## 4 Multi-Attribute Machine Learning Model Selection
### Evaluation of Machine Learning Models Fitting Capabilities
However, even if the process of generating 1000 samples from FEA is more computationally expensive, GPR performs at its best for 1000 samples, enhancing, compared with the 750 samples case, the fitting capabilities, especially for the third most influential torque and back emf harmonics. For this case, the obtained \(R^{2}\) score is 0.95 and MSE is 5% for 12 th TH, and the \(R^{2}\) score is 0.99 and the MSE is 1% for the 11 th BmfH. Figure 7 shows the most influential torque and back emf harmonics over their actual target (original) values obtained from the GPR 1000 samples machine learning model. The capability of the tested regression methods to predict the modes and their corresponding natural frequencies are presented in Table 5, where the \(R^{2}\) and the MSE scores are presented for each individual regression method, SVR, GBR, GPR, sequentially trained using 250, 500, 750 and 1000 samples. As can be observed, the machine learning models under test are able to predict very well even when they are trained using 250 samples. By analysing the presented values, it can be seen that except the GPR method, that keeps its scores constant, the SVR and the GBR show an improvement of the \(R^{2}\) score and a reduction of the MSE error when the number of samples is increased. Moreover, their performance starts to saturate beyond 750 samples. This behaviour is emphasized when the data size is increased from 750 to 1000 samples. Comparing the results, it is clear that the \(R^{2}\) and the MSE are the same in both cases, and an expansion of data size beyond 750 does not influence the accuracy of the developed models. Actually, the models developed for 750 samples perform well, their \(R^{2}\) values being 0.99 for the SVR and GBR models and 1 for GPR model. Regarding the MSE, its values are between 0% and 1% for SVR and GBR, while for GBR it is 0%. The GPR model fits the natural frequencies more accurately than SVR and GBR. Even starting with 250 samples, the model is able to be the best performer. The low number of training samples necessary for a good performance is due to the fact that the global modes and their natural frequency are highly affected by the yoke and tooth thickness and are not as sensitive as the electromagnetic targets when the tooth tip angle, tooth tip height and the slot opening are adjusted within the set range. Therefore, less designs are needed to build the data-driven model and the structural characteristics obtained by imposing 250 input geometrical parameters are sufficient to obtain a good generalisation capacity that allows the characterisation of a new design. Figure 8 shows the model capability to  
Figure 7: GPR 1000 samples accuracy. Predicted vs. original values for the DC, 6 th and 12 th torque and 1 st, 3 rd and 11 th back emf harmonics orders. approximate the motor structural characteristics. Specifically, the frequency at which the first six mode-shapes appear is estimated with a high accuracy by the GPR 250 samples machine learning model. The fitting ability to predict the losses targets can be identified in Table 6, where the scores for all types of motor losses are presented (i.e., total--Tot, winding losses--Wind, iron--Iron, stator back iron--SBI, rotor back iron--RBI and magnet--Mag). The results for \(R^{2}\) and MSE show that SVR performs better than the GBR for all dataset sizes. The performances of all three models are improved by increasing the size of the training data, the worst prediction being obtained at 250 samples for stator tooth losses. Both SVR and GBR manage to increase the \(R^{2}\) to 1 and 0.99 and MSE to a value under 2 for 1000 samples case, but the model that is more suitable to accurately fit the losses data is, also in this case, GPR. This algorithm manages to predict, even for 250 samples, the losses with a maximum MSE of 9%, minimising the MSE value to 1% for 500 samples and 750 cases and to 0% for 1000 samples case.