## 4 Quantitative Analysis of the Internal Leakage in a Valve
### Comparison of Different Quantitative Models for Internal Leakage Rate  
Firstly, the effectiveness of sample dimensionality reduction in improving the prediction performance of the model was verified. A wavelet-BP neural network model based on the sample feature set without dimensionality reduction was established, and the optimal structure of the valve leakage rate quantitative model is 8-6-4-1. It means the input layer of the model contains eight neurons; the model contains two hidden layers, of which the first hidden layer contains six neurons, and the second hidden layer contains four neurons; the output layer contains one neuron. The comparison between the prediction results of the quantitative model of the wavelet-BP neural network based on the sample feature set by dimensionality reduction and without dimensionality reduction is shown in Figure 7.  
Figure 6: Optimal structure of the wavelet-BP neural network prediction model for valve internal leakage. Figure 7: The influence of dimensionality reduction on model prediction results. It can be seen intuitively from Figure 7 that the prediction results of the quantitative model of the valve leakage rate by the wavelet-BP neural network based on the sample feature set reduced by the factor analysis have a high degree of agreement with the actual leakage rate. It is better than the prediction results of the wavelet-BP neural network based on the sample feature set without dimensionality reduction. It can also be seen from the consistency of the prediction results of the test set that the prediction model based on the dimensionality-reduced sample set has a better prediction performance. The main reason is that after the dimensionality reduction process of the sample feature set, the redundant information is removed, the complexity of the model is reduced, and the prediction accuracy is improved. In addition, the wavelet-BP neural network quantitative model based on the sample feature set after factor analysis is less volatile, and the model has better robustness. The prediction error of the wavelet-BP neural network model based on the sample feature sets by dimensionality reduction and without dimensionality reduction is shown in Figure 8. It can be seen from Figure 8b that the prediction error of the wavelet-BP neural network model based on the dimensionality-reduced sample feature set is mainly concentrated within \(\pm 10\%\), and only several sample points are greater than \(10\%\), which indicates that the model has a higher prediction accuracy, and less volatility in prediction results. Compared with the wavelet-BP neural network model based on the sample feature set without dimensionality reduction, factor analysis of the sample feature set reduces the dimension, which increases the anti-interference ability of the model and improves the prediction accuracy of the model. It can also be seen from Figure 8b that when the leakage rate is small, the prediction percentage error of the model is significantly larger. The main reason for this is that when the leakage rate is small, the noise of the signal is relatively small, and the deviation caused by noise is large. Besides, under the same condition that the relative errors are consistent, the smaller the reference leakage rate, the greater the percentage error. Secondly, the effectiveness of wavelet decomposition on the performance improvement of the quantitative model of the internal leakage rate was verified. For the sample feature set without wavelet decomposition, the optimal structure of the quantitative model for the valve internal leakage rate is 3-1-2-1. It means the input layer has three neurons; the first hidden layer has one neuron and the second hidden layer has two neurons; the output layer has one neuron.