Identification Method of Optimal Copula Correlation Characteristic for Geological Parameters of Roof Structure

Abstract

Limited by the actual investigation of coal mine engineering, the measured data obtained are often based on small sample characteristics. How to probabilistically de-integrate the prior information to obtain meaningful statistical values has received increasing attention from geotechnical engineers. In this study, an optimal copula function identification method for multidimensional geotechnical structures of coal mine roofs under the Bayesian approach is proposed. Firstly, the characterization method of multidimensional roof parameter correlation structures is proposed based on copula theory, and 167 sets of measured data from 24 coal mines at home and abroad are collected to study the measured identification results using the Bayesian method. Secondly, Monte Carlo simulation is utilized to compare the correct recognition rates of the commonly used AIC criterion and the Bayesian approach under different correlation structures. Finally, the influencing factors affecting the successful recognition rate of the Bayesian approach are analyzed. The results show that compared with the traditional AIC criterion, the Bayesian approach has more marked advantages in correctly recognizing the multidimensional parameter structures of roofs, and the number of measured samples, the strength of correlation coefficients, and the prior information have a major effect on the correct recognition rate of the optimal copula function under different real copula functions. In addition, the commonly used Gaussian copula has a better characterization effect in characterizing the multidimensional parameter correlation structure of the coal mine roofs, which can be prioritized to be used as a larger prior probability function in the evaluation process.

1. Introduction

The strength parameters of the coal mine roof geotechnical body are key factors in the study of the stability of coal mine roofs, and the geotechnical strength parameter directly affects the safety in the process of roof modeling in the process of assessing the reliability of the roof and determining the roof damage criterion [1,2,3]. The borehole parameters during excavation can likewise have a significant impact on the mechanical behavior of geotechnical structures [4,5]. In roof stability analysis, each strength parameter is usually regarded as a deterministic variable [6]. However, in practical engineering, there may be correlations between parameters. For example, a large amount of data indicates a statistically negative correlation between shear strength parameters [7,8]. Therefore, to accurately assess the effect of parameter correlation on geotechnical structures, it is necessary to establish the joint probability density distribution function among parameters [9]. Meanwhile, with the influence of restricted exploration conditions in coal mine engineering, the statistically obtained data are characterized by small samples [10], and thus, it is impossible to further establish the joint probability distribution function of the parameters.

Copula theory, which has been rapidly developed in hydrological and geotechnical engineering in recent years, provides an effective way to solve the parameter correlation problem [11,12,13,14]. The basic content of Copula theory can be summarized as follows: identify the marginal distribution of each variable as well as pick a copula function to link the marginal distribution functions together. Therefore, the key to constructing a copula function is to select the function that can best characterize the correlation structure among parameters from a large number of copula functions. Commonly used identification methods include the AIC criterion [15], the BIC criterion [16], and the least square Euclidean distance method [17]. However, the accuracy of the identification results of the above methods depends on large-sample data, and the identification results for small-sample of data are unstable. In addition, the relevant parameters of candidate copula functions need to be obtained in the calculation process, and the prior information of the actual engineering cannot be taken into account. It is well known that the value ranges of the relevant parameters of different copula functions are often different, and the calculation also involves the iteration of the dual integration, which has a high computational cost. Generally speaking, the characterization of geotechnical parameters in coal mine engineering relies on a priori information such as engineering experience and preliminary exploration tests [18,19]. Due to the complexity of the underground space and the spatial variability of the soil, researchers are confronted with the challenge of probabilistically de-integrating the prior information from small-sample data, often failing to generate meaningful statistics to probabilistically characterize the roof structure [20]. In order to quantify engineering experience in prior information, Bayesian theory can be combined with small-sample data to transform fuzzy engineering judgments into prior information for probabilistic characterization.

In recent years, Bayesian theory has been continuously applied in engineering practice to characterize the probabilistic features of geotechnical parameters. Wang et al. [21] introduced random field theory by combining Bayesian theory with a cone penetration test to provide an approach for evaluating probabilistic characterization of effective friction angle in sandy soils. In addition, they probabilistically characterized the undrained Young’s modulus of clays using standard penetration tests based on Markov chain Monte Carlo simulation (MCMCS) [22]. On this basis, Cao and Wang [23] identified the statistically most probable number, thickness, and properties of mean soil layers by the Bayesian method. Cao and Wang [24] characterized the probabilistic characteristics of the undrained shear strength of clays using a limited number of mobility index test data using the Bayesian method for small- and medium-scale geotechnical projects, which provides a reference for the determination of eigenvalues in other design specifications. Zhang et al. [25] modeled the uncertainty in c and φ based on the copula approach. The best-fit copula is recognized using the Bayesian method and verified in conjunction with the Xiaolangdi project.

In this study, a multi-dimensional parameter modeling method of coal mine roofs based on copula theory is proposed, and the structural characterization method of multi-dimensional parameter correlation is given based on 167 sets of measured roof data. Bayesian theory is introduced to identify the optimal copula functions of the measured data, and Monte Carlo simulation (MCS) is utilized to identify the best-fit copula functions under diverse real copula. Moreover, the identified results are compared with those of the AIC criterion. Finally, the discussion analyzes the different factors that affect the Bayesian recognition accuracy.

2. A Multi-Dimensional Parameter Model Construction Method for Roofs Based on Copula Theory

2.1. Copula Theory

The copula theory [11], which has received widespread applications in geotechnical engineering in recent years, provides a favorable approach to solving the correlation problem of geotechnical parameters. The key idea of copula theory is that any multivariate joint distribution can be decoupled into the marginal distributions of its parameters with a copula function that can link them together. According to copula theory, when F₁(x₁), F₂(x₂), ···, F_n(x_n) are the marginal distribution functions of the desired n-dimensional parameters, respectively, then a copula function C must exist which can be expressed as an n-dimensional joint distribution function F(x₁, x_2, ···, x_n):

$$F\left(x_1,x_2,\cdots ,x_n\right)=C\left(F_1\left(x_1\right),F_2\left(x_2\right),\cdots ,F_n\left(x_n\right)\right)$$

(1)

Its joint probability density function f(x₁, x_2, ···, x_n) can be expressed as:

$$f\left(x_1,x_2,\cdots ,x_n\right)=f_1\left(x_1\right)f_2\left(x_2\right)\cdots f_n\left(x_n\right)D\left(F_1\left(x_1\right),F_2\left(x_2\right),\cdots ,F_n\left(x_n\right);\theta \right)$$

(2)

where f_n(x_n) denotes the probability density function of the variable x_n; D(·) denotes the density function of C; and θ is the correlation parameter of the copula function.

2.2. Coal Mine Roof Strength Parameters

During the construction of underground coal mine projects, it is often difficult to obtain timely and adequate measured data to assess the stability of the coal mine roof. Therefore, preliminary exploration sampling and laboratory testing of coal mine roof structures are usually conducted to provide risk assessment results. From a large amount of field exploration data and laboratory testing experiments, in addition to the shear strength parameters of soils obtained using common test methods such as the triaxial compression test, in situ testing test, direct shear test, etc., parameters such as the modulus of elasticity, compression modulus, Young’s modulus, etc., are also regarded as the most basic and common indicators of the strength of soils. The statistical eigenvalues of these soil strength parameters are often critical in governing the physical and mechanical behavior of geotechnical structures. Therefore, it is significant to study the correlation of roof structures with multiple parameters as variables simultaneously in coal mine engineering. In this study, the elastic modulus (E) and the shear strength parameters (c and φ) are taken as the study variables, and the roof multidimensional parameter correlation structure is constructed based on copula theory.

2.3. Multidimensional Correlation Structure Characterization Methods

As mentioned above, the steps of constructing the copula function are mainly divided into two steps: firstly, determining the marginal distribution function of each variable, i.e., (E, c, φ), and then determining a copula function that can characterize the correlation between the variables. In determining the copula function, it is required to first determine the correlation parameter θ. Since the range of values of the correlation parameter varies from different copula functions, the same value of θ represents different degrees of correlation in different copula functions. Therefore, it is necessary to use a unified metric to represent the correlation parameter among different copula functions. The Kendall rank correlation coefficient τ, as a metric that can describe the consistency of the variable’s trend, represents different degrees of correlation by calculating the rank of the variable’s data. In this section, 167 sets of measured data containing E, c, and φ are collected from the roofs of 24 coal mines at home and abroad in order to construct the multidimensional correlation structure of the roofs of the coal mines, and the information of the roofs is shown in

Table 1. Basic physical parameters of frozen soil in the vertical and horizontal directions.

Name of Coal Mine	Roof Geological Condition	Sample Data Size
Pingdingshan No. 11 coal mine [26]	Compound roof	2
Shangwan mine [27]	Cracked roof under hallow mining conditions	3
Jinchuan nickel mine [28]	Nickel mine roof	5
Xinjulong coal mine [29]	Deep coal mine roadway roof located in fault zone	5
Gaohe coal mine [30]	Paste false roof	9
Great Wall #3 coal mine [31]	Thin roof	2
He caogou coal mine [32]	Longwall coal mining	5
Sanshandao gold mine [33]	Undersea roof	3
No. 2130 coal mine [34]	Multi-layer gangue roof	11
Boertai coal mine [35]	Thick and hard roof	34
Sheilei coal mine [36]	Hard roof	4
Yuwu coal mine [37]	Compound roof	5
Gaokeng coal mine [38]	Compound roof	4
Lishi coal mine [39]	Extra-thick compound mudstone roof	8
Dongtan coal mine [40]	Thick and hard roof	7
Sangshuping coal mine [41]	Hard roof	7
Zhangshuanglou coal mine [42]	Hard roof	5
Yungaishan coal mine [43]	Hard roof	9
Zhuxianzhuang coal mine [44]	Hard thick roof	7
a coal mine in Guizhou [45]	Compound roof	12
Pingdingshan No. 2 coal mine [46]	Hard roof	1
Linsheng coal mine [47]	Hard roof	10
Sheilei coal mine [48]	Compound roof	4
Dianping coal mine [49]	Hard and thick roof	5

.

To visualize the correlation between the multidimensional parameters, three bivariate graphs are used to represent the correlation structure between the three parameters, and the statistically obtained empirical data are shown in Figure 1. The measured data are transformed into empirical distributions according to rank order, and the three sets of empirical distributions obtained are shown in Figure 2. In Figure 2, it can be clearly visualized that the scatter of the empirical distributions of the variables (E, c) is mainly distributed near the 45° diagonal, and there exists an obvious positive correlation. The uniformly distributed scatter of the variables (c, φ) is mainly located near the 135° diagonal, and there exists an obvious negative correlation. Therefore, in this study, (E, c) and (c, φ) are chosen to be studied as positive and negative correlation structures, respectively.

Based on the above statistical data, the corresponding statistical characterization data of each parameter can be easily calculated. The mean, variance and coefficient of variation of the modulus of elasticity are 14.8 GPa, 292 (GPa)², and 1.16; those of the cohesion are 5.35 MPa, 38.1 (MPa)², and 1.15; and those of the internal friction angle are 31.13°, 22(°)², and 0.15, respectively. It can be recognized that the coefficients of variation for the modulus of elasticity and cohesion are relatively large except for the internal friction angle. The reasons for this are that the roofs collected are widely distributed, the actual geological conditions of each project vary greatly, and the physical and mechanical properties of the soil may also differ greatly. After obtaining the statistical eigenvalues of each parameter, the four common marginal distributions in

Table 2. Four kinds of statistical distribution functions.

Distribution Type	F(x)	f(x)	Mean Value μ, Variance σ2
Normal	$\mathsf{\Phi }(\frac{x-p}{q})$	$\frac{1}{q\sqrt{2\pi }}\times \exp [-\frac{{(x-p)}^2}{2q^2}]$	µ = q σ2 = q2
Lognormal	$\mathsf{\Phi }(\frac{\ln x-p}{q})$	$\frac{1}{q\sqrt{2\pi }}\times \exp [-\frac{{(\ln x-p)}^2}{2q^2}]$	µ = exp(p + 0.5q2) σ2 = [exp(q2) − 1] × exp(2p + q2)
Extreme value type I	$1-\exp \left\{-\exp \left[-\left(x-p\right)/q\right]{\left.\right\}}^{}\right.$	(1/q)exp[(x − p)/q] × exp[−exp(x − p)/q]	µ = p + 0.5772q σ2 = π2q2/6
Gamma	$\frac{1}{\Gamma (p)}\mathsf{{\rm Y}}(p,\frac{x}{q})$	$\frac{1}{\Gamma (p)q^p}{x}^{p-1}{e}^{-x/q}$	µ = pq σ2 = pq2

Note: Φ(·) is the standard normal distribution function; $\mathsf{{\rm Y}}\left(\cdot \right)$ is a low order incomplete gamma function; $\Gamma \left(\cdot \right)$ is a gamma function.

are selected to perform the AIC test on the parameters, and the calculation results are shown in

Table 3. Results of the marginal distribution test for each parameter.

Distribution Type	E	c	φ
Normal	1681.4	1247.7	1142.4
Lognormal	−623.449	−528.2	7068.8
Extreme value type I	1878	1774.1	1178.8
Gamma	16,160	3668	10,232

. In this case, the AIC criterion is defined as the sum of negative two times the logarithmic sum of the values of the copula probability density function at the measured data points and the sum of the number of correlation parameters of the two times Copula function [15]:

$$\mathrm{AIC}=-2{\displaystyle \sum _{i=1}^N\ln D\left(u_{1i},u_{2i};\theta \right)}+2k$$

(3)

where k is the number of copula function parameters; when the copula function is n-dimensional, k = n(n − 1)/2, (u_1i,u_2i) is the uniform distribution value of the measured samples (x_1i,x_2i), and the specific calculation process is as follows:

$$\left\{\begin{array}{c}{u}_{1i}=\frac{\mathrm{rank}\left(x_{1i}\right)}{N+1}\\ u_{2i}=\frac{\mathrm{rank}\left(x_{2i}\right)}{N+1}\end{array}\right.,i=1,2,\cdots ,N$$

(4)

where rank(·) denotes ascending rank.

From the identification results with minimum values in Table 3, E, c and φ obey lognormal, lognormal, normal distributions, respectively.

There are numerous types of copula functions, and in this study, three copula functions that have excellent ability to characterize the positive and negative correlation structure of parameters are selected, i.e., the Gaussian, Plackett, and Frank copula. In addition, the Clayton copula is selected to characterize the structure of the positive correlation between the parameters, and the No. 16 copula to describe the negative correlation structure. The positive and negative correlation coefficients modeled by the above copula functions are [−1, 0] and [0, 1], respectively. The selected copula functions and density functions are listed in

Table 4. Summary of the selected copula functions.

Copula	C(u1,u2; θ)	D(u1,u2; θ)	Range of θ
Gaussian	$\begin{array}{l}C\left(u_1,u_2;\theta \right)={\displaystyle {\int }_{-\infty }^{{\mathsf{\Phi }}^{-1}{(u_1)}_{{}_{}}}{\displaystyle {\int }_{-\infty }^{{\mathsf{\Phi }}^{-1}(u_2)}}}\frac{1}{2\pi \sqrt{1-{\theta }^2}}\\ \times \exp (-\frac{x_1^2-2\theta x_1{x}_2+x_2^2}{2\left(1-{\theta }^2\right)})d{x}_1{x}_2\end{array}$	$\begin{array}{l}D\left(u_1,u_2;\theta \right)=\frac{1}{2\sqrt{1-{\theta }^2}}\exp \left(\frac{{\varsigma }_1{}^2-2\theta {\varsigma }_1{\varsigma }_2+{\varsigma }_2{}^2}{2\left(1-{\theta }^2\right)}\right)\\ {\varsigma }_1={\mathsf{\Phi }}^{-1}\left(u_1\right),{\varsigma }_2={\mathsf{\Phi }}^{-1}\left(u_2\right)\end{array}$	[−1, 1]
Plackett	$\begin{array}{l}C\left(u_1,u_2;\theta \right)=\frac{S-\sqrt{S^2-4u_1{u}_2\theta (\theta -1)}}{2(\theta -1)}\\ S=1+\left(\theta -1\right)(u_1+u_2)\end{array}$	$\begin{array}{l}D\left(u_1,u_2;\theta \right)=\\ \frac{\theta \left[1+\left(\theta -1\right)\left(u_1+u_2-u_1{u}_2\right)\right]}{\left\{{\left[1+\left(\theta -1\right)\left(u_1+u_2\right)\right]}^2{\left.-4u_1{u}_2\theta \left(\theta -1\right)\right\}}^{3/2}\right.}\end{array}$	$\left(0,\infty \right)\backslash \left\{1\right\}$
Frank	$C\left(u_1,u_2;\theta \right)=-\frac{1}{\theta }\ln \left(1+\frac{\left(e^{-\theta u_1}-1\right)(e^{-\theta u_2}-1)}{(e^{-\theta }-1)}\right)$	$D\left(u_1,u_2;\theta \right)=\frac{-\theta \left(e^{-\theta }-1\right)e^{-\theta \left(u_1+u_2\right)}}{{\left[\left(e^{-\theta }-1\right)+\left(e^{-\theta u_1}-1\right)\left(e^{-\theta u_2}-1\right)\right]}^2}$	$\left(-\infty ,\infty \right)\backslash \left\{0\right\}$
Clayton	$C\left(u_1,u_2;\theta \right)={\left(u_1{}^{-\theta }+u_2{}^{-\theta }-1\right)}^{-1/\theta }$	$D\left(u_1,u_2;\theta \right)=\left(1+\theta \right){(u_1{u}_2)}^{-\theta -1}{\left(u_1{}^{-\theta }+u_2{}^{-\theta }-1\right)}^{-2-1/\theta }$	$\left(0,\infty \right)$
No.16	$\begin{array}{l}C\left(u_1,u_2;\theta \right)=\frac{1}{2}\left(S+\sqrt{S^2+4\theta }\right)\\ S=u_1+u_2-1-\theta \left(\frac{1}{u_1}+\frac{1}{u_2}-1\right)\end{array}$	$\begin{array}{l}D\left(u_1,u_2;\theta \right)=\frac{1}{2}\left(1+\frac{\theta }{u_1{}^2}\right)\left(1+\frac{\theta }{u_2{}^2}\right)S^{-1/2}\\ \times {\left\{-S^{-1}\left[u_1+u_2-1-\theta \left(\frac{1}{u_1}+\frac{1}{u_2}-1\right)\right]\right.}^2\left.+1\right\},\\ S={\left[u_1+u_2-1-\theta \left(\frac{1}{u_1}+\frac{1}{u_2}-1\right)\right]}^2+4\theta \end{array}$	$[0,\infty )$

.

3. Identification of Multidimensional Roof Structures Based on Bayesian Approach

3.1. Bayesian Theory

When identifying the parameter optimal correlation structure using Bayesian theory, the first step is to obtain the set of candidate copula functions characterizing the multidimensional correlation structure, i.e., S = {s_i, i = 1, 2, 3, ···, N}. N indicates the maximum number of candidate copula functions, and s denotes the candidate copula function. Subsequently, the probability of occurrence P_r(s_i|C) of the selected copula function is calculated for the provided measured data C = {(x_i,y_i), i = 1, 2, 3, ···, n}. According to the calculation results, the copula function with the maximum occurrence probability P_r(s_i|C) is calculated as the optimal copula function. On the basis of Bayesian theory, P_r(s_i|C) can be expressed by the following equations [21,22,23]:

$$P_r\left(s_i|C\right)=\frac{P_r\left(C|s_i\right)P_r\left(s_i\right)}{P_r\left(C\right)}$$

(5)

where P_r(C) = ${\displaystyle {\sum }_{i=1}^N{P}_r\left(C|s_i\right)P_r\left(s_i\right)}$, and P_r(s_i) denotes the prior probability of the copula function s, i.e., at the initial stage of the computation, one has the information about the probability of the possible occurrence of the copula function under the condition that it satisfies ${\displaystyle {\sum }_{i=1}^N{P}_r\left(s_i\right)=1}$. When the prior information of the copula function is not available in engineering, it is generally assumed that that the candidate copula functions have equal prior probabilities, i.e., P_r(s_i) = 1/N. Hence, to compute the maximum probability of occurrence P_r(s_i|C) for each copula function, it is sufficient to compute P_r(C|s_i). P_r(C|s_i) is normally expressed in terms of a full probability formula that includes the correlation parameter θ:

$$P_r\left(C|s_i\right)={\displaystyle {\int }_{{\theta }_{\min }}^{{\theta }_{\max }}{P}_r\left(C|\theta ,s_i\right)}\times P_r\left(\theta |s_i\right)d\theta$$

(6)

where P_r(C|θ,s_i) denotes the probability of occurrence of the measured data C knowing s_i and its correlation parameter θ. P_r(θ|s_i) denotes the probability density function of θ, which is typically assumed to be P_r(θ|s_i) = 1/(θ_max − θ_min) using the maximum entropy principle. In fact, the range of values of the correlation parameter θ varies due to various candidate copula functions. Consequently, choosing the rank correlation coefficient τ that can uniformly capture the range of values between different copula functions can address this issue. The following relationship exists between the Kendall coefficient τ and the correlation parameter θ:

$$\tau =4{\displaystyle {\int }_0^1{\displaystyle {\int }_0^{1}C\left(u,v;\theta \right)}}dC\left(u,v;\theta \right)-1$$

(7)

where C(u,v;θ) denotes the copula function characterizing the relationship between the variables u,v. Equation (6) can be reduced to an integral containing the correlation coefficient τ:

$$P_r\left(C|\tau ,s_i\right)={\displaystyle \prod _{i=1}^n{D}_i(u_i,v_i;\tau )}$$

(8)

Note that in the calculation process, it is necessary to transform the measured data C = {(x_i,y_i), i = 1, 2, 3, ···, n} into the standard uniformly distribution variable U = {(u_i,v_i), i = 1, 2, 3, ···, n}. Combining Equation (6) and Equation (8), we can derive the expression of P_r(C|τ,s_i) as:

$$P_r\left(C|\tau ,s_i\right)={\displaystyle {\int }_{{\tau }_{\min }}^{{\tau }_{\max }}{\displaystyle \prod _{i=1}^n{D}_i(u_i,v_i;\tau )}}\times \frac{1}{{\tau }_{\max }-{\tau }_{\min }}d\tau$$

(9)

Accordingly, the general steps for identifying the optimal copula function using the Bayesian method are first determining the class and number of alternative copula functions, determining the prior probability information of each copula function to obtain P_r(s_i), and ascertaining the upper and lower bounds of the correlation coefficient τ. Equation (9) can be used to obtain P_r(C|τ,s_i), which, combined with P_r(s_i), can be substituted into Equation (5) to obtain the probability P_r(s_i|C) of each copula function occurring under the given measured data C. The copula function with the maximum probability value is regarded as the best-fit copula function. Typically, the correlation coefficients take values between [−1, 1].

3.2. Optimal Copula Function Identification Results

From the results of the empirical distributions between parameters in Figure 2, it can be concluded that there exists a significant positive correlation between parameters E and c, and a significant negative correlation between parameters c and φ. In this study, the Gaussian, Plackett, Frank and Clayton copula functions are selected as candidate copula functions to characterize the positive correlation structure (E, c), i.e., S₁ = {Gaussian, Plackett, Frank, Clayton}, in this case N = 4. The Gaussian, Plackett, Frank and No.16 copula functions are selected as candidate copula functions characterizing the correlation structure between the parameters (c, φ), i.e., S₂ = {Gaussian, Plackett, Frank, No.16}, in this case, N = 4. Restricted by the survey conditions of the top plate’s measured data, each alternative copula function in the set of alternative copula functions is regarded as an equal prior probability in practical engineering, i.e., P_r(Gaussian) = P_r(Plackett) = P_r(Frank) = P_r(Clayton) = P_r(No.16) = 1/4. Combined with the data in the empirical distributions U₁ = {(u_Ei,v_ci), i = 1, 2, 3, ···, 167} and U₂ = {(u_ci,v_φi), i = 1, 2, 3, ···, 167}, from Equation (8), the correlation coefficients of (E, c) and (c, φ) are 0.332 and −0.339, respectively. Substituting them into Equation (9) and combining them with Equation (5), the posterior probabilities of different copula functions under the two sets of parameters can be obtained, and the calculation results are shown in

Table 5. Posterior probability under Bayesian approach.

Parameter	Gaussian	Plackett	Frank	Clayton	No.16
(E, c)	90.21%	0	9.79%	0	-
(c, φ)	46.61%	22.72%	30.66%	-	0.01%

. It is worth noting that the range of correlation coefficients for the positive correlation parameter group in this paper is set to [0, 1] and the range of correlation coefficients for the negative correlation parameter group is set to [−1, 0] in the calculation process.

From Table 5, we can visibly establish that the Gaussian copula is the optimal copula function for both parameters (E, c) and (c, φ). In particular, the positive correlation parameter (E, c) is recognized as the Gaussian copula function with a probability of 90.21%. Therefore, the Gaussian copula can be considered as the preferred copula to characterize the correlation structure of the correlation parameter groups with an obvious positive correlation in future coal mine engineering.

4. Comparison of Optimal Copula Recognition Methods

In this section, the recognition effectiveness of the Bayesian method in identifying the optimal copula in the multidimensional parametric model of coal mine roofs is verified by means of Monte Carlo simulation. Firstly, a large number of parameter sets obeying the real copula function are simulated according to the statistical characteristics of the measured samples, and the number of times the real copula function can be recognized is calculated using the Bayesian method and the AIC criterion. The recognition results under different numbers of measured samples, different correlation coefficients and different real copula functions are compared to evaluate the recognition effectiveness of the Bayesian method.

4.1. Determination of the Number of MCS

The effectiveness of the simulation results is often strongly related to the number of simulations N. The larger the number N of Monte Carlo simulations, the closer the calculated results are to the actual values. However, as the number of simulations increases, the cost of computation increases accordingly, and the time spent also increases. To obtain a reasonable number of simulations N, so that the selected number of simulations N can not only reflect the simulation results under the real copula function, but also minimize the computational cost, the selection of the real function for the Gaussian copula simulation of different numbers is performed to obtain the recognition results, as shown in Figure 3. It can be recognized that the Bayesian method can better identify the copula function in the real case. With the increase in the number of simulations, the successful recognition rate is characterized by a large up-and-down fluctuation to a smooth fluctuation around 77.5%. Consequently, it is assumed that the calculated result R₁₀₀₀ = 77.5% at the number of simulations of 1000 is almost equal to the approximate value. We take R₁₀₀₀ × (1 ± 2%) as the tolerable convergence interval. In Figure 3, when the number of simulations is greater than 700, the calculated results are within the convergence interval. When the number of simulations is greater than 1000, the results tend to stabilize. In this study, N = 1000 is chosen as the number of simulations for the simulation calculation.

4.2. Analyses and Comparison of Identification Results

To compare the accuracy of identifying the optimal copula function under different testing methods, the AIC criterion is used in this section as a comparison method to evaluate the identification effectiveness of the Bayesian method. Note that in identifying the real copula function using the Bayesian method, equal prior probability is assumed, i.e., the prior probability of each copula function is 1/4. Consistent with the AIC method, the real copula is considered successfully identified when the number of times that the real copula has been identified as the optimal copula is greater than any of the candidate copula functions.

Table 6. Number of successful identifications of the real copula of positive correlation structures using the Bayesian method in 1000 simulations.

Real Copula	n	τ = 0.2	τ = 0.3	τ = 0.4	τ = 0.5	τ = 0.6	τ = 0.7
Gaussian	20	370	460	488	539	574	599
	50	463	533	579	618	663	684
	100	561	658	664	694	730	765
	150	606	728	752	746	780	852
	200	664	761	766	764	826	894
Plackett	20	241	402	553	599	658	735
	50	300	493	682	778	807	894
	100	346	583	835	891	914	967
	150	404	637	885	951	968	984
	200	442	682	925	974	979	996
Frank	20	326	342	378	402	419	455
	50	397	408	436	544	628	736
	100	452	496	568	711	776	871
	150	489	563	614	765	807	916
	200	526	619	703	821	879	945
Clayton	20	578	593	649	658	672	679
	50	669	756	798	805	828	831
	100	790	877	899	920	910	897
	150	833	925	946	949	955	947
	200	899	951	970	976	970	968

and

Table 7. Number of successful identifications of the real copula of positive correlation structures using the AIC criterion in 1000 simulations.

Real Copula	n	τ = 0.2	τ = 0.3	τ = 0.4	τ = 0.5	τ = 0.6	τ = 0.7
Gaussian	20	342	451	486	512	556	572
	50	434	550	559	628	635	736
	100	565	642	672	688	726	781
	150	597	700	744	754	802	797
	200	671	755	782	788	808	872
Plackett	20	239	362	548	638	632	705
	50	294	491	674	756	782	884
	100	344	570	794	902	910	949
	150	390	619	857	947	953	982
	200	413	677	905	972	980	996
Frank	20	180	192	199	207	301	346
	50	281	283	377	462	562	716
	100	402	394	559	697	611	836
	150	464	557	606	762	766	893
	200	505	605	698	808	856	915
Clayton	20	507	571	632	639	661	707
	50	663	735	791	803	811	807
	100	773	876	892	910	904	909
	150	830	925	942	945	939	933
	200	891	955	965	968	969	959

show the identification results of the Bayesian approach with the AIC criterion using 1000 Monte Carlo simulations when the parameters are positively correlated, respectively. Emboldened numbers in these tables indicate that the real copula is not recognized as the optimal copula function for that particular number of measured data and correlation conditions. Underlined numbers indicate that the number of correct copula identified using the Bayesian approach is lower than the AIC criterion. In Table 6 and Table 7, the probability of being able to correctly identify the real copula function increases as the correlation increases. Similarly, the probability of correctly identifying the real copula function increases as the number of samples of measured data increases. Among the four true copula functions, the Clayton copula, as the real copula function, is more likely to be recognized as the optimal copula function when the correlation is weak. Hence, the results obtained are relatively reliable when the identification results are Clayton copula under the weak positive correlation of the parameters. When the correlation is strong, the Plackett copula and Clayton copula functions are more prone to be recognized as optimal copula functions when they are used as real copula functions, respectively. Therefore, with strong positive parameter correlation, the results obtained are relatively reliable when the identification results are the Clayton copula or Plackett copula functions.

When the Bayesian method is used to recognize the four real copula functions, the recognition fails only when τ = 0.2, the number of samples of measured data is 20, and the real copula is the Plackett copula function, with a failure rate of 0.83%. When using the AIC criterion to identify the four real copula functions, there are 10 cases not identified as real copula functions, including one set when the real copula function is the Plackett copula and nine sets when the real copula function is the Frank copula, which results in a failure rate of 8.33%. The failure rate for recognizing the real copula function using the AIC criterion is 10 times higher than the Bayesian approach. In terms of numerical magnitude, the number of recognitions calculated using the Bayesian method is lower than the AIC criterion in only 14 of the 120 cases calculated, and it is mainly concentrated in the case of recognition of the Gaussian copula. Therefore, the results of the optimal copula identified using the Bayesian approach are superior to those of the AIC criterion in the same cases. When the real copula is the Gaussian copula, both Bayesian and AIC methods can recognize the optimal copula perfectly, and the accuracy of the recognition by the two methods is relatively close.

Table 8. Number of successful identifications of the real copula of negative correlation structures using the Bayesian method in 1000 simulations.

Real Copula	n	τ = −0.2	τ = −0.3	τ = −0.4	τ = −0.5	τ = −0.6	τ = −0.7
Gaussian	20	424	512	557	581	595	600
	50	546	607	598	633	649	695
	100	659	659	666	681	733	786
	150	676	689	738	723	786	839
	200	731	758	778	799	829	879
Plackett	20	269	281	307	366	389	512
	50	365	371	382	466	479	630
	100	388	436	439	554	561	757
	150	419	455	481	621	630	805
	200	445	471	509	659	687	859
Frank	20	312	335	361	392	412	445
	50	385	392	415	535	623	721
	100	422	462	496	702	786	873
	150	475	536	587	762	819	902
	200	512	601	698	823	883	951
No.16	20	805	759	689	608	558	550
	50	927	913	819	691	630	581
	100	978	965	857	778	720	671
	150	993	989	905	823	765	712
	200	991	990	945	873	810	782

and

Table 9. Number of successful identifications of the real copula of negative correlation structures using the AIC criterion in 1000 simulations.

Real Copula	n	τ = −0.2	τ = −0.3	τ = −0.4	τ = −0.5	τ = −0.6	τ = −0.7
Gaussian	20	502	529	545	564	573	593
	50	567	587	630	644	658	684
	100	643	667	671	680	737	794
	150	683	714	720	733	768	838
	200	710	757	765	772	827	851
Plackett	20	265	271	275	342	379	491
	50	336	349	360	461	479	606
	100	377	419	449	535	549	749
	150	414	453	480	610	602	803
	200	443	458	491	634	659	856
Frank	20	162	178	193	213	295	352
	50	272	312	362	415	509	679
	100	362	452	476	612	698	827
	150	392	527	561	703	782	873
	200	424	576	669	812	832	928
No.16	20	790	742	671	580	450	228
	50	910	882	802	620	590	310
	100	958	929	831	750	686	660
	150	962	958	881	790	735	701
	200	981	979	940	809	792	762

show the identification results of the Bayesian method and the AIC criterion using 1000 Monte Carlo simulations when the parameters are negatively correlated, respectively. For the Gaussian copula, Plackett copula and Frank copula, the probability of being capable of correctly recognizing the real copula function increases as the correlation strengthens. Conversely, the No.16 copula has the highest recognition accuracy when the correlation is weak. As the correlation strengthens, the number of correctly recognized No.16 copula shows a decreasing trend, but the overall correct recognition rate is still above 50%. Analogously, the probability of correctly recognizing the real copula function increases as the number of samples of measured data rises. Among the four real copula functions, the No.16 copula as a real copula function is more likely to be recognized as the optimal copula function when the correlation is weak. Hence, the results obtained are relatively reliable when the identification results are the No.16 copula function under a weak positive correlation of parameters. When the correlation is strong, the Plackett copula and Clayton copula functions are more likely to be recognized as optimal copula functions when they are used as true copula functions, respectively. Thus, the results obtained are relatively reliable when the identification results are the Gaussian copula functions under strong positive correlation of parameters.

When using the Bayesian approach to identify the four real copula functions, the identification fails only when τ = 0.2, 0.3, 0.4, and the number of samples of measured data is 20, and the real copula is the Plackett copula, with a failure rate of 2.5%. When the AIC criterion is used to identify the four real copula functions, there are 18 cases not identifying the real copula function, including seven groups when the real copula function is the Plackett copula function, nine groups when the real copula function is the Frank copula function, and two groups when the real copula function is No.16, with a failure rate of 15%. The failure rate for recognizing the true copula function using the AIC criterion is six times larger than when using the Bayesian approach. In terms of numerical magnitude, the number of recognitions calculated by the Bayesian approach is lower than the AIC criterion in only 13 out of the 120 cases calculated, and it is mainly concentrated in the cases where the true copula is the Gaussian copula. Accordingly, the results of the optimal copula function identified using the Bayesian method are superior to those of the AIC criterion in the same cases. When the real copula is the Gaussian copula, both the Bayesian method and the AIC criterion can identify the optimal copula perfectly, and the accuracy of identification by the two methods is relatively close. Under the same circumstances, the Bayesian method is considered to be preferred.

5. Factors Affecting the Recognition Accuracy of Bayesian

As can be recognized from the previous section, compared with the commonly used AIC criterion, the Bayesian approach not only has an advantage in the number of correct identifications, but also the overall identification accuracy is greater than that of the AIC criterion. As such, this section analyzes the factors affecting recognition accuracy under different conditions for the Bayesian approach. Figure 4, Figure 5, Figure 6 and Figure 7 show the recognition results when varying the number of samples, with positive and negative correlation coefficients, and with prior information, respectively.

5.1. Changing the Number of Measured Samples

As can be obtained from Figure 4, with the increase in the initial number of samples of the measured data, both with positive and negative correlation parameters, the corresponding successful recognition rate shows a clear ascending trend. Especially when increasing from sample number from 20 to 100, the increment of successful recognition increase rate is more obvious than the sample number from 100 to 180. Consequently, the accuracy of the recognition rate is more sensitive when the initial number of samples is small. In practical engineering, obtaining as much measured data as possible helps to improve the recognition success rate. In addition, the recognition rate of the negative correlation parameter is generally higher than that of the positive correlation parameter group, and this difference presents a tendency to decrease with the increase in the number of samples.

5.2. Changing the Correlation Coefficient

Figure 5 depicts the number of successful identifications of the optimal copula with increasing positive correlation coefficients for different samples when the real copula is the Gaussian copula, Plackett copula, Frank copula and Clayton copula, respectively. It is evident that in addition to the number of samples, the change in correlation coefficient affects the rate of successful identification significantly. As the correlation coefficient enhances, the correct recognition rate increases accordingly. This is due to the fact that all of these copula functions converge to independent copula functions when the correlation coefficient tends to 0. Thus, the closer the correlation coefficients are to 0, the smaller the difference between the correlation structures of the copula functions in characterizing the correlation parameter, and the lower the correct recognition rate. Apart from this, the overall accuracy of identification when the Clayton copula is used as the real copula is consistently above 50%, especially when N > 150, when the identification accuracy can reach more than 90%, the difference between the number of identifications of the optimal copula function and the other copula functions is the largest. Thereby, the larger the difference between the number of optimal copula functions obtained through identification and the number of other copula functions obtained through identification, the more accurate the results of identification, the smaller the number of real samples required, and the easier it is to perform identification.

Figure 6 portrays the number of successful identifications of the optimal copula with the increase in the negative correlation coefficient for different samples when the real copula function is the Gaussian copula, Plackett copula, Frank copula and No.16 copula, respectively. Inconsistently with the pattern of change of the positive correlation parameter, when the real copula is No.16, the recognition correctness decreases as the correlation coefficient is enhanced. This is due to the fact that the No.16 copula, unlike other copula functions, does not converge to an independent copula function when the correlation coefficient tends to zero. Consequently, among the copula functions selected in this paper, only the No.16 copula is used as the real copula function. With the enhancement of the negative correlation coefficient, the difference between the correlation structures of each copula function among the characterization parameters decreases, and the correct identification rate decreases. As an exception to this, when N > 100, using the No.16 copula as the real copula function can keep the recognition accuracy rate above 60%, and the difference between the number of identifications of the optimal copula function and the other copula functions is the largest.

5.3. Changing the Prior Information

In Section 4, due to the scarcity of prior information, when using the Bayesian approach to discriminate the best-fit copula function, it is assumed that the candidate copula prior probabilities are equal, i.e., Pr(s_i) = 1/4. However, in practical engineering, if sufficient information or adequate experience is available to indicate that the probability of a certain copula function being the optimal copula function is higher than that of the other copula functions, the probability of this copula function being greater than that of the other candidate copulas can be chosen for calculation when using Equation (5). As illustrated in Figure 7, the black dashed line and the blue dashed line indicate the results of the successful identification rate of the positive correlation parameter and negative correlation parameter with the number of samples under equal prior probability (both equal to 0.25), respectively. The black solid line and the blue solid line indicate the results of the successful identification rate of the positive correlation parameter and negative correlation parameter with the number of samples under unequal prior probability (true copula probability is 0.4, the rest of the copula is 0.2), respectively. Taking the correlation coefficient obtained from the calculation of real parameters as an example, the Gaussian copula is chosen as the real copula function. The successful recognition rate obtained by considering increasing the prior probability of the real copula function is significantly higher than that obtained by simply treating all the alternative copula functions as equal-probability prior information. Hence, in practical coal mine engineering, comprehensive roof-related structural information should be obtained as much as possible to provide reliable prior information for Bayesian recognition.

6. Conclusions

Based on Bayesian theory, this paper proposes an optimal copula function identification method for the structural correlation of multidimensional parameters of coal mine roofs. On the basis of 167 sets of measured data containing (E, c, φ) from the roofs of 24 coal mines at home and abroad, the accuracy of the Bayesian method is compared with that of the commonly used AIC criterion in the optimal copula identification. The main conclusions obtained are as follows:

(1) The Bayesian theory can effectively identify the optimal copula function characterizing positive and negative correlation structures under the multidimensional parameters of coal mine roofs, which can fully consider the prior information of the parameters in the practical engineering, and provides reference experience for the identification of correlation structures of other geotechnical multidimensional parameters in the practical engineering.

(2) The copula theory provides a favorable approach for solving the joint distribution model of multidimensional geotechnical parameters. In the identification of correlation structures of geotechnical multidimensional parameters in coal mine roofs, the Gaussian copula, as a commonly used copula function, still has some advantages in characterizing the multidimensional geotechnical parameters of roofs.

(3) Compared with the commonly used AIC criterion, the Bayesian approach has an advantage in correctly identifying the number of multidimensional geotechnical parameter structures of the roofs, and the overall identification accuracy is superior to that of the AIC criterion under different numbers of measured samples and different correlation coefficients, and hence the use of the Bayesian approach can be prioritized in the practical engineering.

(4) The number of measured samples, the strength of the correlation coefficient, and the prior information have a vital effect on the correct identification rate of the optimal copula function under different real copula functions. Except when the real copula function is the No.16 copula, the larger the number of measured samples, the stronger the correlation coefficient, the higher the prior probability, and the greater the correct recognition rate.