1 Introduction

Roof fall is a crucial component in underground mine safety because it has a great impact on the cost and production of a mine. It accounted for 18.18 % of all fatal accidents in coal mines, contributing to about 35.29 % of all fatal accidents in below-ground operations in 2005 [1]. For this reason, various studies are conducted to find the relationship between the roof falls and different parameters that have a significant effect on events of falling roofs in coal mines. Table 1 lists several studies that formulate the relationship between the roof fall and other effective factors.

Table 1 A list of several studies on modeling roof fall
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However, the aforementioned models are not capable of simultaneously handling both complexity and inherent uncertainty connected with the roof fall problem. Hence, with respect to importance of the roof fall problem, applying a proper technique can help designers to analyze the problem more accurately and precisely. Adaptive neuro-fuzzy inference system (ANFIS) has been a powerful tool for formulating complex problems over the last decade and has demonstrated its capabilities and effectiveness as a problem-solving tool in modeling different aspects of engineering and management problems [16–30]. Some applications of ANFIS in geotechnical engineering can be found in [31]. This method is found to be a viable contender in competition with various conventional models [18, 22, 25].

The unique features of ANFIS can be on account of the existing advantages in the two methods, artificial neural network (ANN) and fuzzy inference system (FIS), that form the ANFIS structure. However, ANN is capable of modeling all types of the existing complexity and non-linearity in the structure of the data under consideration. Likewise, FIS is successful in the face of uncertain data and can take into account the human knowledge in modeling.

The main aim of the study is to develop an ANFIS model for predicting the roof fall rate in order to obtain a more accurate, precise, and sure equation. To show the capability and effectiveness of the constructed model, the results are compared with multivariate regression (MVR) based on three indices, including coefficient of determination (R 2), mean absolute error (MAE), and root mean square error (RMSE).

The rest of this paper is organized as follows: multivariate regression is briefly illustrated in the next section. The basic concepts of adaptive neuro-fuzzy inference system (ANFIS) are presented in Sect. 3. Section 4 explains the indices employed for evaluating model performance. Dataset is described in Sect. 5. In Sect. 6, development of the ANFIS model is implemented to formulate the roof fall rate and its performance is investigated in comparison with the MVR model. In Sect. 7, results and discussion are discussed. Section 8 includes the conclusions of the study.

2 Multivariate regression analysis

Regression analysis is widely used for modeling relationship between inputs and output variables. Based on basic concepts of multivariate regression (MVR), the relationship between input variables and the output variable is generally explored by the function that is fitted to a dataset. The standard form of the MVR model in statistical analysis can be defined as

$$ y = b_{1} x_{1} + b_{2} x_{2} + \cdots + b_{n} x_{n} + c $$
(1)

where y is the dependent variable, x n (n = 1, 2, …, n) is independent variable, b 1, b 2, …, b n are the regression coefficients, which represent the amount the dependent variable changes when the corresponding independent variables change 1 unit. c is the constant term, which represents the amount the dependent variable will be when all the independent variables are 0.

The major conceptual limitation of all regression techniques is that one can only ascertain relationships, but never be sure about the underlying causal mechanism [25].

3 ANFIS modeling

A fuzzy inference system applied in the form of a neuro-fuzzy system with crisp functions in consequents is the Takagi–Sugeno-type fuzzy system, well known as ANFIS. ANFIS was first introduced by Jang [32]. The ANFIS can be trained to tune its parameters and learn the existing structures in the dataset. The relationships between input and output variables are represented by means of fuzzy if–then rules with unclear predicates. The ANFIS model is established by adapting the antecedent parameters and consequent parameters so that a specified objective function (usually a difference between the model output and the actual output) is minimized [33]. Several methods are developed for learning rules [32, 34]. In this paper, the hybrid learning algorithm developed by Jang [32], that is a combination of least square estimation and back-propagation algorithms, is applied.

Suppose that the rule base contains of the following two Sugeno-type fuzzy if-then rules:

  • Rule 1 : If x is A 1 and y is B 1, then \( f_{1} = p_{1} x + q_{1} y + r_{1} \)

  • Rule 2 : If x is A 2 and y is B 2, then \( f_{2} = p_{2} x + q_{2} y + r_{2} \)

where x and y are two inputs; A i and B i are the terms which are represented by fuzzy sets. f i is the output variable within the fuzzy region specified by the fuzzy rule the membership function parameters of which are premise parameters. p i , q i , and r i are designing parameters which are obtained during the learning process [35]. The ANFIS architecture with two rules is depicted in Fig. 1. The output of each node in every layer is indicated by O l i (ith node output in lth layer). The performance of each layer can be described as follows:

Fig. 1
figure 1

ANFIS architecture with two rules

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The first layer is the fuzzifying layer in which A i and B i are the linguistic terms. The output of the layer is the membership functions of these linguistic terms are given as

$$ O_{i}^{l} = \mu_{{A_{i} }} (x) $$
(2)
$$ O_{i}^{l} = \mu_{{B_{i} }} (y) $$

In the second layer, the rules’ firing strengths are calculated by multiplying each node with each other as presented in the following equation:

$$ w_{i} = \mu_{{A_{i} }} (x)\mu_{{B_{i} }} (y),\quad i = 1,2 $$
(3)

where \( \mu_{{A_{i} }} \) and \( \mu_{{B_{i} }} \) are the membership functions of the input variables x and y, respectively.

In the third layer, the firing strengths obtained in the previous layer of the nodes are normalized. Every node in the layer computes the ratio of the ith rule’s firing strength to the sum of all rules’ firing strengths by using Eq. (4):

$$ \bar{w}_{i} = \frac{{w_{i} }}{{w_{1} + w_{2} }},\quad i = 1,2 $$
(4)

In the fourth layer, every node calculates a linear function where the function coefficients are adapted using the back-propagation algorithm of the artificial neural networks [36]. The output of this layer is derived from multiplication of normalized firing strength obtained in the third layer by the first order of the Sugeno fuzzy rule.

$$ \bar{w}_{i} f_{i} = \bar{w}_{i} (p_{i} x + q_{i} y + r_{i} ),\quad i = 1,2 $$
(5)

In the fifth layer, the overall output is calculated as a summation of all the incoming signals through Eq. (6):

$$ \sum\limits_{i = 1}^{2} {\bar{w}_{i} f_{i} } = \frac{{\sum\nolimits_{i = 1}^{2} {w_{i} f_{i} } }}{{\sum\nolimits_{i = 1}^{2} {w_{i} } }} $$
(6)

The first and fourth layers in ANFIS structure are adaptive layers. The consequent coefficients (p i , q i , and r i ) are continuously adjusted using fuzzy membership functions in order to minimize the errors between the model outputs and the observations [36].

4 Model performance

Since there are no unique and more appropriate unbiased estimators applied to see how far the model is able to forecast the values of roof fall, several measures of accuracy are employed. For this reason, the models are evaluated by three estimators containing the coefficient of determination (R 2), the root mean square error (RMSE), and the mean absolute error (MAE).

The above-mentioned estimators are obtained by

$$ R^{2} = 1 - \frac{{\sum\nolimits_{i = 1}^{N} {(A_{i} - P_{i} )^{2} } }}{{\sum\nolimits_{i = 1}^{N} {(A_{i} - \bar{A}_{i} )^{2} } }} $$
(7)
$$ {\text{RMSE}} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{N} {(A_{i} - P_{i} )^{2} } }}{N}} $$
(8)
$$ {\text{MAE}} = \frac{{\sum\nolimits_{i = 1}^{N} {\left| {A_{i} - P_{i} } \right|} }}{N} $$
(9)

where P i is the predicted value, A i is the observed value, \( \bar{A}_{i} \) is the average of observed set, and N is the number of datasets.

R 2 shows how much of the variability in the dependent variable can be explained by independent variable(s). R 2 is a positive number that can only take values between zero and one. A value for R 2 close to one shows a good fit of the forecasting model and a value close to zero presents a poor fit.

MAE would reflect if the results suffer from a bias between the actual and modeled datasets [37]. RMSE is a used measure in order to calculate the differences between values predicted by a model and the values observed from the thing being modeled. RMSE and MAE are non-negative numbers that for an ideal model can be zero and have no upper bound.

5 Dataset

The dataset involved in this study includes 109 observations of the roof fall rate against its affecting parameters [38]. In order to construct ANFIS model for the roof fall rate, the existing dataset is separated into training and test sets. For achieving the aim, 82 observations (75 %) are randomly selected to formulate the model and the rest of the observations (25 %) are used to reflect the performance of the different constructed models. Based on the MVR model, the affecting parameters on the roof fall rate are extracted as described in the following part.

One of the most important steps in developing a successful predicting model is the selection of the input variables, which determine the architecture of the model. Based on the literature, five input parameters for predicting the roof fall rate (RFR) are identified, including CMRR (i.e., the quality of roof rock plays a significant role in the rate of roof fall [38]), PRSUP (i.e., roof fall rate is influenced by roof bolt density [15]), IS (i.e., most of the falls in the total database occur in intersections, so that they are much more likely to fall than entry or crosscut segments between intersections [38]), DOF (i.e., increasing depth leads to increase in virgin stress levels in the rock mass [15]), and MH (i.e., mining height influences the rate of roof fall [39]). Basic descriptive statistics on the dataset involved in modeling are presented in Table 2.

Table 2 Statistical parameters of each dataset
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6 Prediction model

Based on classical assumptions of the linear model, the best forecast of Y (variable Y regressed on X) is \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{Y} \) can be calculated by Eq. (10):

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{Y} = X(X^{T} X)^{ - 1} X^{T} $$
(10)

here, Y is RFR and X = [1, CMRR, PRSUP, IS, DOF, MH] is the set of independent variables defined, respectively, as the constant term, Coal Mine Roof Rating, Primary roof support, Intersection diagonal span, Depth of cover, and Mining height, so that the model is results in the following:

$$ {\text{RFR}} = 5. 2 6- 0. 1\left[ {\text{CMRR}} \right] + 0. 4 8 2\left[ {\text{PRSUP}} \right] + 0.0 1 1 4\left[ {\text{IS}} \right] - 9. 8 5 4 {\text{E}} - 5\left[ {\text{DOF}} \right] - 0. 1 3 6\left[ {\text{MH}} \right] $$

The performance measures of the MVR model for the dataset are listed in Table 3. From the last row of Table 3, it can be seen that the MVR model is not satisfactory. Therefore, the ANFIS model is implemented in the following part.

Table 3 Comparison among performances of different constructed models
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According to the basic concepts of ANFIS design and using productive algorithm in MATLAB 7.11 package software, the development of the ANFIS model is implemented. In order to obtain the optimum network architecture, several network architectures are established to select the best-fitted ANFIS model. The established model is schematically depicted in Fig. 2.

Fig. 2
figure 2

Structure of fuzzy model

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Before constructing the ANFIS model, all variables are normalized to the interval of 0 and 1 using Eq. (11):

$$ X_{\text{norm}} = (X - X_{\hbox{min} } )/(X_{\hbox{max} } - X_{\hbox{min} } ) $$
(11)

There are different methods to generate the fuzzy rules. Two of the most popular methods are Grid partition [32] and subtractive clustering [40]. The subtractive clustering method is more used when there are many input variables. For instance, let it be assumed that there are 10 input variables and three MFs for each input variable; the rules will be 310 (59049 rules) that results in the computations being long and time-consuming. With this reason, the authors use a subtractive fuzzy clustering to generate the relationship between the input and output variables. This method uses the given search radius to measure the density of data points in the feature space [40]. A small cluster radius will usually yield many small clusters in the data and lead to generating many rules, and a large cluster radius will usually result in a few large clusters in the data and cause fewer rules [41–44]. In this paper, the most appropriate value for the cluster radius is identified by a trial and error approach by changing the cluster radius value from 0.1 to 0.9 (in increments of 0.1). The results of testing data, as seen in Table 3, show that the optimum value for the cluster radius is 0.4 since it produces the highest R 2 of 0.856 with the lowest MAE and RMSE of 1.215 and 1.738, respectively (As highlighted in Table 3). The optimal number of rules for the best-fitting model is 71.

After forming the initial ANFIS structure, the training stage is accomplished. In order to train the ANFIS model, the number of iteration of the hybrid algorithm for correction of model parameters and objective error is taken into account as 30 and 10−9, respectively. Figure 3 shows the trend of errors when the number of epochs is increased. A sample of rules generated by the ANFIS model is presented in Fig. 4.

Fig. 3
figure 3

Trend of errors during epochs

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Fig. 4
figure 4

A sample of rules generated for the ANFIS model

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Then, the constructed model is checked by the testing dataset. For achieving the aim, the input vectors from the test dataset are presented to the trained network and the predicted outputs are compared with the actual ones for the performance measurement as shown in Table 4.

Table 4 Testing dataset applied for performance measurement
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The interdependency of input and output parameters derived from the rules generated by subtractive clustering can be shown using the control surface as depicted in Fig. 5. As seen in the figure, Fig. 5a shows the interdependency of RFR on CMRR and PRSUP, Fig. 5b depicts the interdependency of RFR on CMRR and IS, Fig. 5c shows the interdependency of RFR on CMRR and DOF, and Fig. 5d depicts the interdependency of RFR on CMRR and MH.

Fig. 5
figure 5

Control surface of RFR on a CMRR and PRSUP; b CMRR and IS; c CMRR and DOF; and d CMRR and MH

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7 Results and discussion

The performance measures of the ANFIS and MVR models for testing dataset are listed in the last three rows of Table 4. It can be seen that the R 2 value for the ANFIS model is 0.856, which is dramatically greater than that obtained in MVR, which is 0.039 (R 2 < 60 % is unacceptable).

The MAE value for the ANFIS model is 1.215, which is significantly smaller than that resulting from MVR, which is 2.834. The RMSE value for the ANFIS model is 1.738, which is smaller than that yielded by MVR, which is 4.034. According to the yield results, the capability and efficiency of the ANFIS model for predicting RFR is more accurate and precise than the MVR model. This is due to the fact that the problem of RFR is such a complex and vague problem that linear methods are not capable to model such behaviors.

8 Conclusion

Roof fall is an important and critical problem in underground coal mines, which has a significant impact on the activity continuity of mining and producing. Therefore, accurate prediction of the problem can help decision makers reduce or eliminate risks associated with the problem of roof fall. This problem is a vague and uncertain issue because there are a number of parameters that can affect the problem of roof fall and often these parameters are ill defined or immeasurable. Therefore, according to the sophisticated structure of the problem, imprecise data, less of information, and inherent uncertainty, the usage of the fuzzy sets can be useful. On the other hand, ANN is widely used for modeling different problems of science and engineering. Fuzzy logic is integrated with ANN, well known as ANFIS, to take into account advantages of the two methods for modeling complex problems under a fuzzy environment.

In this paper, using the ANFIS model based on the subtractive clustering method, a model is constructed to predict roof fall rate. The results of the established model demonstrate that the ANFIS is a more powerful and capable tool than the MVR to improve the safety of coal mines. The output of the model can be used for selecting the proper reaction strategy in order to reduce or eliminate the risk of roof fall.