Application of the FCM algorithm and improved grey target theory on the prediction of rock burst

Abstract

Rock bursts represent a high-risk geological hazard in deep underground engineering, posing severe threats to both operational safety and structural integrity. Although the conventional gray target model is widely recognized for evaluating rock burst intensity, its predictive accuracy is often compromised by inherent correlations among assessment indicators, resulting in potential information redundancy and distorted assessment outcomes. To overcome these limitations, this study proposes a novel hybrid methodology that integrates the fuzzy C-means (FCM) clustering algorithm with optimized gray target theory, enabling more accurate and robust intensity quantification. A practical engineering case validates the proposed model, demonstrating its feasibility and superior predictive performance compared to the traditional gray target approach, and it can mine and develop data to the greatest extent using information from a few samples and limited information. These findings suggest that this hybrid methodology offers a robust alternative for rock burst intensity assessment, offering both theoretical advancements and practical guidance for future deep mining operations.

1 Introduction

Rock bursts are a prevalent geological hazard in high-stress zones of the deep underground; they are distinguished by their sudden onset, high hazard potential, and difficulty in prevention (Gu et al., 2022b). These events manifest as violent ejection of surrounding rock, projectile-like fragmentation, and spalling (Gu and Wu, 2019), posing a serious threat to personnel safety and construction and equipment integrity in underground engineering projects. Rock bursts frequently result in casualties and substantial property losses, making accurate prediction a critical priority for ensuring operational safety in both underground and rock engineering construction (Wang et al., 2025).

In recent years, scholars have proposed various methods for rock burst prediction, including the gray theory method (Gu et al., 2021), the particle swarm optimization projection pursuit method (Xu et al., 2026), the genetic-RBF neural network-based method (Xu et al., 2025), the distance discrimination method (Xu et al., 2023), the numerical simulation-based 3D geostress measurement method (Ji-liang et al., 2009), and the acoustic emission (AE) time series method (Bin et al., 2011). Although these methods offer scientific insights into rock burst prediction, they each exhibit notable limitations. For instance, the gray theory method is highly sensitive to noise in the original data, compromising prediction accuracy. The particle swarm optimization projection pursuit method relies on precise measurements of indicators from actual engineering projects, and its optimization becomes challenging as the number of discriminative factors increases (Weizhang and Guoyan, 2022; Cai et al., 2018). The genetic-RBF neural network approach demonstrates limited extrapolation capability and requires an adequately large sample distribution space to ensure reliable predictions (Sun et al., 2021). The distance discrimination method is strongly influenced by the representativeness and accuracy of the input data, while the numerical simulation-based 3D geostress measurement method demands high-precision, comprehensive geostress measurements in the target area, making practical implementation difficult. Additionally, the AE time series method suffers from slow convergence and limited applicability (Shun‐chuan et al., 2019).

In rock burst grade evaluation, challenges such as the highly nonlinear relationships between rock bursts and evaluation factors, difficulties in satisfying grade-specific criteria simultaneously, measurement errors, and subjective expert weighting contribute to ambiguity and uncertainty in classification outcomes. The traditional gray target theory often relies on a single dataset and lacks comprehensiveness. Meanwhile, its determination of the target center exhibits certain subjectivity and static nature, thereby affecting the accuracy and reliability of model predictions (Gu et al., 2022a; Gu and Wu, 2019).

Some researchers have begun to adopt machine learning methods. Machine learning does not require any prior knowledge about the relationship between input and output variables and reduces the impact of human intervention. For example, classic machine learning algorithms such as support vector machine (SVM) (Xiaoyu et al., 2025), logistic regression classifier (Shirani Faradonbeh et al., 2024), decision tree (DT) (Dianshang et al., 2025), Bayesian network (BN) (Mao et al., 2023), random forest (RF), K-nearest neighbors (KNN) (Wang et al., 2024), and light gradient boosting machine (LGBM) have been employed for rock burst risk prediction. However, these methods suffer from the problem of low accuracy, which can be primarily attributed to three aspects: first, the models themselves have limitations. For example, relatively simple or outdated structures fail to effectively extract feature information for training; second, some datasets are relatively homogeneous and lack comprehensiveness, which restricts the generalization capability of the models; third, the feature selection process in certain studies is overly subjective and insufficient and lacks systematic and scientific analysis.

To overcome these limitations, this study introduces a hybrid evaluation framework that combines the fuzzy c-means (FCM) clustering algorithm with an optimized gray target theory for improved rock burst intensity assessment. This synergistic methodology exhibits superior performance in analyzing complex, multidimensional, and highly correlated geomechanical systems. The integration strategy incorporates two methodological innovations to improve upon the traditional gray target model (Xia et al., 2022; Kamran et al., 2022): ① Distance metric optimization: The correlation coefficient matrix is substituted for the conventional covariance matrix in Mahalanobis distance calculations, effectively decoupling interrelated indicators while preserving their inherent relationships. This modification enhances dimensionality reduction efficiency and reduces information redundancy. ② Target center calibration: Positive and negative ideal solutions from the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) model are adapted to define reference target centers, providing more discriminative benchmarks for intensity classification. This refinement improves the model’s ability to distinguish between different hazard levels.

Although it still has some shortcomings, this hybrid methodology offers a robust alternative for rock burst intensity evaluation and brings both theoretical innovation and practical guidance for future deep mining operations.

The remainder of this article is organized as follows: in Section 2, the methodology based on the FCM algorithm and gray target theory is presented; in Section 3, the engineering background is introduced. In Section 4, an assessment model is constructed, and the assessment results are analyzed. In Section 5, conclusions are drawn.

2 Methodology

2.1 Fuzzy c-means algorithm and rough set theory

First, the data uncertainty is determined, and fuzzy sets are generated through the fuzzy c-means method. Then, the dependency degree of conditional attributes on decision attributes using rough set theory is calculated. Finally, weight values are obtained through normalization.

Rough set theory is an analysis theory of discrete data proposed by the Polish mathematician Pawlak in 1982 to handle fuzzy and uncertain knowledge (Yari et al., 2023; Zhou et al., 2020). It is supposed that a quadruple is a knowledge representation system, where the universe of discourse and the attribute set are both non-empty finite sets. , where and are the conditional attribute set and the decision attribute set, respectively. is the value domain set of attributes, and is the information function. This type of knowledge representation system with conditional and decision attributes is called a decision table. In rough set theory, a decision table is applied to describe objects in the universe of discourse. The decision table is a two-dimensional table that has been discretized based on the FCM algorithm. Each row describes an object, and each column describes an attribute of the object.

Let and Then, the support degree of the decision attribute with respect to (or 's support over with regard to) the conditional attribute is defined as follows (Yan et al., 2007):

where is the degree to which decision attribute depends on conditional attribute . Its range of values is [0, 1]. When , is fully dependent on ; when , is fully independent of ; and when , is roughly dependent on .

denotes the positive region of under ; is the cardinality or size of the set, which represents the number of elements in the set for finite sets. The importance of a conditional attribute subset , with respect to the decision attribute , is defined as follows:

Based on the magnitude of the importance of conditional attribute subsets, one can discern their strength of influence on the decision attribute. The weights of various conditional attribute subsets can be expressed as follows:

2.2 Calculating the Mahalanobis distance

To accurately feature the relationship between the assessment indicators, the correlation coefficient matrix is substituted for the covariance matrix in the Mahalanobis distance, improving on the traditional theory. Simultaneously, the Mahalanobis distance is applied to determine the positive and negative target center distances. Their calculative process is listed as follows:

2.2.1 Building of the sample matrix

Suppose that there exist m samples in the set . assessment indicators are constructed as the index set . The magnitude of the sample is . Then, the corresponding matrix of to is

2.2.2 Normalizing the sample matrix

To reduce the dimensions, it can be expressed as follows:

For the cost-type indicator:

For the benefit-type indicator:

where .

Based on the features of the normalization matrix, a standardized matrix can be expressed as follows:

2.2.3 Determining negative and positive target centers

Suppose that , then

where is the positive ideal solution.

Assuming that , then

where is the positive ideal solution.

2.2.4 Determining the positive and negative target center distance

where and are the coefficient differences between sample and the positive and negative target centers, respectively.

The positive target center distance is calculated as follows:

The negative target center distance is

where denotes the correlation coefficient matrix.

2.2.5 Determining the relative target center distance

The relative target distance of indicator is calculated as follows:

2.2.6 Classifying the quality grade

The set is defined as the ordered intervals obtained by partitioning set based on quality levels, namely, . Let , where is a positive integer. The quality grade , then the level classification is (Gu et al., 2022c)

3 Engineering background

The Cangling Tunnel is located in the Kuocangshan Mountain area within Zhejiang Province (see the map in Figure 1), with a maximum burial depth of 768 m. It primarily traverses slightly weathered tuff and potassium feldspar granite porphyry, with local sections passing through fault zones and lithofacies contact zones. In areas away from faults, the rock mass is relatively intact, and the uniaxial compressive strength of the rock is 160 MPa. Results from in situ stress testing and inversion analysis indicate that the maximum principal stress in the engineering area where the tunnel is situated ranges from 7.9 MPa to 27.3 MPa, and it has the potential for rock bursts.

FIGURE 1

4 Establishing the assessment model

4.1 Establishing the indicator systems

Taking into account the lithological factors, geostress factors, and energy factors related to rock burst occurrence, the manuscript selects uniaxial compressive strength (), stress coefficient (), elastic energy index (), and brittleness index () as the set of rock burst evaluation indicators. Referring to existing research on rock burst criterion indicator systems (Gong et al., 2020), the classification standards for rock burst grades corresponding to each indicator are presented in Table 1.

Thirty groups of data were selected as the model test set to verify and evaluate the accuracy of the model, as shown in Table 2.

The first 22 samples shown in Table 2 are selected as the training data, and the final eight samples are selected as the testing data.

4.2 Determining weight coefficients

Data discretization was performed on Table 2 based on the FCM algorithm, where represents the universe of discourse, and denotes the attribute set. In this article, continuous attribute values are discretized according to the classification levels of rock burst categories based on various influencing factors, with the corresponding relationships shown in Table 3.

In Table 3, the condition attribute is , and its number of clusters is 4; the decision attribute is , and its number of clusters is 4. For example, the indicator is selected as the instance, then , , , , and . Then, based on Equations 1–3, the weight coefficients can be obtained as follows:

4.3 Standardizing the sample matrix

The sample data are standardized. Uniaxial compressive strength (), stress coefficient (), and elastic energy index () are benefit-type indicators; brittleness index () is a cost-type indicator. They are standardized based on Equations 4–8, respectively. The matrix is shown in Table 4.

4.4 Determining the relative target center distance

Based on Table 4, the positive target center of the samples is determined as , and the negative target center is ; based on the standardized sample indicator matrix R, the correlation coefficient matrix among evaluation indicators can be obtained using the SPSS software, as shown in Table 5.

Table 5 shows that the evaluation indicator has strong correlation with . To eliminate the impact caused by the correlation among evaluation indicators, the correlation coefficient matrix is introduced into the calculation of the relative target-heart distance.

According to Equations 9–14, the target center distance is shown in Table 6.

4.5 Determining the classification standard

Based on Tables 2, 6, in combination with Equations 15–17, the standard of the relative target center distance is shown in Table 7.

Its classification target is plotted in Figure 2.

FIGURE 2

4.6 Model predictions

The latter eight sets of data in Table 1 are chosen as the training samples. The risk grades of these training samples can be determined based on the risk grade classification presented in Table 7 and the corresponding target center distances. Notably, the grade levels derived from the improved gray target model align precisely with the actual risk levels. The actual risk grade level of the testing sample was established by referencing geological data and exploration information. The comparative results are summarized in Table 8.

Based on the comparative results of the assessment model presented in Table 8, it can be concluded that the outcomes derived from the proposed method are in complete alignment with the actual risk levels for all eight samples. The accuracy of the proposed method reaches 100%. Consequently, this conclusion supports the feasibility of employing this approach to estimate rock burst intensity. Compared to the traditional gray target theory (Gong et al., 2020), the proposed method demonstrates superior accuracy (with the traditional theory achieving an accuracy of only 75%). This discrepancy primarily arises because the traditional gray target model overlooks the correlation between evaluation indicators during the assessment process, leading to redundant computation of information and a consequent decline in the precision of evaluation results. For instance, the positive correlation coefficient between two evaluation indicators reaches 0.848. As evident from Table 8, the risk levels of rock bursts vary among samples 23–30. Specifically, samples 23, 28, and 29 exhibit a risk level of IV, indicating a strong propensity for rock bursts. In contrast, sample 30 displays a medium risk level. Therefore, corresponding consolidation measures should be implemented for these samples. For the remaining samples, which exhibit either no rock burst or a weak rock burst, no specific measures are deemed necessary.

5 Conclusion

6 Research limitations and prospects

Although the suggested model has improved rock burst prediction, it still has some limitations, for example, insufficient sample size, single validation scenario, and inadequate indicator system. Specifically, overfitting problems should be valued. Therefore, in future research, sample data should be expanded; more influential factors should be considered, and the dynamic prediction in combination with real-time monitoring data should be performed. In addition, to solve the overfitting problems, the data should be denoised and outliers must be removed. After regularization techniques are introduced, cross-validation should be performed for the evaluation results to avoid randomness caused by data splitting. These improvements will enhance the suggested model and will be the focus of future work on assessing rock burst intensity. It will bring both theoretical innovation and practical guidance for future deep mining operations.

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