# Behaviour Investigation of SMA-Equipped Bar Hysteretic Dampers Using Machine Learning Techniques
## 5 Machine Learning Approaches
### ML Proposed Models Performances  
To evaluate the performance of the ANN and GMDH-NN models, Equations (26)-(31) were used, which are typical criteria for measuring error and model performance [100], including the correlation coefficient (R) and the coefficient of determination (R\({}^{2}\)), mean square error (MSE), root-mean-square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE). Detailed results are presented in Table 6 for each developed model. \[R=\frac{\sum\limits_{i=1}^{n}{{{{\left({{A_{i}}-\overline{A}\right)}\left({{F_{ i}}-\overline{F}}\right)}}}}}{\sqrt{\sum\limits_{i=1}^{n}{{{{\left({{A_{i}}- \overline{A}\right)}\left({{F_{i}}-\overline{F}}\right)}^{2}}}}}} \tag{26}\]  
\[R^{2}=\left(\frac{\sum\limits_{i=1}^{n}{{{{\left({{A_{i}}-\overline{A}\right)} \left({{F_{i}}-\overline{F}}\right)}}}}}{\sqrt{\sum\limits_{i=1}^{n}{{{{\left({{A _{i}}-\overline{A}\right)}\left({{\sum\limits_{i=1}^{n}{{{{\left({{F_{i}}- \overline{F}}\right)}^{2}}}}}}\right)}^{2}}}}}}\right)^{2} \tag{27}\]  
\[MSE=\frac{1}{n}{\sum\limits_{i=1}^{n}{{{{\left({{A_{i}}-{F_{i}}}\right)}^{2}}}}} \tag{28}\]  
\[RMSE=\sqrt{\frac{1}{n}{\sum\limits_{i=1}^{n}{{{{\left({{A_{i}}-{F_{i}}}\right)} ^{2}}}}}} \tag{29}\]  
\[MAE=\frac{1}{n}{\sum\limits_{i=1}^{n}{{{\left|{{A_{i}}-{F_{i}}}\right|}}}} \tag{30}\]  
\[MAPE=\frac{1}{n}{\left[\frac{\sum\limits_{i=1}^{n}{{{{\left|{{A_{i}}-{F_{i}}} \right|}}}}}{\sum\limits_{i=1}^{n}{{{{\left|{{A_{i}}}\right|}}}}}\right]}\times 100 \tag{31}\]  
In Equations (26)-(31), \(A_{i}\) indicates the analysed value, and \(F_{i}\) represents the estimated value, \(n\) is the number of the considered data, \(\overline{A}\) is the mean analysed values, and \(\overline{F}\) is the mean estimated values. As shown in Table 6, R and R\({}^{2}\) for the ANN models (ANN (\(\beta\)) and ANN (\(\alpha\))) in both training and testing steps are more than the GMDH-NN models (GMDH-NN (\(\beta\)) and GMDH-NN (\(\alpha\))). In the same manner, the evaluated error criteria demonstrate that the ANN models became more accurate than corresponding GMDH-NN models. In particular, the ANN (\(\beta\)) model with R\({}^{2}\) values of 0.9949 and 0.9834 and MAPE values of 0.91 and 1.18, respectively, in training and testing stages, outperforms the corresponding GMDH-NN (\(\beta\)) \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Parameter** & **Method** &
\begin{tabular}{c} **Data** \\ **Partition** \\ \end{tabular} & **R** & **R\({}^{2}\)** & **MSE** & **RMSE** & **MAE** & **MAPE (\%)** \\ \hline \multirow{3}{*}{\(\beta\)} & ANN (\(\beta\)) & Training & 0.9974 & 0.9949 & 0.00008 & 0.00903 & 0.00572 & 0.91 \\  & Testing & 0.9917 & 0.9834 & 0.00024 & 0.01538 & 0.00773 & 1.18 \\ \cline{2-9}  & GMDH-NN (\(\beta\)) & Training & 0.9726 & 0.9459 & 0.00554 & 0.07441 & 0.05507 & 3.55 \\  & Testing & 0.9744 & 0.9494 & 0.00423 & 0.06507 & 0.05080 & 3.03 \\ \hline \multirow{3}{*}{\(\alpha\)} & ANN (\(\alpha\)) & Training & 0.9986 & 0.9972 & 0.00004 & 0.00597 & 0.00444 & 2.23 \\  & Testing & 0.9985 & 0.9971 & 0.00003 & 0.00578 & 0.00464 & 2.21 \\ \cline{1-1} \cline{2-9}  & GMDH-NN (\(\alpha\)) & Training & 0.9597 & 0.9210 & 0.00100 & 0.03159 & 0.02334 & 10.90 \\ \cline{1-1} \cline{2-9}  & Testing & 0.9685 & 0.9379 & 0.00070 & 0.02642 & 0.01991 & 8.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The errors and performances of ANN and GMDH-NN models. model, with R\({}^{2}\) values of 0.9459 and 0.9494 and MAPE values of 3.55 and 3.03, respectively, in training and testing stages. Moreover, regarding the \(\alpha\) estimation models, ANN (\(\alpha\)) model with R\({}^{2}\) values of 0.9972 and 0.9971 and MAPE values of 2.23 and 2.21, respectively, in training and testing stages, outperforms corresponding GMDH-NN (\(\alpha\)) model, with R\({}^{2}\) values of 0.9210 and 0.9379 and MAPE values of 10.90 and 8.16, respectively, in training and testing stages.