Prediction of Grade Classification of Rock Burst Based on PCA-SSA-PNN Architecture

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Introduction
Rock burst is a kind of dynamic failure phenomenon which often occurs in the high ground stress area of deep buried tunnels. When the mechanical equilibrium state of rock mass is broken, the accumulated energy in the rock mass is released in a sudden and violent form, resulting in a dynamic instability phenomenon [1][2][3][4][5]. The accurate assessment of rock burst grade classification is an important content of rock burst prevention measures.
The prediction methods of rock burst grade used in practical engineering are usually based on rock strength, such as the Russenes criterion [6], Erlang Mountain criterion [7], and Barton criterion (Barton et al. [8]). Furthermore, in recent years, some scholars have further studied the rock burst prediction method also based on the rock strength. For example, Afraei et al. [9] analyzed the contribution rate of rock burst influencing factors to rock burst grade prediction and found that maximum tangential stress and uniaxial compressive strength of rock significantly contributed to rock burst grade prediction; He et al. [10] modified the previous rock burst grade prediction method by introducing the gradient stress and recognized that the range of the rock strength stress ratios used for rock burst grade prediction was not uniform; Wu et al. [11] put forward the rock burst criterion of Lalin railway tunnel, and compared with the Russenes criterion and Erlang Mountain criterion, the rock burst criterion of Lalin railway tunnel is more consistent with the actual situation of Lalin railway; Wang et al. [12] established a rock burst prediction model based on rock mechanical properties and in situ stress, which could be reliably applied to rock burst prediction. These above prediction methods of rock burst grade consider the stress condition and mechanical properties of rock well and promote the research of rock burst prediction method.
However, the occurrence of an actual rock burst is more closely related to rock mass structure and strength, and there are some reports about predicting rock burst using rock mass structure and strength. For example, Qiu et al. [13] discussed the influence of rock mass structure on the velocity of flying rock when rock burst occurred and realized that floor deflection is an important focal mechanism that causes rock velocity and serious rock burst damage; Chen et al. [14] and Ma et al. [15] considered the impact of rock mass integrity on rock burst and found that the strength of rock mass is the main controlling factor for the prediction of rock burst grade; Du et al. [16] analyzed the influence of structural plane strength on rock burst and held that structural plane strength is the main factor to judge whether instantaneous rock burst or hysteretic rock burst occurs in rock mass; Mohamad et al. [17] considered that the joint spacing and apertures are the main causes that determine the number and distance of flying rocks as rock burst occurs; Feng et al. [18] and Zhou et al. [19] believed that rock mass structure is an important factor affecting the occurrence of rock burst, and the structural surface where rock burst occurs is mostly rigid surface. These above researches take the rock mass structure as the important factor of inducing rock burst, which is consistent with the actual situation.
In practice, most of the rock burst grade classification prediction methods based on the strength of rock materials or rock mass were based the single-index methods, and these methods were usually summarized from the specific engineering cases with the relatively less evaluation information. Considering the actual complexity, many scholars adopted the multi-index method to estimate the rock burst grade. For example, Zhou et al. [20] considered that genetic algorithm and particle swarm optimization algorithm could speed up the parameter optimization search of support vector machine (SVM), and the proposed method of rock burst grade prediction has strong robustness; Dong et al. [21] found that compared with SVM, the random forest algorithm had a lower misjudgment rate of rock burst grade; Wang et al. [22] established a multi-index method for rock burst prediction based on the fuzzy matter-element theory, information entropy theory, and proximity rule and found that the established method is more reliable than the traditional method; Zhang et al. [23] made a comprehensive prediction of rock burst based on the rock elastic energy index, rock strength, and principal stress, which could make up for the deficiency of single-index rock burst prediction method; Li et al. [24] proposed a rock burst prediction network based on genetic algorithm and extreme learning machine, and the prediction results show that the maximum relative error of the proposed method is 4.71%; Xu et al. [25,26] put forward a new rock burst grade evaluation using the ideal point theory, and the error rate is 5%, and the average crossover error rate is 13.33%; Liang et al. [27] found that gradient-boosted decision tree algorithm could be applied to short-term rock burst prediction with an accuracy of more than 90%; Meng et al. [28] believed that BP (back propagation) neural network prediction and least square method may reduce the influence of subjective judgment on the prediction results and could obtain the prediction results in the first time; Chen et al. [29] utilized the Bayesian method to estimate the rock burst grade and found that Bayesian statistical learning model has robustness and generalization in rock burst risk assessment; Gao et al. [30] held that the radial basis neural network optimized by hybrid particle swarm optimization algorithm may take into account individual optimization and global optimization and could predict the rock burst grade correctly and effectively; Gong et al. [31] established a deep learning rock burst prediction model based on dropout and Adam algorithm, and the model avoids the problem of determining index weights and is completely data-driven; Liu et al. [32] found that the rock burst prediction network based on histogram gradient-enhanced tree algorithm still has a high prediction ability for incomplete rock burst data, with an accuracy of nearly 80%. In these above researches, machine learning and deep learning methods were adopted to establish a multiindex rock burst grade prediction network, and the accuracy of the prediction results was significantly improved than that of the single-index rock burst prediction method. The multiindex rock burst grade prediction methods could consider the influencing factors of rock burst in many aspects, reduce the interference of human factors, and make the prediction result more close to the actual situation.
Actually, the multi-index rock burst prediction methods may be used to reveal the mechanism of rock burst in more detail. However, the existing multi-index-based predictions usually focused on the rock strength and ignored the influence of rock mass strength. For the high complexity and unpredictability of a rock burst, it is necessary to conduct a new multiindex evaluation method of rock burst grade considering the rock mass strength. In this study, the field data of a tunnel in western China and the case data of diversion tunnel project in Pakistan [33] and Erlangshan tunnel project in China [7,34] were used as the data set for rock burst grade evaluation. The data set was then divided into training set and test set. The principal component analysis (PCA) method was furthermore used both to reduce the dimension of the rock burst data set and to eliminate the linear correlation between different indexes. The sparrow search algorithm was also conducted to optimize the smoothing factor in the probabilistic neural network (PNN). The prediction results from the improved PNN algorithm were thereafter compared with those from other existing rock burst prediction methods, including singleand multi-index rock burst prediction methods.  Specht in 1990 [35], mainly composing of input layer, pattern layer, summation layer, and output layer as shown in Figure 1. The PNN has the characteristics of fast convergence, high stability, and no local optima and is suitable for rock burst grade classification. The PNN is briefly described below. same as that of training samples. The Gaussian function was introduced into this layer as the activation function. The distance from the input layer was used as the input in the pattern layer. The output of the pattern layer is where G ij ðX k Þ represents the output value of class i and group j in the pattern layer; σ represents the smoothing factor, a key parameter for PNN training; and l represents the dimension of the sample vector.

Summation
Layer. The number of cells in this layer is the same as the number of the PNN target classes. In this layer, the output of pattern layer was added separately based on different classes, and the output of the summation layer is as follows [36,37]: where N i is the group number of class i in the training sample and F i ðXÞ reflects the probability that the input vector X k is judged as class i.

Output
Layer. The number of cells in the layer is 1.
The class corresponding to the largest one in F i ðX k Þ was output as 1, and the rest was output as 0.
2.2. SSA. Sparrow search algorithm (SSA) is an intelligent optimization algorithm to simulate the foraging and antipredation behavior of sparrow population, which was proposed in 2020 [38]. This algorithm has the characteristics of strong optimization ability and fast convergence speed. Sparrow populations were divided into finders and followers, and the finders provided foraging directions for the followers. The sparrow population was set up with a certain percentage of sparrows aware of danger to avoid attacks, usually 10 to 20 percent. SSA is shown in Figure 2 and could be divided into seven steps as follows.
2.2.1. Set the SSA Initial Parameters. The number of sparrows, the maximum number of iterations, the ratio of discoverers to followers, the proportion of sparrows aware of danger of sparrows, and warning value are set.

Calculate the Fitness Function
Value. The fitness function value was calculated by using the initial parameters of SSA. The fitness function is usually the error function between the estimated value and the actual one.

Update the Finder's Location Information
Based on the Calculated Alarm Value. The position information of finders in the population was updated by using [39][40][41] where k represents the current number of iterations; iter max represents the maximum number of iterations; X k ij represents the position information of the ith sparrow in the jth dimension of the sparrow population; R 2 represents a random number ranging from 0 to 1; ST represents the earlywarning value initially set, generally 0.5 to 1; Q represents a random number subject to normal distribution; and L represents a matrix of 1 × D, where D represents the dimension of the sparrow population. When R 2 < ST, that is, there are no predators around the foraging environment, the finder could perform extensive search operations; when R 2 ≥ ST, that is, some sparrows have already found predators, all sparrows need to quickly move to somewhere safe to feed.
where M represents the number of population, X k worst represents the worst position information in the sparrow population at the k iteration, X k F represents the best position information of the finders in the sparrow population at the k iteration, and A is a matrix of 1 × d, where each element in A is randomly assigned as 1 or -1 and When i > M/2, that is, the ith follower with a lower fitness value has not received food and needs to fly somewhere else to forage for food.

Update the Location Information of the Sparrows
Aware of Danger. Update the position information of the sparrows aware of danger in the population by using where X k best represents the optimal position information in the sparrow population at the k iteration, β represents a random number following a normal distribution, K represents a random number from -1 to 1, F i represents the fitness value of the ith sparrow, F g represents the current optimal fitness value, F w represents the current value of the worst fitness, and ε represents a very small constant in preventing the denominator from returning to zero. When F i < F g , the sparrow is at the edge of the population and is extremely vulnerable to predators; when F i = F g , that is, sparrows are aware of danger and need to be close to other sparrows to minimize their risk of predation.

Update the Fitness Values and Record the Optimal
Parameters. Using the position information of sparrow population in Equations (4), (5), and (6), the fitness values were recalculated and reordered. The fitness values and the best and worst position information of the sparrow were also recorded.

Whether the Maximum Number of Iterations Is
Achieved. If the number of iterations does not reach the maximum number of iterations at this time, the iterative calculation will continue until the number reaches the maximum of iterations, and then, the optimal fitness value and corresponding optimal position information of the sparrow will be output.

Geofluids
Erlang Mountain tunnel of China, 43 groups of data [7,14,25,26,33] were selected. Each group of the data contains seven rock burst influencing factors, including maximum in situ stress σ max , maximum tangential stress σ θ , rock strength σ ci , rock mass strength σ cm , the ratio of σ θ to σ ci , the ratio of σ ci to σ max , and the ratio of σ cm to σ max . Among them, σ max , σ θ , and σ ci were obtained by indoor and outdoor tests, while σ cm was estimated by the Hoek-Brown strength criterion. These seven factors constitute a prediction index system of rock burst grade. Using the method of Considering that the Gaussian function in the PNN needs to be independent of each index of the rock burst prediction, the principal component analysis (PCA) method was used to reduce the dimension of the seven indexes of rock burst grade prediction, and four new independent indexes of rock burst grade prediction were obtained: PCA 1 , PCA 2 , PCA 3 , and PCA 4 . Figure 3 shows the structure of the new multi-index rock burst grade prediction method PCA-SSA-PNN. The original and new predicted indexes are shown in Table 2.
As shown in Figure 3, the 43 groups of new index data of rock burst in Table 2 were standardized by where x s represents the standardized data, x p represents new index data of rock burst, and x p and x pσ represents the mean value and variance of the new index data of rock burst, respectively. The 29 of the 43 groups of new index data were randomly selected as the training set, and the remaining 14 groups were selected as the test set. The smoothing factor σ is a key parameter in determining the performance of PNN. The network is easy to be overfitted if the smoothing factor is too small, while the details may be impossible to be distinguished if the smoothing factor is too great [43]. In the current study, the SSA was used to optimize the smooth factor in the PNN, the training set was again divided  5 Geofluids into optimization training set and optimization test set, and the estimated loss value of rock burst grade in optimization test set was calculated. The estimated loss value was minimized by constantly updating the smooth factor, and the optimal smooth factor value was then recorded and output, and the optimum smooth factor was finally input into the PNN to obtain the rock burst grade by using the prediction network PCA-SSA-PNN.  The new rock burst index data in Table 2 were utilized as the original data set of rock burst grade prediction network, and the total errors of training set and test set were considered as the fitness function value of rock burst grade prediction network. The SSA was used to optimize the PNN, and the optimal smoothing factor (1.0217) was obtained. Therefore, a multi-index rock burst grade prediction network based on PCA-SSA-PNN was constructed. The PCA-SSA-PNN-based architecture was adopted to predict the rock burst grades of the test set. Figure 4 shows the prediction results. It can be seen from Figure 4 that there is a misjudgment in group 10 of the test data set, and the prediction was one grade higher than the actual rock burst, with an error ratio of less than 8%; the prediction results of PCA-SSA-PNN-based architecture are generally consistent with the actual rock burst grades. In order to further analyze the performance of PCA-SSA-PNN, the statistical results in Figure 4 are shown in Table 3. Table 3 shows the comparison between the prediction results of PCA-SSA-PNN and actual grades in detail. It can be seen from Table 4 that the PCA-SSA-PNN has a higher prediction accuracy of no rock burst, slight rock burst, and strong rock burst with the accuracy ratios of 100%; the prediction accuracy of PCA-SSA-PNN for medium rock burst is close to 90%, which could meet the needs of engineering; the PCA-SSA-PNN may overestimate the medium rock burst by 11.11%; the average overprediction ratio of rock burst prediction by PCA-SSA-PNN is less than 8%; the PCA-SSA-PNN does not underestimate the rock burst grades; the average accuracy of rock burst prediction by PCA-SSA-PNN reaches 90% (here 92.86%), indicating that the established PCA-SSA-PNN has strong ability in estimating rock burst grades.

Comparison between PCA-SSA-PNN and Other
Prediction Methods. In exploring the feasibility of the above-established PCA-SSA-PNN for estimating rock burst grades, three single-index methods, or Russenes' [6], Barton's [8], and Xu et al.'s [25,26] methods, were conducted to estimate the rock burst grade for the test set data. The single-index prediction methods are shown in Table 4. Furthermore, four multi-indexes, or back propagation (BP) neural network, support vector machine (SVM), random forest (RF), and standard PNN (smoothing factor is set to 0.5), were also conducted to estimate the rock burst grades.
It can be seen from Table 4 that in predicting the rock burst grades, Barton's and Russenes' methods are based on the rock strength, while Xu et al.'s method is based on the rock mass strength. Considering the consistency in comparison, the cases of σ cm /σ max greater than 0.15 were considered as minor rock burst.
Using these above single-and multi-index prediction methods, the rock burst grade predictions were obtained (see Table 5).
The statistics of prediction results of rock burst grades by using the single-and multi-index prediction methods are conducted and shown in Table 6.
As shown in Table 6,

Geofluids
Russenes' method are based on the rock strength and have the poorer performances with the prediction accuracy less than 50%, and Xu et al.'s method is based on the rock mass strength and has a better per-formance with the prediction accuracy of greater than 70%, while the established PCA-SSA-PNN in this study has the best performance; Russenes' method may overestimate the rock bursts grade in    In conclusion, the multi-index methods is more suitable for the prediction of rock burst grade than the single-index prediction methods; compared with the other four multiindex rock burst prediction methods, the established PCA-SSA-PNN is more reasonable to predict the grade classification of rock burst.

Discussions
In order to analyze the performance when the proposed model face with the new conditions, a new rock burst case was introduced. Mufeiling tunnel of Hangzhou-Wenzhou railway is located in Tonglu County, Hangzhou City, Zhejiang Province, China. The starting mileage is DK74 +702.93, the ending mileage is DK84+943.27, and the central mileage is DK79+823.08, with a total length of 10240.34 m. The tunnel site is located in the middle and low mountainous area, with large relief and natural slope of 25~35°. The highest elevation of the tunnel site is about 889 m, and the maximum depth is 619 m. According to statistics, rock bursts have occurred 421 times on six working faces during the construction of the Mufeiling tunnel.
Rock burst has caused serious damage to Mufeiling tunnel, increased the difficulty of the initial support and the amount of engineering, and seriously hindered the construction progress. Therefore, it is necessary to determine the rock burst grade of Mufeiling tunnel in order to take appropriate preventive measures. PCA-SSA-PNN was used to predict rock burst at DK77+500 and DK79+050 locations of Mufeiling tunnel, respectively, and the results are shown in Table 7.
It can be seen from Table 7 that the predicted results are basically consistent with the actual rock burst grades. The rock burst grade prediction method based on PCA-SSA-PNN has been well applied in Mufeiling tunnel.

Conclusions
In this work, the field data of a tunnel in western China and the rock burst data of the Pakistan diversion tunnel project and the Erlang Mountain tunnel project in China were used as the original input data in the rock burst prediction network. Different influencing factors of rock burst were considered to establish the index system for the rock burst grade prediction. The principal component analysis (PCA) was used to reduce the dimension of the rock burst data set and eliminate the linear correlation among different influencing factors. The sparrow search algorithm (SSA) was used to optimize the smoothing factor in the probabilistic neural network (PNN), and a multi-index rock burst prediction network PCA-SSA-PNN was thereafter obtained. The comparison of the prediction results by PCA-SSA-PNN with those by the single-and other multi-index rock burst prediction methods was furthermore conducted, respectively. It shows the following: (1) Among the single-index rock burst prediction methods, the method considering the strength of rock mass has higher prediction accuracy (2) The accuracy of multi-index methods is often higher than that of single-index methods in predicting the grade classification of rock burst (3) Among the multi-index rock burst prediction methods, in the order from large to small, the overprediction rates are random forest, back propagation, support vector machine, standard PNN, and PCA-SSA-PNN, while the underprediction rates are RF, BP, standard PNN, SVM, and PCA-SSA-PNN (4) The established rock burst prediction method based on the PCA-SSA-PNN architecture considers the influence of rock mass strength, and the rock burst grade estimated by the method is in good agreement with the actual rock burst grade, which could be used for the rapid prediction of rock burst in practice The multi-index rock burst grade prediction method established in this study takes into account the influence of rock mass strength on rock burst. However, the rock mass strength cannot be directly obtained by testing methods at 9 Geofluids present, so it is often estimated by empirical formula, and there may be errors between estimated results and actual values. Moreover, the number of different rock burst grades in PCA-SSA-PNN training is small, which will have some influence on the generalization ability of the network. Therefore, the category and quantity of rock burst grade data need to be increased in the subsequent research, and the selection method of rock mechanical characteristic parameters and the application range of the new multi-index prediction method also need to be further studied.

Data Availability
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
The authors declare that there is no conflict of interest.