## 3 Application of Theoretical Framework
### Results of Network Trainings  
The maximum story drifts obtained for each case study are used as the dataset for the network training process by the LM method. For the network architecture, the parameters of height, slenderness ratio, fundamental period, distributed mass, stiffness ratio, terrain categories and basic wind speed represent seven input values needed to estimate an output, which tends to predict the maximum story drifts. With the aim of evaluating the efficiency of the network against new data (not analyzed), a small percentage (15%) taken randomly of the dataset is excluded in the training process. Table 3 shows the mean square error (MSE) generated for the ANNs with different number of neurons in the hidden layer. It is observed that the error is reduced when the number of neurons in the hidden layer increases. A larger number of hidden neurons allows for modeling any variation in the data; however, the decrees of the error are not significant from 20 hidden neurons.  
The cross-validation is a technique to evaluate the independence of the results of a prediction model with relation to the training and testing dataset selected. Table 4 shows the cross-validation of the ANN with 20 hidden neurons using, in each iteration, different training and evaluation dataset taken randomly that represent the 85 and 15 percent, respectively. Small variations of MSE between each dataset and iteration are observed. Therefore, the dataset selected generates in an independent way results very close to each other.  
Figure 8 shows the relation between the target values and the output values of the ANNs with 1, 10 and 20 hidden neurons considering training and evaluation data. It is observed that the increase in the number of neurons in the hidden layer helps to improve the linear relation close to 99%.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Hidden** & **Training** & **Evaluation** & **Training** & **MSE Training** & **MSE Evaluation** \\
**Neurons** & **Data** & **Data** & **Iterations** & **Data** & **Data** \\ \hline
1 & 21,760 & 3840 & 1000 & 101.72 & 108.96 \\
2 & 21,760 & 3840 & 1000 & 65.48 & 62.96 \\
5 & 21,760 & 3840 & 1000 & 23.92 & 29.22 \\
10 & 21,760 & 3840 & 1000 & 8.89 & 9.06 \\
12 & 21,760 & 3840 & 1000 & 5.73 & 5.88 \\
20 & 21,760 & 3840 & 1000 & 2.59 & 2.91 \\
30 & 21,760 & 3840 & 1000 & 2.05 & 2.21 \\ \hline \end{tabular}
\end{table}
Table 3: Results of network trainings with different number of neurons in the hidden layer.  
Table 5 shows the memory and computer time required in the tasks of computational processing for the case H = 150 m, R\({}_{\text{e}}\) = 5, M\({}_{\text{d}}\) = 500 Kg/m\({}^{2}\), F\({}_{\text{k}}\) = 1, T\({}_{0}\) = 3 s, C\({}_{\text{t}}\) = 2, \(\overline{\text{U}}_{10,\text{II}}\) = 150 Km/h. A greater computer time in the training network is observed than the sum of all phases of the standard procedure; however, once the network has been trained, it is prepared to respond to many possible cases of immediately and using little memory.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Validation Iterations** & **MSE Training Data** & **MSE Evaluation Data** \\ \hline Iteration 1 & 3.51 & 3.71 \\ Iteration 2 & 2.86 & 2.94 \\ Iteration 3 & 2.24 & 3.01 \\ Iteration 4 & 2.58 & 2.82 \\ Iteration 5 & 3.99 & 3.65 \\ Iteration 6 & 3.42 & 3.58 \\ Iteration 7 & 2.79 & 2.74 \\ Iteration 8 & 3.21 & 3.31 \\ Iteration 9 & 2.49 & 2.58 \\ Iteration 10 & 2.73 & 3.00 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Cross-validation of the ANN with 20 hidden neurons.  
Figure 8: Relation between the target values and output values of the ANN with 1, 10 and 20 hidden neurons. (**a**) Using the training data; (**b**) using evaluation data.