, the number of GPC was optimized by using the five-fold cross validation based on the training set. The average accuracy of five-fold cross validation corresponding to different numbers of GPC was shown in Figure 6. It can be seen that the average accuracy does not increase with the number of GPC, and the optimal number of GPC was 12 because of its maximum average accuracy.

After the bagged ensemble of GPCs with the optimal hyperparameter value was fitted on the training set, it was used to evaluate the rockburst damage on the test set. The evaluation results were expressed by the confusion matrix defined by Equation (13), which were indicated as:

$$S=\left[\begin{array}{cccc}17& 4& 0& 1\\ 2& 6& 1& 0\\ 2& 0& 6& 4\\ 0& 1& 3& 2\end{array}\right]$$

Based on Equation (14), the value of accuracy was 63.27%. If the levels $L_2$ and $L_3$ were merged into the low-risk group, and the levels $L_4$ and $L_5$ were merged into the high-risk group, then the accuracy value of low and high risk was 93.55% and 83.33%, respectively.

According to Equations (15)–(17), the values of precision, recall and F₁ corresponding to different levels were obtained, as shown in Figure 7. It can be seen that the evaluation performance for level $L_2$ was the best, while that for level $L_5$ was the worst. After comprehensively considering the values of precision, recall and F1, the ranking of the evaluation performance for different levels was $L_2>L_3>L_4>L_5$.

5. Case Study

The proposed methodology was applied to evaluate the rockburst damage in the Perseverance nickel mine. The main orebody is hosted in ultramafic rocks, which is mined using the sub-level caving method. The hanging wall is composed of stiff Felsic volcanics and metasediments, which are prone to mining-induced seismicity. The microseismic monitoring system has been established in this mine, making it possible to manage rockburst risk based on microseismic data. Because most of the ramp and infrastructure are located in the hanging wall, it is necessary to evaluate the rockburst damage potential.

Heal [35] recorded twelve rockburst damage cases caused by six microseismic events at the depth of 950 m to 1100 m in this mine. The specific data was shown in

Table 1. Rockburst damage cases at Perseverance nickel mine.

Microseismic Event	I1	I2	I3	I4	I5	I6	I8	Actual Level	Results Using Heal’s Method [35]	Evaluation Results in This Study
#1	57.7	5	12.2	0.5	1.62	14	2700	L2	L5	L2
	57.7	5	12.2	0.5	1.62	22	2700	L2	L5	L2
	47	8	6	0.5	1.62	29	2700	L2	L2	L2
#2	47	8	10.3	0.5	1.8	10	2700	L3	L4	L2
	47	8	6.6	0.5	1.8	10	2700	L2	L2	L4
	46.9	5	5.9	0.5	1.8	16	2700	L3	L3	L4
#3	47.5	10	4.8	0.5	1.5	10	2700	L2	L2	L2
	47.5	10	10	1	1.5	10	2700	L2	L2	L2
#4	39.2	5	5	1	1.8	13	2700	L2	L1	L2
	43.4	8	5	1	1.8	13	2700	L2	L1	L2
#5	58	8	12	0.5	1.6	10	2700	L2	L5	L2
#6	58.1	8	11	1	2.2	5	2700	L4	L3	L2

. The Richer magnitude of microseismic events ranged from 1.5 to 2.2, and their real rockburst damage levels were between $L_2$ and $L_4$.

The proposed methodology was used to evaluate the rockburst damage levels for these twelve cases. First, these cases were preprocessed together with the original rockburst dataset. Then, the preprocessed dataset was used as the training set to train the model. Finally, the rockburst damage levels of these twelve cases were obtained using the trained model, as shown in the last column of Table 1.

According to the evaluation results in Table 1, only the rockburst damage levels caused by microseismic events #2 and #6 were not identified. That is, the evaluation results of four cases were inconsistent with the actual situation, and the accuracy is 66.67%. In Heal’s method [35], the evaluation results of seven cases did not match the actual situation, and the accuracy is 41.67%. Therefore, the methodology proposed in this study improved the evaluation accuracy of rockburst damage to a certain extent.

6. Discussions

Since the dataset used in this study is the same as that in Heal [35], Zhou et al. [16], and Li et al. [17], the evaluation results using our methodology are compared with theirs. The comparison results are shown in

Table 2. Comparison of evaluation results in different literatures.

Evaluation Criterion	Level	Method	Accuracy
The evaluation value corresponds to the actual value	$L_1$, $L_2$, $L_3$, $L_4$, $L_5$	EVP [35]	28.0%
	$L_1$, $L_2$, $L_3$, $L_4$, $L_5$	EVP.PPV [35]	24.4%
	$L_2$, $L_3$, $L_4$, $L_5$	Stochastic gradient boosting approach [16]	61.22%
	$L_2$, $L_3$, $L_4$, $L_5$	The proposed method	63.27%
The evaluation value corresponds to the actual value or the neighboring value	$L_1$, $L_2$, $L_3$, $L_4$, $L_5$	EVP [35]	66.1%
	$L_1$, $L_2$, $L_3$, $L_4$, $L_5$	EVP.PPV [35]	72.4%
	$L_2$, $L_3$, $L_4$, $L_5$	The proposed method	91.84%
The evaluation value corresponds to the actual value after combining $L_1$, $L_2$ and $L_3$ (or $L_2$ and $L_3)$ into one group while $L_4$ and $L_5$ into another group	$L_1$, $L_2$, $L_3$, $L_4$, $L_5$	EVP [35]	71.3%
	$L_1$, $L_2$, $L_3$, $L_4$, $L_5$	EVP.PPV [35]	78.0%
	$L_2$, $L_3$, $L_4$, $L_5$	The proposed method	89.80%
The evaluation value corresponds to the actual value after combining $L_2$ and $L_3$ into one group	$L_2$, $L_3$, $L_4$, $L_5$	Rock engineering systems and artificial neural network [17]	71%
	$L_2$, $L_3$, $L_4$, $L_5$	The proposed method	75.51%

. Among them, Heal [35] artificially synthesized 277 samples with level $L_1$ according to the distribution of the existing data. The evaluation criteria of accuracy used in different literatures are dissimilar, which mainly include the following four categories: (1) The evaluation value corresponds to the actual value; (2) the evaluation value corresponds to the actual value or the neighboring value; (3) the evaluation value corresponds to the actual value after combining $L_1$, $L_2$ and $L_3$ (or $L_2$ and $L_3$) into a group while $L_4$ and $L_5$ into another group; and (4) the evaluation value corresponds to the actual value after combining $L_2$ and $L_3$ into a group. According to these four evaluation criteria, the accuracy of the proposed method is 63.27%, 91.84%, 89.80% and 75.51%, respectively. From Table 2, it can be seen that the accuracy of the proposed method is higher than that of other methods under these four evaluation criteria. This verifies the effectiveness of the method proposed in this study to a certain extent.

Moreover, to further illustrate the reliability of the proposed method, it is also compared with the bagged ensemble of GPCs without preprocessing, bagged ensemble of GPCs without under-sampling, and GPC without under-sampling. The evaluation results of different approaches are shown in

Table 3. Evaluation results of different approaches.

Approaches	Confusion Matrix	Accuracy	F1
Bagged ensemble of GPCs without data preprocessing	$\left[\begin{array}{cccc}15& 4& 2& 1\\ 3& 4& 1& 1\\ 5& 1& 4& 2\\ 0& 1& 4& 1\end{array}\right]$	48.98%	[0.6667, 0.4211, 0.3478, 0.1818]
Bagged ensemble of GPCs without under-sampling	$\left[\begin{array}{cccc}20& 1& 0& 1\\ 5& 4& 0& 0\\ 4& 0& 6& 2\\ 3& 1& 2& 0\end{array}\right]$	61.22%	[0.7407, 0.5333, 0.6, 0]
GPC without under-sampling	$\left[\begin{array}{cccc}19& 2& 0& 1\\ 5& 3& 1& 0\\ 4& 0& 6& 2\\ 3& 1& 2& 0\end{array}\right]$	57.14%	[0.7170, 0.4, 0.5714, 0]
The proposed method	$\left[\begin{array}{cccc}17& 4& 0& 1\\ 2& 6& 1& 0\\ 2& 0& 6& 4\\ 0& 1& 3& 2\end{array}\right]$	63.27%	[0.7907, 0.6, 0.5455, 0.3077]

. It can be seen that the bagged ensemble of GPCs without data preprocessing has the lowest accuracy of 48.98%, which shows the importance of data preprocessing. Before preprocessing, the distribution of some indicators is skewed, and there is no clear distribution law for each indicator. When using the Yeo-Johnson transformation, the sample data is mapped from an arbitrary distribution to a Gaussian distribution as close as possible to stabilize variance and minimize skewness. In addition, the influence of diverse dimensions and units on the evaluation results can be avoided after standardization. Therefore, the accuracy is improved by using data preprocessing. The bagged ensemble of GPCs and the GPC without under-sampling can identify level $L_2$ well, but cannot achieve a reliable recognition of level $L_5$. Moreover, the value of F₁ is 0 in these methods, which illustrates that under-sampling has an important influence on the evaluation results. Because the distribution of different rockburst damage levels is relatively unbalanced, the prediction results are biased towards the level with a larger number of samples. When using the under-sampling technique, this influence can be avoided to some extent by balancing the training samples. Compared with GPC, the bagged ensemble of GPCs improves the evaluation accuracy, which indicates the bagging method can improve the evaluation performance. By integrating multiple GPCs using the bagging method, the generalization ability and robustness of the model can be increased to a certain extent. Therefore, the integration of data preprocessing, under-sampling technique, GPC, and bagging method improves the comprehensive performance.

Although the proposed method can evaluate the rockburst damage to some extent, there are still some shortcomings:

(1)

The bagged ensemble of GPCs has better evaluation performance for level $L_2$, but the evaluation performance for level $L_5$ still needs to be improved. The reason may be that the sample size of $L_2$ is the largest, while that of $L_5$ is the least. A large number of samples can make the model fit better, which can improve the evaluation performance in turn. Because the data-driven method is highly dependent on the quality of data, a higher-quality rockburst damage database should be established in the future.

(2)

More indicators for rockburst damage evaluation need to be considered. According to the original rockburst damage database, some samples with the same indicator values have different levels. This shows that some key indicators are ignored, which may be an important reason for restricting the evaluation accuracy of rockburst damage. In the future, some novel evaluation indicators may be proposed from the perspective of focal mechanisms and failure characteristics of rock mass under dynamic and static stress.

(3)

Considering the distribution of some indicators is skewed, the Gaussian process may yield impropriate results. Although the proposed method can obtain relatively good results, the Gaussian process with skewed errors can be further used to investigate the evaluation performance [39,40].

7. Conclusions

To effectively assess rockburst damage potential with an imbalanced dataset, this study proposed a novel model by integrating the under-sampling technique, GPC, and bagging method. Based on the rockburst dataset preprocessed by the Yeo-Johnson transformation and standardization, the reliability of the proposed model was verified. The accuracy values of all samples, low risk ($L_2$ and $L_3$) and high risk ($L_4$ and $L_5$) were 63.27%, 93.55%, and 83.33%, respectively. According to the values of precision, recall and F₁, the ranking of evaluation performance at different levels was $L_2>L_3>L_4>L_5$. By using the evaluation criteria in other literature, the accuracy of the proposed method was highest, which further verified the reliability of our method. Based on the evaluation results of the other three methods (the bagged ensemble of GPCs without preprocessing, bagged ensemble of GPCs without under-sampling and GPC without under-sampling), the comprehensive performance of the proposed method was better. It indicated the effectiveness of the integration of data preprocessing, under-sampling technique, GPC and bagging method in this study. The proposed methodology was applied to assess rockburst damage potential in the Perseverance nickel mine. The evaluation accuracy was 66.67%, which was 25% higher than the method in original literature. Due to the improved performance, the evaluation results provided a valuable guidance for the prevention of rockburst disasters.

In the future, a higher-quality database for the evaluation of rockburst damage potential should be established under a unified data acquisition standard. Because of the complex mechanisms of rockburst, some novel evaluation indicators are worth investigating based on focal mechanisms and failure characteristics of rock mass under dynamic and static stress. The Gaussian process with skewed errors can be obtained to investigate the evaluation performance after considering the skewness of indicator distributions. In addition, the proposed methodology can be applied in other mining and geotechnical engineering fields, such as pillar stability prediction and landslide risk analysis.