# Behaviour Investigation of SMA-Equipped Bar Hysteretic Dampers Using Machine Learning Techniques
## 5 Machine Learning Approaches
### Proposed ANN Models  
According to the proposed method by [PERSON] et al. (2018), in ANN models, the obtained database (containing 389 data) is divided into two training (around 75% of all datasets equal to 292 data) and testing (around 25% of all datasets equal to 97 data) classes. Moreover, to assure the ANN and GMDH-NN models efficiency, the train set (around 75% of dataset equal to 292 data) itself is classified into three subsets, including train, test, and validation, with around 80% (232 data), 10% (30 data) and 10% (30 data) ratios, respectively. In order to determine the structure of the ANN models, the feed-forward back propagation model, which has a single input layer and one hidden layer with the tan-sigmoid activation \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Mechanical Property**} & \multicolumn{3}{c}{**Inputs**} & \multicolumn{3}{c}{**Outputs**} \\ \hline
**Statistical Feature** & \(D\) **(m)** & \(L\) **(m)** & \(N\) & \(E\) **(GPa)** & \(F_{y}\) **(MPa)** & \(\beta\_Bi.\) & \(\alpha\_Bi.\) \\ \hline Min & 0.018 & 0.324 & 1 & 24.6 & 280 & 0.005 & 0.061 \\ Max & 0.050 & 1.362 & 10 & 30 & 350 & 0.040 & 0.720 \\ Ave & 0.037 & 0.728 & 6.286 & 27.658 & 318.274 & 0.022 & 0.268 \\ SD & 0.008 & 0.241 & 2.597 & 2.203 & 28.564 & 0.008 & 0.110 \\ CoV (\%) & 24.016 & 32.901 & 41.303 & 8.010 & 8.977 & 36.725 & 41.720 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical properties of the numerical database. function (\(f_{n}{}^{2}\)), as shown in Equation (14), and an output layer having linear activation function (\(f_{n}{}^{1}\), expressed as \(\alpha\)x + \(b\) where \(a\) and \(b\) are constants) is used. \[f_{n}^{2}=\frac{2}{1+e^{-2x}}-1 \tag{14}\]  
MATLAB was used to train the connecting weights of network neurons using feed-forward backpropagation and the Levenberg-Marquardt method [94]. The literature has supported the usage of one hidden layer to tackle various nonlinear problems [85; 95]. The optimum artificial neural network was created using the trial-and-error approach. In this study, the number of neurons in the hidden layer was assumed to be between 3 and 25. The optimal configuration was determined by using conventional statistical error and performance metrics, such as the correlation coefficient (R), the mean square error (MSE), and the mean absolute percent error (MAPE). The results of the ANN models in estimating the first yield point displacement modification factor (\(\beta\)) and also post-yield stiffness ratio (\(\alpha\)) are shown in Tables 3 and 4, respectively. Figure 16 shows the R and MSE values for different neurons in the hidden layer of ANN models. Figure 14: Adopted bilinear curve for obtaining seismic parameters. Figure 13: Idealisation of hysteresis curve using bilinear envelope curve. As can be seen in Table 3 and Figure 16, the structure of the best ANN model for estimating \(\beta\) contains six neurons in the hidden layer with R values of 0.9972, 0.9958, and 0.9916 in training, testing, and validation data sets and considerably small MSE values of 0.00012 and 0.00047, and MAPE values of 2.60% and 3.31%, in training and testing data sets, respectively. On the other hand, as reported in Table 4 and Figure 16, to estimate the \(\alpha\) values, selecting 10 neurons in the hidden layer is the most proper ANN model with the R values of 0.9986, 0.9966, and 0.9972 in training, testing, and validation data sets and MSE values of 0.00007 and 0.00008, and MAPE values of 2.34% and 2.43%, in training and testing data sets, respectively. It should also be noted that training data division in the network includes one step of verification, which uses 80%, 10%, and 10% for training, testing, and validation, respectively. However, in order to provide a more reliable and efficient model, the training data division is not a good problem. Figure 15: The process of \(Q\_Bi.\) determination.