# A Proposed Soft Computing Model for Ultimate Strength Estimation of FRP-Confined Concrete Cylinders
## 4 Proposing a New Formulation to Predict the Compressive Strength of FRP-Confined Concrete Cylinder
### Using a Model with a K-Fold Cross-Validation Technique in FFBPNN  
In this section of the paper, a K-fold cross-validation (KFCV) technique is applied for the optimization and evaluation of the perfected ANN [80; 81]. In the KFCV technique, the data are divided randomly into K folds. Then, the K-1 folds are used for training, and the last fold is used to test the neural network. In the parametric study conducted, the values for K, changing from two to five and K = 4, are considered. The process of learning and testing is conducted for all the K sections. Therefore, all the K sections contribute to the learning and testing of the ANN. This process is iterated three times for the reduction and variation of KFCV and similar distribution of data in each K. The performance of the neural network for each iteration can be computed by the percentage of correct predictions in the neural network for K folds. In every epoch, the performance evaluation of the neural network is calculated. The curve is the correct classification factor (CCF), it is drawn for three iterations and, finally, it is averaged. In the CCF curve, after a specified epoch, the curve is saturated. Then, the optimal epoch is defined using 10% of the curve plateau. In this study, a neural network with three layers is selected for the sake of simplicity. For optimization of the ANN structure, some neurons in the hidden layer are optimized. For this purpose, the selecting criteria are considered to be the area under the CCF curve (AUCCF). Therefore, the AUCCF is measured until it reaches the optimal epoch. Hence, different neurons, from two to 13 neurons in the hidden layer, are selected and the KFCV process is repeated for the structures. Finally, the structure with the maximum efficiency can be determined by drawing the CCF and calculating the AUCCF. Figure 9 shows the AUCCF curve. As can be seen from this figure, the 6:11:1 structure with 11 neurons in the hidden layer has the highest performance with 86.6%. Figure 10 shows the CCF curve for the optimized ANN structure. As shown in this figure, the optimum epoch is 224. Figure 8: Regressions of training, validating, and testing datasets simulated by ANN 6:15:1. Data that are predicted by the optimized ANN neural network and the training data are plotted in Figure 11. As shown in this figure, the correlation coefficient is equal to 0.9809, which confirms the performance of the optimized ANN structure. Figure 10: The CCF curve of optimized ANN structure. Figure 9: Area under the correct classification factor (AUCCF) curve of the neural network with different structures. The Tansig and Pureline activation functions are selected for the hidden layer and the output layer, respectively. Considering the optimum structure of the neural network, weights, biases, and activation functions, a relation, such as Equation (2), could be extracted:  
\[Output=\left(\frac{2}{1+exp^{(-2\times(Inputclw+b_{1}))}}-1\right)\times Lw+b_{2} \tag{2}\]  
_Input, Output, IW, LW, \(b_{1}\), and \(b_{2}\) in Equation (2) are constant coefficients, which are defined as follows:  
\[