# Rockburst Intensity Level Prediction Method Based on FA-SSA-PNN Model

^{1}

^{2}

^{*}

## Abstract

**:**

_{θ}, σ

_{t}, σ

_{c}, σ

_{c}/σ

_{t}, σ

_{θ}/σ

_{c}, W

_{et}) were selected to build a rockburst intensity level prediction index system. Seventy-five sets of typical rockburst case data at home and abroad were collected, the original data were preprocessed based on factor analysis (FA), and the comprehensive rockburst prediction indexes, CPI

_{1}, CPI

_{2}, and CPI

_{3}, obtained after dimensionality reduction, were used as the input features of the SSA-PNN model. Sixty sets of rockburst case data were extracted as the training set, and the remaining 15 sets of rockburst case data were used as the test set. After the model training was completed, the model prediction results were analysed and evaluated. The research results show that the proposed rockburst intensity level prediction method based on the FA-SSA-PNN model has the advantages of high prediction accuracy and fast convergence, which can accurately and reliably predict the rockburst intensity level in a short period of time and can be used as a new method for rockburst intensity level prediction, providing better guidance for rockburst prediction problems in deep rock projects.

## 1. Introduction

## 2. Methods

#### 2.1. Factor Analysis (FA)

_{1}, X

_{2}, …, X

_{n}, n rockburst prediction variables can be represented by m factors F

_{1}, F

_{2}, …, F

_{m}and the product of an A

_{n}

_{×m}order factor loading matrix plus a special factor ε = (ε

_{1}, ε

_{2}, …, ε

_{n}) (n ≥ m), while the established mathematical model of factor analysis is: X

_{n}= A

_{n}

_{×m}F

_{m}+ ε

_{n}, i.e., Equation (1):

_{1}, F

_{2}, …, and F

_{m}are m independent common factors. The matrix A

_{n}

_{×m}is called the factor loading matrix. a

_{nm}denotes the weight of the nth variable on the mth factor variable, which reflects the importance of the common factor on the variable, and is important for explaining the common factor. The special factors in the model have a small effect relative to the main factor F

_{m}in Equation (1) and are neglected in the study. The steps of the factor analysis are as follows:

_{max}is the maximum value in the sample; X

_{min}is the smallest value in the sample; X

_{ij}

^{*}is the normalized data value of indicator X

_{ij}in the sample, i = 1, 2, …, n; j = 1, 2, …, m.

_{i}of the sample correlation matrix R, which is symmetric,

_{qp}is the correlation coefficient of the variable F

_{m}with F

_{n}.

_{q}, F

_{p}) is the covariance of F

_{q}and F

_{p}; D(F

_{q}) and D(F

_{p}) are the variances of F

_{q}and F

_{p}, respectively. The raw sample data have been normalized so that there is:

_{qp}can be regarded as the correlation coefficient between F

_{q}and F

_{p}, which also exactly indicates the degree of linear combination of X

_{m}and F

_{m}. λ

_{i}(i = 1, 2, …, m) are all the characteristic roots of the sample correlation matrix R, finding all the characteristic root λ

_{i}(i = 1, 2, …, m) of the sample correlation matrix R and the corresponding normalized eigenvectors U

_{i}(i = 1, 2, …, m),

_{1}, U

_{2}, …, U

_{m}), and the load matrix is derived from the characteristic root and eigenvectors A,

_{ij}, then we have:

^{2}

_{ir}, b

^{2}

_{ig}2 sets of data are required to be spread as much as possible, the degree of dispersion is expressed by the variance of the samples (V

_{1}, V

_{2}, …, V

_{n}), and the total variance is required to be maximum, that is, the request:

#### 2.2. Sparrow Search Algorithm (SSA)

_{max}is the maximum number of iterations; Q and α are random numbers; ${x}_{ij}^{t}$ denotes the position of the ith sparrow in the jth dimension at iteration t; R

_{2}and ST are the warning value and safety threshold. When R

_{2}< ST, it means there is no danger in the surrounding environment and the foraging range can continue to be expanded, and when R

_{2}≥ ST, it means the scout detects danger in the surrounding area and sends an alert to the population, telling the population to move to a safe area as soon as possible.

^{+}= A

^{T}(AA

^{T})

^{−1}; x

_{pj}is the optimal position of the discoverer; x

_{wj}is the worst position in the game.

_{bj}indicates the global best position; β denotes the iteration step size; f

_{i}indicates the current level of adaptation; f

_{g}and f

_{w}denote the global optimal and worst adaptation degrees; K is a random number between [−1, 1]; ε is the smallest constant that prevents the denominator from going to zero.

#### 2.3. Probabilistic Neural Network (PNN)

_{i}to obtain the scalar product Z

_{i}input to the model layer, as shown in Equation (15):

_{ij}is the output value of the jth neuron of the ith class of patterns in the pattern layer; σ is the smoothing factor; d is the dimensionality of the sample space data; and Z

_{ij}is the jth center of the ith class of samples.

_{i}is the output of class i, (i = 1, 2, …, n); n is the total number of training sample patterns; L denotes the number of neurons in class i.

## 3. Dataset Preparations

#### 3.1. Selection of Rockburst Prediction Indicators

_{θ}) is selected as the rockburst prediction evaluation indicators.

_{t}) and uniaxial compressive strength (σ

_{c}) are rockburst prediction evaluation indicators.

_{et}) is selected as the rockburst prediction indicators. A number of rockburst cases have shown that the occurrence of rockbursts is closely related to the brittleness of the rock, and the brittleness coefficient of the rock is often used as a rockburst criterion. The stress coefficient is also commonly used as a rockburst criterion; therefore, the brittleness index (σ

_{c/}σ

_{t}) and the stress coefficient (σ

_{θ/}σ

_{c}) are rockburst prediction evaluation indicators.

_{θ}, σ

_{t}, σ

_{c}, σ

_{c/}σ

_{t}, σ

_{θ/}σ

_{c}, W

_{et}) were selected as the rockburst prediction indicators in this paper.

#### 3.2. Sample Library of Rockburst Case Data

## 4. Implementation Process of FA-SSA-PNN Model

#### 4.1. Model Construction Steps

_{1}, CPI

_{2}, CPI

_{3}.

#### 4.2. Test of Applicability of Factor Analysis

#### 4.3. Data Processing

_{θ}and σ

_{c}, σ

_{θ}and σ

_{t}, σ

_{θ}and σ

_{θ}/σ

_{c}, σ

_{c}and σ

_{t}, σ

_{c}and W

_{et}, and σ

_{t}and σ

_{c}/σ

_{t}were all greater than 0.5, indicating that the rockburst prediction evaluation indicators were significantly correlated with each other and the sample data were suitable for factor analysis.

_{1}is significantly positively correlated with the rockburst prediction evaluation indicators σ

_{θ}, σ

_{c}, σ

_{t}, σ

_{θ}/σ

_{c}, indicating that the principal factor F

_{1}concentrates on the maximum tangential stress, compressive strength, compressive strength, and the influence of the stress coefficient on the prediction results of rockburst. The principal factor F

_{2}is only positively correlated with the indicator σ

_{c}/σ

_{t}, indicating that the main factor F

_{2}combines the information of the indicators of the brittleness index, which can be referred to as the brittleness factor. The main factor F

_{3}is positively correlated with the indicator W

_{et}only and can be referred to as the energy factor.

_{1}, Y

_{2}, Y

_{3}and the six rockburst prediction evaluation indicators, as follows (${x}_{i}^{*}$ is the standardized data value of ${x}_{i}$).

_{1}, CPI

_{2}, CPI

_{3}.

#### 4.4. Datasets Segmentation

#### 4.5. Model Parameter Setting and Implementation

_{θ}, σ

_{t}, σ

_{c}, σ

_{c}/σ

_{t}, σ

_{θ}/σ

_{c}, W

_{et}) as the input vectors of the model. The FA-SAA-PNN rockburst prediction model developed in this paper used factor analysis to preprocess the original rockburst prediction evaluation indicators, and the comprehensive rockburst prediction indicators CPI

_{1}, CPI

_{2}, CPI

_{3}obtained after factor analysis were used as the prediction input vectors of the model. The selection of the smoothing factor is the key to the performance of PNN networks, and when the value of the smoothing factor is too small, it tends to cause the network to be overfitted and in essence a nearest neighbor classifier; when the value of the smoothing factor is too large, the details cannot be fully distinguished so close to a linear classifier [39]. This paper makes use of the good global search ability of the SSA algorithm to optimize the smooth factor of PNN neural network. The algorithm has the advantages of being rapid and efficient when optimizing for a single objective, as well as good merit-seeking ability, which solves the problem of selecting the optimal smoothing factor and improves the accuracy of the prediction model.

## 5. Model Performance Evaluation and Comparison

_{1}value (the summed average of precision and recall), macro-average F

_{1}value (the arithmetic mean of F

_{1}for each category), and accuracy rate are used as the evaluation indicators of the models in this paper. F

_{1}value and macro-average F

_{1}value reflect the classification performance of the models for different rockburst intensity levels, and accuracy rate reflects the overall classification performance of the models.

- (1)
- The FA-SSA-PNN model does not improve the F
_{1}value of the primary rockburst compared to the PNN model; the F_{1}value for Level 2 rockburst is increased by 50% (from 50% to 100%); the F_{1}value for Level 3 rockburst is increased by 25.6% (from 66.7% to 92.3%); and the F_{1}value for Level 4 rockburst is increased by 20% (from 80% to 100%). - (2)
- Compared with the original PNN model, the macro-average F
_{1}value reflecting the classification performance of the model for different rockburst intensity level increased by 18.9% (from 69.2% to 88.1%) after the introduction of FA dimensionality reduction, and the macro-average F_{1}value improved but remained low, and then, after the optimization of the PNN neural network by the SSA algorithm, the macro-average F_{1}value increased by another 5% (from 88.1% to 93.1%), and the macro-average F_{1}values of the FA-SSA-PNN model were significantly higher than those of the other five rockburst prediction models. - (3)
- The accuracy of the FA-PNN model after the introduction of FA improved by 13.3% (from 66.7% to 80%) compared with the original PNN model, and then, after the optimization of the PNN neural network by the SSA algorithm, the accuracy of the model improved by another 13.3% (from 80% to 93.3%), and the prediction accuracy of the FA-SSA-PNNN model was significantly higher than that of the other models, verifying the advantages and disadvantages of the FA-SSA-PNN rockburst intensity level prediction model.

## 6. Conclusions

_{1}value, macro-averaged F

_{1}value, and accuracy rate are introduced as the evaluation indexes of rockburst prediction model classification performance. This study proposes a new method for predicting the intensity level of rockbursts, which provides better guidance for the problem of predicting rockbursts in deep underground rock projects and can provide a reference for other geological hazard prediction problems similar to rockburst hazards, with the following main conclusions:

- (1)
- The maximum tangentialstress of surrounding rock (σ
_{θ}), uniaxial tensile strength (σ_{t}), uniaxial compressive strength (σ_{c}), brittleness index (σ_{c/}σ_{t}), stress coefficient (σ_{θ/}σ_{c}), and elastic energy index (W_{et}) of surrounding rock are selected to form a rockburst prediction index system. The characteristic information of the original rockburst prediction indexes was compressed and extracted by the factor analysis method, and three comprehensive rockburst prediction indexes, CPI_{1},CPI_{2}, and CPI_{3}, were obtained. The introduction of factor analysis into the rockburst intensity level prediction eliminates the correlation between indicators and solves the problem of overlapping information of indicators, so that the comprehensive prediction index of rockburst after dimensionality reduction has a broader mathematical expression of Gaussian function in the PNN model. - (2)
- Fifteen sets of rockburst case data were sampled as test data, and the prediction results of the FA-PNN model were analyzed and compared with those of the original PNN model. It was found that the macro-average F
_{1}value and accuracy of the FA-PNN model were improved, with the macro-average F_{1}value reaching 88.1% (from 69.6% to 88.1%) and the accuracy rate reaching 80% (from 66.7% to 80%). - (3)
- The SSA algorithm was used to select the smoothing factors in PNN to avoid the subjectivity and contingency of the existence of artificial preset smoothing factors. The comparison between the prediction results of FA-SSA-PNN rockburst prediction model and those of FA-PNN rockburst prediction model shows that, after the introduction of SSA algorithm, the accuracy of FA-SSA-PNN rockburst prediction model significantly improved, reaching 93.3% (increased from 80% to 93.3%), and the macro-average F
_{1}value is 93.1% (increased from 88.1% to 93.1%). Moreover, the SSA algorithm has good optimization ability and can complete the optimization of smoothing factors in a few seconds. It greatly reduces the operation time of the model and improves the prediction efficiency of the model. - (4)
- The prediction results of the FA-SSA-PNN model were compared and analyzed with those of the FA-PNN model, PNN model, RF model, SVM model, and ANN model, and the results showed that the macro-averaged F
_{1}values and the prediction accuracy of the FA-SSA-PNN model were significantly higher than those of the other five models, which verified the feasibility and effectiveness of the FA-SSA-PNN rockburst prediction model.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Violin diagram of rockburst prediction indicators: (

**a**) σ

_{θ}distribution; (

**b**) σ

_{c}distribution; (

**c**) σ

_{t}distribution; (

**d**) σ

_{θ}/σ

_{c}distribution; (

**e**) σ

_{c}/σ

_{t}distribution; (

**f**) W

_{et}distribution.

**Figure 6.**Rockburst prediction model test results: (

**a**) PNN model; (

**b**) FA-PNN model; (

**c**) FA-SSA-PNN model; (

**d**) ANN model; (

**e**) SVM model; (

**f**) RF model.

Serial Number | Primitive Rockburst Prediction Indicators | Comprehensive Rockburst Prediction Indicators | Actual Level | |||||||
---|---|---|---|---|---|---|---|---|---|---|

σ_{θ} | σ_{c} | σ_{t} | σ_{θ}/σ_{c} | σ_{c}/σ_{t} | W_{et} | CPI_{1} | CPI_{2} | CPI_{3} | ||

1 | 18.8 | 178 | 5.7 | 0.11 | 31.23 | 7.4 | 0.549 | 0.456 | 0.964 | I |

2 | 96.41 | 18.32 | 0.38 | 0.19 | 47.93 | 1.87 | 0.411 | 0.657 | 0.933 | I |

3 | 15.2 | 53.8 | 5.56 | 0.283 | 9.68 | 1.92 | 0.562 | 0.314 | 1.001 | I |

… | … | … | … | … | … | … | … | … | … | … |

61 | 48 | 120 | 1.5 | 0.4 | 80 | 5.8 | 0.606 | 0.998 | 0.746 | III |

62 | 48.75 | 180 | 8.3 | 0.27 | 21.69 | 5 | 0.634 | 0.320 | 0.768 | III |

63 | 105 | 115 | 1.5 | 0.55 | 76.67 | 5.7 | 0.538 | 0.895 | 0.486 | III |

64 | 33.94 | 117.48 | 4.23 | 0.29 | 27.77 | 2.37 | 0.644 | 0.497 | 0.892 | II |

65 | 14.96 | 115 | 5 | 0.1 | 23 | 5.7 | 0.498 | 0.403 | 1.059 | I |

66 | 157.3 | 91.23 | 6.92 | 0.58 | 13.18 | 6.27 | 0.311 | 0.088 | 0.317 | IV |

67 | 91.43 | 157.63 | 11.96 | 0.58 | 13.18 | 6.27 | 0.559 | 0.108 | 0.397 | IV |

68 | 13.9 | 124 | 4.22 | 0.112 | 29.4 | 2.04 | 0.667 | 0.538 | 1.086 | I |

69 | 38.2 | 71.4 | 3.4 | 0.53 | 21 | 3.6 | 0.539 | 0.423 | 0.718 | III |

70 | 39.4 | 69.2 | 2.7 | 0.57 | 25.6 | 3.8 | 0.537 | 0.478 | 0.686 | III |

71 | 52 | 175 | 7 | 0.3 | 25 | 5.2 | 0.615 | 0.368 | 0.744 | III |

72 | 105 | 304.21 | 20.9 | 0.35 | 14.56 | 10.57 | 0.639 | −0.094 | 0.331 | IV |

73 | 35.82 | 127.93 | 4.43 | 0.28 | 28.9 | 3.67 | 0.608 | 0.485 | 0.872 | II |

74 | 69.8 | 198 | 22.4 | 0.35 | 8.84 | 4.68 | 0.763 | −0.062 | 0.570 | II |

75 | 55.4 | 176 | 7.3 | 0.31 | 24.11 | 9.3 | 0.452 | 0.290 | 0.683 | III |

Kaiser-Meyer-Olkin test | KMO value | 0.641 |

Bartlett spherical test | chi-squared test value | 187.075 |

Sig | 0.000 |

Test Method | Range of Values | Factor Analysis Applicability |
---|---|---|

Kaiser-Meyer-Olkin test | >0.9 | Perfect suitable |

0.8~0.9 | Great suitable | |

0.7~0.8 | Relatively suitable | |

0.6~0.7 | Suitable | |

0.5~0.6 | Barely suitable | |

<0.5 | Not suitable | |

Bartlett spherical test | sig ≤ 0.01 | Suitable |

Indicators | σ_{θ} | σ_{c} | σ_{t} | σ_{θ}/σ_{c} | σ_{c}/σ_{t} | W_{et} |
---|---|---|---|---|---|---|

σ_{θ} | 1.00 | 0.411 | 0.449 | 0.410 | −0.114 | 0.541 |

σ_{c} | 0.411 | 1.00 | 0.677 | −0.089 | −0.153 | 0.643 |

σ_{t} | 0.449 | 0.677 | 1.00 | 0.142 | −0.583 | 0.588 |

σ_{θ}/σ_{c} | 0.410 | −0.089 | 0.142 | 1.00 | −0.220 | 0.276 |

σ_{c}/σ_{t} | −0.114 | −0.153 | −0.583 | −0.220 | 1.00 | −0.174 |

W_{et} | 0.541 | 0.643 | 0.588 | 0.240 | −0.174 | 1.00 |

Number of original variables | 5 | 7 | 8 | 9 | 11 |

Number of principal factors | 2 | 3 | 4 | 5 | 6 |

Principal Factor | Load Sum of Squares | Sum of Squared Rotating Loads | ||||
---|---|---|---|---|---|---|

Eigen Value | Variance Contribution | Cumulative Variance Contribution | Eigen Value | Variance Contribution | Cumulative Variance Contribution | |

F_{1} | 2.897 | 48.282% | 48.282% | 2.410 | 40.160% | 40.160% |

F_{2} | 1.186 | 19.769% | 68.051% | 1.367 | 22.785% | 62.945% |

F_{3} | 1.049 | 17.486% | 85.538% | 1.356 | 22.593% | 85.538% |

Indicators | Factor Loading before Rotation | Factor Loadings after Rotation | ||||
---|---|---|---|---|---|---|

F_{1} | F_{2} | F_{3} | F_{1} | F_{2} | F_{3} | |

σ_{θ} | 0.874 | −0.150 | −0.315 | 0.907 | −0.130 | −0.173 |

σ_{c} | 0.823 | −0.100 | 0.281 | 0833 | −0.072 | 0.260 |

σ_{t} | 0.769 | −0.512 | 0.126 | 0.704 | −0.622 | 0.570 |

σ_{θ}/σ_{c} | 0.712 | 0.278 | 0.423 | 0.628 | 0.460 | −0.606 |

σ_{c}/σ_{t} | 0.344 | 0.875 | 0.123 | −0.061 | 0.965 | −0.118 |

W_{et} | 0.489 | −0.221 | 0.813 | −0.029 | −0.158 | 0.934 |

Indicators | Factor Score Coefficients | ||
---|---|---|---|

F_{1} | F_{2} | F_{3} | |

σ_{θ} | 0.243 | 0.227 | 0.409 |

σ_{c} | 0.444 | 0.061 | −0.266 |

σ_{t} | 0.212 | −0.376 | −0.103 |

σ_{θ}/σ_{c} | −0.166 | −0.050 | 0.736 |

σ_{c}/σ_{t} | 0.185 | 0.793 | 0.006 |

W_{et} | 0.363 | 0.136 | 0.095 |

Serial Number | Parameters | Parameter Values |
---|---|---|

1 | Number of neurons in the input layer | 3 |

2 | Number of neurons in the pattern layer | 60 |

3 | Number of neurons in summation layer | 4 |

4 | Number of neurons in the output layer | 4 |

5 | Mode layer activation function | Gauss function |

6 | Optimization parameters | Spread Value |

7 | Number of populations of SSA | 100 |

8 | Maximum number of iterations of SSA | 20 |

9 | Proportion of discoverers | 70% |

10 | Scout’s ratio | 20% |

11 | Early warning values | 0.6 |

Evaluation Indicators | Intensity Level | PNN | FA-PNN | SSA-FA-PNN | ANN | SVM | RF |
---|---|---|---|---|---|---|---|

Accuracy rate | I | 0.667 | 0.500 | 0.667 | 0.500 | 1.000 | 0.667 |

II | 0.400 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |

III | 0.800 | 1.000 | 1.000 | 0.857 | 0.778 | 0.875 | |

IV | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.667 | |

Recall Rate | I | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

II | 0.667 | 0.667 | 1.000 | 0.667 | 0.667 | 0.333 | |

III | 0.571 | 0.857 | 0.857 | 0.857 | 1.000 | 1.000 | |

IV | 0.667 | 1.000 | 1.00 | 0.800 | 0.667 | 0.667 | |

F_{1} value | I | 0.800 | 0.667 | 0.80 | 0.667 | 1.000 | 0.800 |

II | 0.500 | 0.800 | 1.00 | 0.800 | 0.800 | 0.500 | |

III | 0.667 | 0.923 | 0.923 | 0.857 | 0.875 | 0.933 | |

IV | 0.80 | 1.000 | 1.00 | 0.667 | 0.800 | 0.667 | |

Macro average F_{1} value | - | 0.692 | 0.881 | 0.931 | 0.781 | 0.86.9 | 0.725 |

Accuracy | - | 0.667 | 0.800 | 0.933 | 0.800 | 0.800 | 0.867 |

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## Share and Cite

**MDPI and ACS Style**

Xu, G.; Li, K.; Li, M.; Qin, Q.; Yue, R.
Rockburst Intensity Level Prediction Method Based on FA-SSA-PNN Model. *Energies* **2022**, *15*, 5016.
https://doi.org/10.3390/en15145016

**AMA Style**

Xu G, Li K, Li M, Qin Q, Yue R.
Rockburst Intensity Level Prediction Method Based on FA-SSA-PNN Model. *Energies*. 2022; 15(14):5016.
https://doi.org/10.3390/en15145016

**Chicago/Turabian Style**

Xu, Gang, Kegang Li, Mingliang Li, Qingci Qin, and Rui Yue.
2022. "Rockburst Intensity Level Prediction Method Based on FA-SSA-PNN Model" *Energies* 15, no. 14: 5016.
https://doi.org/10.3390/en15145016