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Table 3 Recent proposed correlations between UCS and BTS trapped in local minima—has encouraged the implemen- tation of hybrid techniques such as PSO-based ANN. The PSO component of such a hybrid system is able to find a global minimum and continue searching. Therefore, a hybrid PSO-based ANN model possesses advantages of both PSO and ANN: PSO will search for all of the minima in the search space and ANN will then use them to find the best solution. In a PSO-based ANN, each particle (i.e., ANN weight) is a candidate solution for minimizing the RMSE. After optimizing the problem, the optimized weights are used to train the network. In essence, the objective of implementing PSO in an ANN is to improve the ANN’s training procedure. established in the literature (Mendes et al. 2002). However, the simple evolutionary process obtained using Eqs. | and 2 is the key advantage of PSO that separates it from other optimization algorithms. This was highlighted in a study by Victoire and Jeyakumar (2004), who recommended PSO as an effective computational tool with a reduced memory requirement compared to other similar algorithms.

Table 3 Recent proposed correlations between UCS and BTS trapped in local minima—has encouraged the implemen- tation of hybrid techniques such as PSO-based ANN. The PSO component of such a hybrid system is able to find a global minimum and continue searching. Therefore, a hybrid PSO-based ANN model possesses advantages of both PSO and ANN: PSO will search for all of the minima in the search space and ANN will then use them to find the best solution. In a PSO-based ANN, each particle (i.e., ANN weight) is a candidate solution for minimizing the RMSE. After optimizing the problem, the optimized weights are used to train the network. In essence, the objective of implementing PSO in an ANN is to improve the ANN’s training procedure. established in the literature (Mendes et al. 2002). However, the simple evolutionary process obtained using Eqs. | and 2 is the key advantage of PSO that separates it from other optimization algorithms. This was highlighted in a study by Victoire and Jeyakumar (2004), who recommended PSO as an effective computational tool with a reduced memory requirement compared to other similar algorithms.

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Table 1 Proposed correlations between UCS and J,50)
Table 1 Proposed correlations between UCS and J,50)
Table 2 Some established correlations between UCS and V, Artificial neural networks noting that weathering plays an important role in all of the aforementioned correlations, as indicated in a study by Arikan et al. (2007). They determined the strength prop- erties of rock specimens with weathering states ranging from | to 6 and found an inverse relationship between the degree of weathering and the engineering properties of the rock, such as its porosity and tensile and compressive strengths.
Table 2 Some established correlations between UCS and V, Artificial neural networks noting that weathering plays an important role in all of the aforementioned correlations, as indicated in a study by Arikan et al. (2007). They determined the strength prop- erties of rock specimens with weathering states ranging from | to 6 and found an inverse relationship between the degree of weathering and the engineering properties of the rock, such as its porosity and tensile and compressive strengths.
Network design In this section we discuss the identification of PSO-based ANN parameters, including the number of particles in the swarm (swarm size), the number of iterations, and the network architecture. A parametric study was performed to find the optimum swarm size and number of iterations. The optimum network architecture was subsequently deter- mined using the trial and error method.
Network design In this section we discuss the identification of PSO-based ANN parameters, including the number of particles in the swarm (swarm size), the number of iterations, and the network architecture. A parametric study was performed to find the optimum swarm size and number of iterations. The optimum network architecture was subsequently deter- mined using the trial and error method.
Table 4 Laboratory test results In order to conduct a parametric study, a MATLAB code was prepared. Based on an educated guess, an ANN model consisting of one hidden layer with five nodes was selected as an initial model. The dataset was divided into two subsets: training and testing datasets. Eighty percent of the data was used for training purposes and the other 20 % was used to test the model’s performance. The first step in the parametric study was to determine the swarm size. Although the optimum swarm size depends on the problem of interest, applying more particles increases the computational complexity and training time, while reducing the number particles may negatively affect the model’s performance. Hence, in order to find the optimum swarm size, a wide range of swarm sizes (from
Table 4 Laboratory test results In order to conduct a parametric study, a MATLAB code was prepared. Based on an educated guess, an ANN model consisting of one hidden layer with five nodes was selected as an initial model. The dataset was divided into two subsets: training and testing datasets. Eighty percent of the data was used for training purposes and the other 20 % was used to test the model’s performance. The first step in the parametric study was to determine the swarm size. Although the optimum swarm size depends on the problem of interest, applying more particles increases the computational complexity and training time, while reducing the number particles may negatively affect the model’s performance. Hence, in order to find the optimum swarm size, a wide range of swarm sizes (from
that the maximum number of iterations was fixed at 800. Figure 4 suggests that there was no significant change in the RMSE after 450 iterations, so 450 iterations was selected as the maximum number of iterations and thus the termination criterion. It is worth mentioning that in all of the analyses conducted in this study, acceleration constants (C, and C,) equal to 2 were used to find the optimum PSO structure (Shi and Eberhart 1998; Mendes 2002). small to large, i.e., 25-300) were examined, and the coefficient of determination (R’) as well as the RMSE were measured in each analysis. It is worth mentioning that other PSO parameters were kept constant during this step. Fig- ures 2 and 3 show the values of R* and RMSE obtained for different swarm sizes. The results from the testing dataset indicate that a swarm size of 125 yields the best perfor- mance in terms of both R? (0.991) and RMSE (0.077). A swarm size of 125 was chosen as the optimum number of particles for subsequent analyses. It is worth mentioning that, when developing the predictive model, the data were normalized to values between —1 and 1. that the maximum number of iterations was fixed at 800. After obtaining the appropriate PSO parameters, the network structure was designed. To determine the optimum network architecture, ten hybrid models were considered with various numbers of nodes in different hidden layers. Each model was trained and tested five times to find t average values of R? and RMSE. However, the R* values the testing dataset were considered when evaluating t model’s performance. Although an ANN model with one hidden layer can approximate any continuous function, t! effect of two hidden layers on network performance was also investigated in this study. Either 6, 9, 12, or 15 nodes were trialled in each hidden layer in order to determine t he of he ne ne The second step of the parametric study involved determining a termination criterion for the analysis. The termination criterion is a condition that, when met, ends the iterative procedure. For the problem at hand, the termina- tion criterion was considered to be a maximum number of iterations. A series of analyses were performed to find the optimal maximum number of iterations. For this reason, a model similar to that used in the first step was used, except
that the maximum number of iterations was fixed at 800. Figure 4 suggests that there was no significant change in the RMSE after 450 iterations, so 450 iterations was selected as the maximum number of iterations and thus the termination criterion. It is worth mentioning that in all of the analyses conducted in this study, acceleration constants (C, and C,) equal to 2 were used to find the optimum PSO structure (Shi and Eberhart 1998; Mendes 2002). small to large, i.e., 25-300) were examined, and the coefficient of determination (R’) as well as the RMSE were measured in each analysis. It is worth mentioning that other PSO parameters were kept constant during this step. Fig- ures 2 and 3 show the values of R* and RMSE obtained for different swarm sizes. The results from the testing dataset indicate that a swarm size of 125 yields the best perfor- mance in terms of both R? (0.991) and RMSE (0.077). A swarm size of 125 was chosen as the optimum number of particles for subsequent analyses. It is worth mentioning that, when developing the predictive model, the data were normalized to values between —1 and 1. that the maximum number of iterations was fixed at 800. After obtaining the appropriate PSO parameters, the network structure was designed. To determine the optimum network architecture, ten hybrid models were considered with various numbers of nodes in different hidden layers. Each model was trained and tested five times to find t average values of R? and RMSE. However, the R* values the testing dataset were considered when evaluating t model’s performance. Although an ANN model with one hidden layer can approximate any continuous function, t! effect of two hidden layers on network performance was also investigated in this study. Either 6, 9, 12, or 15 nodes were trialled in each hidden layer in order to determine t he of he ne ne The second step of the parametric study involved determining a termination criterion for the analysis. The termination criterion is a condition that, when met, ends the iterative procedure. For the problem at hand, the termina- tion criterion was considered to be a maximum number of iterations. A series of analyses were performed to find the optimal maximum number of iterations. For this reason, a model similar to that used in the first step was used, except
optimum number of nodes. The results of the analyses are tabulated in Table 5. The results presented in this table suggest that the fourth PSO-based ANN model (with 12 nodes in the hidden layer) outperformed the other models. Hence, the architecture of this model was considered to be optimum for UCS prediction. schematically in Fig. 5. The R* of the proposed correlation demonstrates its reliability. Results and discussion In addition to the previous correlations, a new corre- lation between UCS and BTS is also proposed, as dis- played in Fig. 7. In comparison with the correlations highlighted in the literature, the new proposed correlation, with R* = 0.821, is quite reliable. This correlation is as follows: UCS prediction are proposed in the following paragraphs cant cant * — & Based on laboratory observations, a new correlation between UCS and V, is proposed (Eq. 3). It is shown
optimum number of nodes. The results of the analyses are tabulated in Table 5. The results presented in this table suggest that the fourth PSO-based ANN model (with 12 nodes in the hidden layer) outperformed the other models. Hence, the architecture of this model was considered to be optimum for UCS prediction. schematically in Fig. 5. The R* of the proposed correlation demonstrates its reliability. Results and discussion In addition to the previous correlations, a new corre- lation between UCS and BTS is also proposed, as dis- played in Fig. 7. In comparison with the correlations highlighted in the literature, the new proposed correlation, with R* = 0.821, is quite reliable. This correlation is as follows: UCS prediction are proposed in the following paragraphs cant cant * — & Based on laboratory observations, a new correlation between UCS and V, is proposed (Eq. 3). It is shown
It is worth mentioning that, strictly speaking, the sug- gested reliabilities of the proposed equations only apply when the data fall within the range of the data used in this study. Although the proposed equations can yield reason- able UCS estimates for other rock types, such generaliza- tion of the proposed equations may decrease the confidence level of the estimated UCS, due to the site-specific behavior of rock. While the use of correlations to predict the UCS is a
It is worth mentioning that, strictly speaking, the sug- gested reliabilities of the proposed equations only apply when the data fall within the range of the data used in this study. Although the proposed equations can yield reason- able UCS estimates for other rock types, such generaliza- tion of the proposed equations may decrease the confidence level of the estimated UCS, due to the site-specific behavior of rock. While the use of correlations to predict the UCS is a
Table 6 Performance indices of different models used for UCS prediction no. 4), as presented in Figs. 8 and 9, are the best results obtained, whereas the tabulated results listed in Table 5 are the average values of five repeated runs.
Table 6 Performance indices of different models used for UCS prediction no. 4), as presented in Figs. 8 and 9, are the best results obtained, whereas the tabulated results listed in Table 5 are the average values of five repeated runs.
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