As a complex dynamic disaster in coal mining, rockburst is very hard to be accurately predicted (Qiao et al., 2021; Zhu et al., 2022). The occurrence of rockbursts is influenced by various factors and is characterized by a regional distribution (Wang et al., 2021a; Cao et al., 2021; Chen et al., 2021). As the depth of mining deepens, the number and frequency of rockbursts in mines increase (Yu et al., 2021; Xue et al., 2022). There are distinctive rockburst modes under different conditions of mining areas, mines, coal seams, structures, and stresses. Traditional linear data analysis is not accurate enough under complex mining conditions (He et al., 2020; Zhang and Jiang, 2020; Lin et al., 2022). Based on the non-linear relationship between rockburst hazard and its influencing factors, the probability prediction value of unit hazard is determined (Wu et al., 2021a; Chen et al., 2021; Yang et al., 2021; Yang and Zhang, 2021). According to the magnitude of the risk probability value of each cell, the engineering area is divided into four classes, and the regional and quantitative rockburst prediction can be significantly improved by using the multi-factor pattern recognition method.

In the prediction of mine rockburst hazards, machine learning methods have been proposed to predict rockburst hazards with good results (Ullah et al., 2022; Wojtecki et al., 2022; Xiao et al., 2022). Machine learning is a complex and cross-cutting discipline. In the prediction of rockburst hazards in mines, data from multiple sources are analyzed, and then machine learning algorithms are used to continuously learn from previous rockburst events and train computer models. The study of “neural network + machine learning” artificial intelligence prediction techniques allows monitoring and predicting the likelihood of rockburst hazards in coal mines (Wang et al., 2021b; Ke et al., 2021; Zhang et al., 2021). In order to accurately predict rockburst hazards under complex conditions, a rockburst hazard prediction model based on the Gaussian process for binary classification (GPC) was proposed (Hui and Zhang, 2020; Davis et al., 2021).

For the rockburst situation in Jixian Coal Mine, a GPC-based rockburst hazard prediction, prevention, and control technology system was established based on theoretical analysis (Iwata and Tanaka, 2022), and the intrinsic relationship between multiple influencing factors and rockburst was determined using a multi-factor pattern identification method. By dividing the engineering area into prediction units and determining the pattern identification criteria and unit hazard probability values (Gladyr et al., 2021), the rockburst hazard area of the on-site engineering area was classified, and corresponding management measures were proposed (Wu et al., 2021b).

Statistically, the Gaussian process is a stochastic process: the distribution of any finite variable set is a Gaussian distribution. In other words, for any integer n ≥ 1 and any family of random variables X, the joint probability distribution of the corresponding process state f(X) at time t obeys the n-dimensional Gaussian distribution. All statistical characteristics of the Gaussian process are determined by its mean and covariance function. In the field of machine learning, the Gaussian process refers to a machine learning method based on the Gaussian stochastic process and Bayesian learning theory.

The Gaussian process for binary classification (GPC) model is a kind of classification model based on the machine learning principle of the Gaussian process. In the GPC model, let an input x correspond to the output value of the binary classification mark y, y ∈ {−1,1}, and the observation data set is D = { ( x i , y i ) | i = 1 , … , m } . The GPC model aims to predict the classification y* corresponding to the new test input x* (Ahmad et al., 2022).

For a given x, the p ( y | x ) distribution is the Bernoulli distribution, and the probability of y = 1 is p ( y = 1 | x ) = Φ ( f ( x ) ) , where f(x) is the potential function, and Φ(•) is the cumulative probability density function of the standard Gaussian distribution. Generally, the sigmoid function is taken as Φ(z) = 1/(1 + e^−z). The function of the sigmoid function is to convert the f( x ) constrained by intervals into the value of [0,1], so as to ensure that the probability value ranges in [0,1]. For simplicity, let f _i = f( x _i), f = [f ₁,…,f _m]^T, y = [y₁,…,y_m]^T, X = [x₁,…,x_m]^T.

For a given potential function, the observed value is an independent Bernoulli distribution variable, whose likelihood function is p ( y | f ) = ∏ i = 1 m p ( y i | f i ) = ∏ i = 1 m Φ ( y i f i ) . (1)

The prior distribution of potential functions is b e i p ( f | X , θ ) = N ( 0 , K ) , (2) where K is a covariance matrix of order m × m, K _ij = k(x _i ,x _j ,θ), k() is a positive definite covariance function related to θ, and θ is a hyper-function.

The covariance function of the Gaussian process model needs to be satisfied: a non-negative positive definite covariance matrix can be generated for any point set. The commonly used covariance function is the squared exponential function, namely, k y ( x p , x q ) = σ f 2 exp ( − 1 2 l 2 ( x p − x q ) 2 ) , (3) where the hyper-function θ = { σ f ,l}; the optimal hyper-parameters can be estimated by the maximum likelihood method, as described in the literature.

According to Bayes’ rule, after obtaining the actual observation value, a posterior distribution of the potential function f is obtained as follows: p ( f | D , θ ) = p ( y | f ) p ( f | X , θ ) p ( D | θ ) = N ( 0 , K ) p { D | θ } ∏ i = 1 m Φ ( y i f i ) . (4)

The aforementioned equation is the learning process of GPC, and the following is the prediction process of GPC. The conditional probability of the potential function value f _* corresponding to x _* is p ( f ∗ | D , θ , x ∗ ) = ∫ p ( f ∗ | f , X , θ , x ∗ ) p ( f | D , θ ) d f . (5)

The prediction probability of y _* is p ( y ∗ | D , θ , x ∗ ) = ∫ p ( y ∗ | f ∗ ) p ( f ∗ | D , θ , x ∗ ) d f ∗ . (6)

When the predicted probability value of Y_* is greater than 0.5, then y _* = 1; otherwise, y _* = −l. Eqs 5 and 6 have no analytical solutions. Approximate solutions can be obtained by using Laplace’s method and expectation propagation method (Villacampa–Calvo and Hernández–Lobato, 2020; Chakir et al., 2022). Let m and A be the mean and variance of the approximate solutions, respectively, and the approximate Gaussian distribution of the posterior distribution of the potential function f is p ( f | D , θ ) ≈ q ( f | D , θ ) = N ( m , A ) . (7)

Similarly, the posterior distribution of f _* can be set as an approximate Gaussian distribution: q ( f ∗ | D , θ , x ∗ ) = N ( μ ∗ , σ ∗ 2 ) . (8)

The mean and variance are μ ∗ = K ∗ T K − 1 m , (9a) σ ∗ 2 = k ( x ∗ , x ∗ ) − k ∗ T ( K − 1 − K − 1 A K − 1 ) k ∗ , (9b) where k _* = [k(x₁,x_*),…,k(x_m,x_*)]^T represents the prior covariance vector between x _* and training input X . x _* belongs to classification 1 of the prediction probability: q ( y ∗ = 1 | D , θ , x ∗ ) = Φ ( μ ∗ 1 + σ ∗ 2 ) . (10)

By using the probability prediction method with multi-factor pattern recognition, the sub-unit probability prediction of coal seam hazard was realized, and the results were classified in line with the regional prediction grade. In the actual mining, when excavation roadways or working faces enter different prediction units, the potential rockburst hazard of the location can be obtained in advance, and corresponding preventive measures can be taken in advance. According to the hazard probability value of the prediction unit, when the conditions were suitable, regional measures were first chosen to relieve the hazard, or the corresponding local measures were taken to reduce the hazard. Regional prediction of rockburst provides a scientific basis for taking prevention and control measures against the rockburst, so as to ensure safe production in coal mines. For different hazard grades, the corresponding prevention and control measures were adopted, as shown in Table 2.

For the problem of rockburst hazard area prediction under complex mining conditions, traditional linear data analysis is not an ideal solution. In order to improve the precision and accuracy of prediction results, this article studies the non-linear mapping relationship between rockburst hazard and its influencing factors through machine learning of rockburst hazard training samples and draws the following conclusion based on the principle of the binary Gaussian process and the main influencing factors of rockburst:

(1) Aiming at the prediction of rockburst hazard under complex conditions, a multi-factor pattern recognition method of rockburst hazard prediction was proposed based on GPC. By learning the training samples, the non-linear mapping relationship between rockburst hazard and its influencing factors was established, and the probability prediction value of the unit hazard was determined.