## 4 Example Verification
### The Establishment of the Health Assessment Model for Hydropower Units  
The health model of the hydropower unit is established by using ten monitoring parameters in the initial three months of operation. Each monitoring parameter has 1200 groups of data, and the data sampling time is 10 min. The first 1000 sets of data are used for training the GPR model, and the remaining 200 sets of data are used for model verification. The validation effect of the GPR model is shown in Figure 5a. The predicted curve by the GPR model is in good agreement with the actual monitoring data. To highlight the superiority of the GPR model, the back propagation neural network (BPNN), and the radial basis function neural network (RBFNN) model are used for comparison. BPNN can realize the nonlinear mapping between input samples and output samples, and has the characteristics of self-organization and self-learning, which makes it very effective in nonlinear approximation. In addition, the potential relationship between these data can be summarized through the automatic learning process. So far, there is no definite method to determine the number of nodes in the hidden layer of BPNN. [PERSON] points out that the number of hidden layers should be between \(\sqrt{2m+1}\) and \(2m+n\), [PERSON]  
Figure 4: Time-domain diagram of from 1 to 6 stator coil temperatures. thinks that the number of nodes is \(2m+1\)[40], and Some scholars think that the number of nodes is \(\sqrt{m+n}+a\), where a is a constant between \(1\) and \(10\)[41]. In these methods, \(m\) and \(n\) are the number of nodes in the input layer and the output layer respectively. Therefore, according to these empirical formulas, the number of nodes in the hidden layer should be between 4 and 19. Considering the complexity of monitoring parameters, the transfer functions _logsig_ and _purelin_ are used as the transfer functions of hidden layer and output layer. _Tainglm_ adaptive learning algorithm is used to complete network training and prediction, and is programed and run in MATLAB 2019a environment. The main parameters of the network are as follows: the target error is \(6.5\times 10^{-4}\), the learning rate is \(0.005\), the maximum number of calculations is \(5000\), and the additional momentum factor is \(0.90\). The test error of typical network performance with different number of hidden layer nodes is shown in Table 5. Therefore, the best prediction model of BPNN is \(9\times 19+1\), and the research results of the BPNN model is seen in Figure 5b. The research results of the RBFNN model are seen in Figure 5c. The comparison between the results obtained with the GPR model to the results of BPNN and RBFNN models is showed in Table 6.  
In this paper, Equation (16) is used to quantitatively test the above prediction results. \[\sigma_{MAE}=\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\left|\frac{\overline{y}(i)-y(i)}{ y(i)}\right| \tag{16}\]  
In Equation (16), \(\sigma_{MAE}\) is the absolute value of average relative error. \(N_{1}\) is the number of test samples \(\overline{y}(i)\) is the predicted value and \(y(i)\) is the real value of the monitoring data. According to Table 6, the \(\sigma_{\text{MAEs}}\) of the GPR model, the BPNN model, and the RBFNN model are 2.1158%, 6.6455%, and 5.1104%, respectively. Obviously, the predicted curve by BPNN model diverges largely from the actual monitoring data. For the monitoring data with strong randomness and high noise of hydropower units, compared with BPNN and RBFNN, the covariance function is used to describe the random distribution of the monitoring data of hydropower units in GPR modeling, and the noise of the data can be effectively identified and separated. Therefore, the GPR model is very suitable for the health assessment model of hydropower units. In the field of condition monitoring, there are many research results on determining the alarm threshold of equipment abnormal operation by analyzing the monitoring data collected by equipment status [38; 39; 42].