## 4 Multi-Attribute Machine Learning Model Selection
### Evaluation of Machine Learning Models Fitting Capabilities  
The machine learning models that are most suitable to predict the motor multi-physical characteristics (electromagnetic, losses and structural targets) are selected based on their capacity to accurately fit the target values at the lowest computational costs. The computational cost must be cheaper than the FEA. Since the training process time is negligible, the most suitable machine learning model for each physics involved is chosen based on its accuracy and the number of samples used for the training process, looking for the one that uses the smallest dataset size. Usually, the accuracy of a machine learning model depends on the number of training samples, hence its behaviour was tested for different sample sizes (i.e., 250, 500, 750 and 1000 samples). The first values that were fitted were the torque and back emfs harmonics. The ability of the tested regression methods to predict the most influential torque harmonic orders, (i.e., DC TH, 6 th TH and 12 th TH), and the back emf harmonics (i.e., 1 st BmfH, 3 rd BmfH and 11 th BmfH harmonic orders) is presented in Table 4. Here, the \(R^{2}\) and the MSE scores are presented for each individual regression method (i.e., SVR, GBR, GPR) sequentially trained using 250, 500, 750 and 1000 samples. By analysing the values, it can be seen that all tested methods show an ascandant trend for the \(R^{2}\) score and a reduction of the MSE error when the number of samples is increased. In terms of machine learning models, SVR model performs much better than the GBR model for all types of samples, the GBR presenting a higher MSE and a lower \(R^{2}\) than the SVR. At the same time, it can be observed that the GPR model fits the targets more accurately, for each training sets. Starting with 750 samples, the GPR presents excellent results, its \(R^{2}\) score being higher than 0.93 and MSE lower than 7%. In particular, \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Samples/**} & \multicolumn{2}{c}{**DC TH**} & \multicolumn{2}{c}{**6 th TH**} & \multicolumn{2}{c}{**12 th TH**} & \multicolumn{2}{c}{**1 st Bmf H**} & \multicolumn{2}{c}{**3 rd Bmf H**} & \multicolumn{2}{c}{**11 th Bmf H**} \\ \cline{2-13} \multicolumn{2}{c}{**Models**} & \(R^{2}\) & **MSE** & \(R^{2}\) & **MSE** & \(R^{2}\) & **MSE** & \(R^{2}\) & **MSE** & \(R^{2}\) & **MSE** & \(R^{2}\) & **MSE** \\ \hline \multirow{3}{*}{250} & SVR & 0.96 & 4 & 0.96 & 4 & 0.8 & 19 & 0.95 & 6 & 0.96 & 4 & 0.74 & 26 \\  & GBR & 0.92 & 8 & 0.94 & 6 & 0.82 & 18 & 0.93 & 7 & 0.94 & 5 & 0.76 & 23 \\  & GPR & 1 & 0 & 0.98 & 1 & 0.86 & 13 & 0.99 & 1 & 0.99 & 1 & 0.89 & 11 \\ \hline \multirow{3}{*}{500} & SVR & 0.98 & 2 & 0.98 & 2 & 0.87 & 13 & 0.97 & 3 & 0.98 & 2 & 0.88 & 12 \\  & GPR & 0.96 & 4 & 0.96 & 4 & 0.88 & 11 & 0.97 & 3 & 0.97 & 3 & 0.76 & 23 \\  & GPR & 1 & 0 & 0.99 & 1 & 0.95 & 5 & 1 & 0 & 0.99 & 1 & 0.94 & 6 \\ \hline \multirow{3}{*}{750} & SVR & 0.98 & 2 & 0.98 & 2 & 0.9 & 9 & 0.97 & 3 & 0.98 & 2 & 0.9 & 10 \\  & GPR & 0.97 & 3 & 0.97 & 3 & 0.9 & 10 & 0.97 & 2 & 0.96 & 4 & 0.84 & 16 \\ \cline{1-1}  & GPR & 1 & 0 & 0.99 & 1 & 0.93 & 7 & 1 & 0 & 0.99 & 1 & 0.94 & 6 \\ \hline \multirow{3}{*}{1000} & SVR & 0.99 & 1 & 0.98 & 2 & 0.9 & 9 & 0.98 & 2 & 0.99 & 2 & 0.98 & 2 \\ \cline{1-1}  & GPR & 0.97 & 3 & 0.94 & 6 & 0.9 & 10 & 0.98 & 2 & 0.98 & 2 & 0.97 & 3 \\ \cline{1-1}  & GPR & 1 & 0 & 0.99 & 1 & 0.95 & 5 & 1 & 0 & 1 & 0 & 0.99 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Machine learning models performance to predict electromagnetic targetsfor 750 samples, GPR succeeds to improve the scores for the 12 th TH, \(R^{2}\) from 0.86 (for 250 samples) to 0.93 and the MSE from 13% (for 250 samples) to 7%. The same situation applies for the 11 th BmfH, the \(R^{2}\) value increases from 0.89 to 0.94 and the MSE reduces from 11% to 1%.