# Utilizing Soft Computing Techniques to Estimate the Axial Permanent Deformation of Asphalt Concrete
## 5 Results and Analysis  
In this study, a total of 3214 data points were utilized and randomly divided into 70% training, 15% validation, and 15% testing data sets by the software. The weights and bias values in the network are trained and determined using the training data set. The network generalization performance is assessed through a cross-validation process using the validation data. The accuracy of the developed model is evaluated through the testing data set. Several iterations have been conducted using a trial-and-error approach to allocate the ideal number of hidden neurons in the network. The model was sequentially examined with three, five, seven, and nine hidden neurons. The results of each iteration were statistically evaluated using coefficient of determination values R\({}^{2}\) and mean square errors of MSE, as results in Table 6. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Neurons**} & \multicolumn{2}{c}{**Training**} & \multicolumn{2}{c}{**Validation**} & \multicolumn{2}{c}{**Test**} & \multicolumn{2}{c}{**All Data**} \\ \cline{2-9}  & **R\({}^{2}\)** & **MSE** & **R\({}^{2}\)** & **MSE** & **R\({}^{2}\)** & **MSE** & **R\({}^{2}\)** & **MSE** \\ \hline
3 & 0.86564 & 0.0069 & 0.71538 & 0.0087 & 0.87049 & 0.0057 & 0.87265 & 0.0078 \\ \hline
5 & 0.89643 & 0.0058 & 0.91623 & 0.0049 & 0.89246 & 0.0051 & 0.89814 & 0.0052 \\ \hline
7 & 0.93412 & 0.0038 & 0.92064 & 0.0043 & 0.90592 & 0.0041 & 0.9305 & 0.0039 \\ \hline
9 & 0.92602 & 0.0038 & 0.89284 & 0.0067 & 0.93586 & 0.0042 & 0.9213 & 0.0041 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistical performance of trial neurons in the network. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Inputs/Output** & \(X_{min}\) & \(X_{max}\) & \(Y_{min}\) & \(Y_{max}\) \\ \hline n & 0 & 10,000 & \(-1\) & 1 \\ Temp. & 20 & 70 & \(-1\) & 1 \\ AC & 3.4 & 5.6 & \(-1\) & 1 \\ Vis & 0.35 & 0.71 & \(-1\) & 1 \\ Stress & 6.5 & 30 & \(-1\) & 1 \\ PP1/2 & 68 & 96 & \(-1\) & 1 \\ PPNo.4 & 43 & 58 & \(-1\) & 1 \\ AV & 1.87 & 10.49 & \(-1\) & 1 \\ VMA & 9.36 & 20.54 & \(-1\) & 1 \\ Gmm & 2.37 & 2.52 & \(-1\) & 1 \\ GmbH & 2.22 & 2.47 & \(-1\) & 1 \\ Gsb & 2.53 & 2.64 & \(-1\) & 1 \\ \(\varepsilon\)p & 0.05 & 103,000 & \(-1\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Ranges of the used data sets. The results indicate that the accuracy of models improves by increasing the number of neurons in the hidden layer. However, the best performance of the model was achieved when seven neurons were used in the network, where the R\({}^{2}\) and MSE values were converged and consistent across the data groups. Therefore, a network model with seven neurons was adopted in this study for predicting the (\(\varepsilon\)p) values. Figure 5 shows a plot of all the predicted versus experimental (\(\varepsilon\)p) data points. Figure 6 shows the high degree of convergence for the measured MSE of this model's training, validation, and test datasets, where the best performance is achieved at epoch 79. The weights and bias results of the proposed model were extracted from the software and summarized in Table 7. In this table, the notation w\({}_{1}\)(i,j) represents the connection weights between the hidden layer (i) neurons and inputs (j), whereas w\({}_{2}\)(i) represents the connection weights between the neurons in the hidden layer (i) and the output. As illustrated in Figure 4, each neuron in the hidden layer (i) has a bias value of B\({}_{1}\)(i), whereas the output has a bias value (B\({}_{2}\)) equal to \(-0.7068\). Figure 5: Predicted versus experimental (\(\varepsilon\)p) values. Figure 6: Network training process with seven neurons. While the ANN model is developed in this study to evaluate the influence of various parameters on the axial permanent strain (\(\epsilon\)p) measurements, the model sensitivity is further validated by analyzing the relative importance (contribution) and the impact of each input in the model.