# Behaviour Investigation of SMA-Equipped Bar Hysteretic Dampers Using Machine Learning Techniques
## 5 Machine Learning Approaches
### Proposed ANN Models
\begin{table}
\begin{tabular}{r r r r r r r r} \hline \hline
**NN** & **MSE\_Tr** & **MSE\_Ts** & **MAPE\_Tr** & **MAPE\_Ts** & **R\_train** & **R\_test** & **R\_valid** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: ANNs performances for optimised \(\beta\) model. testing data division, including 97 data (25% out of 389 data), are entirely unseen for the models and provide a double-check step for the models. Using the weights and bias acquired from the proposed ANN models, the following formula was developed to establish the mathematical relationship between input (\(D\), \(L\), \(N\), \(E\), and \(F_{y}\)) and outputs (\(\beta\) and \(\alpha\)) variables:  
\[Y=f_{n}^{1}\Bigg{\{}b_{0}+\sum_{k=1}^{h}\Bigg{[}w_{k}f_{n}^{2}\Bigg{(}b_{hk}+ \sum_{i=1}^{m}w_{ik}X_{i}\Bigg{)}\Bigg{]}\Bigg{\}} \tag{15}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**NN** & **MSE\_Tr** & **MSE\_Ts** & **MAPE\_Tr** & **MAPE\_Ts** & **R\_train** & **R\_test** & **R\_valid** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 4: ANNs performances for optimised \(\alpha\) model. Figure 16: The R and MSE values at different neurons in hidden layer of ANN model: (**a**) \(\beta\); (**b**) \(\alpha\). where \(Y\) denotes the values corresponding to output parameters (\(\beta\) and \(\alpha\)), \(f_{n}\)1 and \(f_{n}\)2 are the transfer functions, \(h\) indicates the number of neurons in the hidden layer (in this paper equals to 6 and 10 for ANN(\(\beta\)) and ANN(\(\alpha\)), respectively), \(X_{i}\) is the input values of the network (\(D\), \(L\), \(N\), \(E\), and \(F_{y}\)), \(m\) is the number of the input variables (equals to 5), \(W_{ik}\) indicates the link weight between the \(i\)th input layer and \(k^{th}\) neuron in the hidden layer, \(w_{k}\) is the link weight between \(k\)th neuron in the hidden layer and the independent output layer, \(b_{lk}\) represents a bias in the \(k\)th neuron of the hidden layer, and \(b_{0}\) is the bias value in the output layer. Footnote 1: The \(\beta\) and \(\alpha\) values are presented in Table 5.  
Therefore, to ensure other researchers may profit from these results, network weights and bias values were supplied in this study. The estimated \(\beta\) and \(\alpha\) using ANN models, respectively, can be achieved by Equation (15), and the results are presented in Table 5.  
The \(\beta\) and \(\alpha\) estimations based on ANN models are expressed as following with respect to Equation (15), and the weights are presented in Table 5:  
\[ANN(\beta)=10.051A_{1}+3.784A_{2}-1.853A_{3}-5.004A_{4}+1.520A_{5}-3.428A_{6}+ 10.821 \tag{16}\]  
\[ANN(\alpha)= -3.141C_{1}-0.520C_{2}-0.252C_{3}-3.149C_{4}+0.741C_{5}-0.250C_{6}-2. 342C_{7} \tag{17}\] \[-0.353C_{8}-4.083C_{9}-2.730C_{10}+3.533\]  
where \(A_{1}\) to \(A_{6}\) and \(C_{1}\) to \(C_{10}\) are the response of the hidden neurons which feed the network output and are calculated as \(\overline{A}\) and \(\overline{C}\)array elements (Equation (18)) using Equations (19) and (20). \[\overline{A}^{T}=\left[ \begin{array}{cccccccc}A_{1}&A_{2}&A_{3}&A_{4}&A_{5} &A_{6}\end{array} \right];\overline{C}^{T}=\left[ \begin{array}{cccccccc}C_{1 }&C_{2}&C_{3}&C_{4}&C_{5}&C_{6}&C_{7}&C_{8}&C_{9}&C_{10}\end{array} \right] \tag{18}\]  
\[\overline{A}=Tansig\big{(}W_{\beta}I+B_{\beta}\big{)} \tag{19}\]  
\[\overline{C}=Tansig\big{(}W_{\alpha}I+B_{\alpha}\big{)} \tag{20}\]  
where \(I\) is the input array, \(W_{\beta}\) and \(B_{\beta}\), and \(W_{\alpha}\) and \(B_{\alpha}\) are, respectively, corresponding \(\beta\) and \(\alpha\) weight and bias matrices which are presented in Equations (21)-(23). \[I^{T}=\left[ \begin{array}{cccccccc}D&L&N&E&F_{y}\end{array} \right] \tag{21}\]  
\[W_{\beta}=\left[ \begin{array}{cccccccc}3.965&-20.960&-0.619&0.187&-0.620\\ -1.108&4.541&0.132&0.415&-0.837\\ -1.193&5.374&0.216&1.576&-1.959\\ -2.051&8.078&0.261&0.024&-0.998\\ 1.008&0.108&0.645&0.208&-0.059\\ 2.392&-9.337&-0.315&1.584&-0.635\end{array} \right];B_{\beta}=\left[