## 4 Quantitative Analysis of the Internal Leakage in a Valve
### Dimensionality Reduction of Acoustic Emission Signals by Factor Analysis  
Six parameters were extracted from the acoustic emission signals collected in the experiment, including entropy, energy, maximum value, root mean square value, root square amplitude and kurtosis. Meanwhile, the upstream and downstream pressures corresponding to each signal were also recorded. These high-dimensional features contain a lot of redundant features. If regression prediction analysis is conducted directly, it will be time-consuming and the prediction accuracy will be low. Therefore, before quantitative evaluation of the leakage rate, the high-dimensional features of the acoustic emission signal were reduced based on the factor analysis method. It is expected that a few common factors can be used to explain the complex relationship between multiple variables to be observed.  
First, the principal component analysis method was used to extract common factors. Because of the strong correlation in the original data structure, the percentages of variance explained by the first three common factors were 69.635%, 13.823% and 8.658%, respectively. The cumulative variance percentages of these three common factors reached 92.116%, indicating that the loss of valve leakage signal characteristic variable data was very limited, and most of the useful information was retained. Therefore, the first three factors were selected as common factors.  
Figure 4: Block diagram of the valve internal leakage rate prediction method.  
Then, the extraction rate analysis was performed on the three extracted common factors. The common factor variances of the factor analysis are given in Table 2. It can be seen from Table 2 that the information extraction rate of the five variables--energy, maximum value, root mean square value, square root amplitude and kurtosis--is higher than 95%, indicating that most of the information in the variables were extracted by the common factors. The information extraction rate of upstream pressure, downstream pressure and entropy also reached more than 75%, indicating that the common factors can explain the information of the original sample data commendably. It is shown that the result of factor analysis is effective.  
Next, the maximum variance method, a kind of orthogonal rotation method, was used to rotate the three extracted common factors so as to increase their explanation ability. The principal factor analysis method was used to extract the common factor load, and then Kaiser standardized maximum variance method was used to rotate the factor load four times until it converged iteratively. The factor load matrix after rotation is given in Table 3. It can be seen from Table 3 that Factor 1 has a high load on the five variables of energy, maximum value, root mean square value, root square amplitude and downstream pressure, indicating that these five variables can be explained by a common factor. Factor 2 has a high load on the two variables--upstream pressure and entropy, indicating that these two variables can be explained by a common factor. Factor 3 has a high load on the kurtosis variable, indicating that the kurtosis can be explained by Factor 3 alone.