# Prediction of Blade Tip Timing Sensor Waveforms Based on Radial Basis Function Neural Network
## 4 Validating the Prediction Accuracy of RBF Neural Network
### Predicted Static Radial Characteristic Curve  
When the blade cutting angle \(\alpha\) is 17.825\({}^{\circ}\) and 35.65\({}^{\circ}\), respectively, the RBF neural network model, Kriging prediction model, BP neural network model, and polynomial function are adopted for prediction comparison to verify their accuracy. The blades in the middle position range from the centerline to the edge of the sensor when the blade cutting angle \(\alpha\) is 17.825\({}^{\circ}\). Moreover, in the angle under the BTC in 0.45 mm after the blade is unable to cut the eddy current sensor magnetic induction line, there is no output voltage signal. Therefore, the BTC measurement range is 0.1-0.45 mm under this angle. As shown in Figure 5a-f, the comparison plots among the RBF model prediction curves, Kriging model prediction curves, BP model prediction curves, and polynomial curves against the static radial characteristic curves for the blade cutting angle of magnetic susceptibility at 17.825\({}^{\circ}\) reveal that the prediction curves under the different models possess a high overlapping degree and are basically similar to the static radial characteristic curves. The prediction curves of the Kriging model show pointwise errors, but with the same orientation. The overlap between the prediction curves of the BP model and the polynomial fitting curves shown in Figure 5c,f is slightly lower compared to the curves plotted from the empirical values, but there is a higher overlap for the RBF model compared to Figure 5a. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Points** & **RBF NN** & **Kriging Model** \\ \hline \multirow{3}{*}{327 data points} & Neurons = 44, MSE = 12.9223 & \multirow{3}{*}{MSE = 9.2249 \(\times\) 10\({}^{-11}\)} \\  & Neurons = 48, MSE = 1.36114 & \\  & Neurons = 108, MSE = 1.9125 \(\times\) 10\({}^{-9}\) & \\ \hline \multirow{3}{*}{654 data points} & Neurons = 46, MSE = 10.4189 & \multirow{3}{*}{MSE = 2.1877 \(\times\) 10\({}^{-10}\)} \\  & Neurons = 52, MSE = 1.53358 & \\  & Neurons = 109, MSE = 1.20741 \(\times\) 10\({}^{-10}\) & \\ \hline \multirow{3}{*}{872 data points} & Neurons = 56, MSE = 8.25402 & \multirow{3}{*}{MSE = 2.6446 \(\times\) 10\({}^{-10}\)} \\  & Neurons = 64, MSE = 1.00306 & \\ \cline{1-1}  & Neurons = 109, MSE = 2.84326 \(\times\) 10\({}^{-19}\) & \\ \hline \multirow{3}{*}{1090 data points} & Neurons = 65, MSE = 9.94967 & \multirow{3}{*}{MSE = 3.0617 \(\times\) 10\({}^{-10}\)} \\  & Neurons = 73, MSE = 1.08635 & \\ \cline{1-1}  & Neurons = 109, MSE = 1.31676 \(\times\) 10\({}^{-23}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relative performance of RBF model and Kriging model with different groups of data. Further, Figure 6 shows the RBF model, Kriging model, BP model and polynomial function fitting predicted curves against the static calibration data, from which it is found that there is a small fluctuation between the curves at the clearance of 0.25 mm. The prediction errors of each model are shown in Figures 7 and 8. The absolute errors of the RBF model, Kriging model and BP model are less than 40 mV. With the same number of sampling points in the same group, the RBF model's prediction accuracy is high when the clearance is small and decreases compared to the Kriging model when the clearance becomes large, and the BP model has a higher accuracy in the localized range of 0.1-0.45 mm in Figure 7a,b. The prediction accuracies of the above models are within the measurement errors of the sensors. The absolute error predicted by the polynomial function is lower compared to the RBF model and Kriging model, however it is less than 60 mV, which is still within the allowed range. The training data curves of the BP model are shown in the upper half of Figure 8. With its training set, test set and function fitting curves highly overlapping, it presents an excellent training data state.