# Research on Prediction Method of Hydraulic Pump Remaining Useful Life Based on KPCA and JITL
## 2 Wavelet Packet Denoising Method and KPCA Method
### Determining Performance Degradation Evaluation Index  
Owing to the different types and degrees of hydraulic pump faults, the extracted pressure signal characteristic parameters in the time domain are also different. In general, dimensionless indicators are not directly affected by the operating state of a mechanical equipment; they are only determined by its corresponding probability density functions. However, dimensional indicators are directly affected by the operating state of a mechanical equipment. Both dimensional and dimensionless indicators can directly or indirectly reflect the changing trend of gear pump performance degradation during operation. The time-domain characteristic parameter calculation methods of pressure signals are tabulated in Table 1, where \(f_{\text{dn}}(n)\) represents the denoised pressure signal obtained after the wavelet packet denoising, where \(n=1,2,3,\ldots,N\) and \(N\) is the length of the original signal.  
Feature extraction methods based on the time domain are frequently unable to accurately determine the performance degradation trend of mechanical equipment. Therefore, an extraction method based on frequency-domain features is used. First, the power spectral density of the collected mechanical equipment time-domain sign  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Characteristic Parameter** & **Formula** & **Characteristic Parameter** & **Formula** \\ \hline Mean value & \(\overline{X}=\frac{1}{N}\sum\limits_{i=1}^{N}f_{\text{dn}}(i)\) & Absolute mean value & \(|\overline{X}|=\frac{1}{N}\sum\limits_{i=1}^{N}\left|f_{\text{dn}}(i)\right|\). \\ Variance & \(\sigma^{2}=\frac{1}{N-1}\sum\limits_{i=1}^{N}\left[f_{\text{dn}}(i)-\overline{ X}\right]^{2}\) & Standard deviation & \(\sigma=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}\left[f_{\text{dn}}(i)-\overline{ X}\right]^{2}}\) \\ Root-mean-square value & \(X_{\text{rms}}=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}f_{\text{dn}}(i)^{2}}\) & Shape factor & \(S_{\text{f}}=\frac{X_{\text{rms}}}{|\overline{X}|}\) \\ Kurtosis & \(K=\frac{1}{N}\sum\limits_{i=1}^{N}f_{\text{dn}}(i)^{4}\) & Crest factor & \(C_{\text{f}}=\frac{X_{\text{rms}}}{X_{\text{rms}}}\) \\ Peak value & \(X_{\text{max}}=\max[|f_{\text{dn}}(i)|]\) & Impulse factor & \(I_{\text{f}}=\frac{X_{\text{rms}}}{|\overline{X}|}\) \\ Root amplitude & \(X_{\text{r}}=\left[\frac{1}{N}\sum\limits_{i=1}^{N}\sqrt{\left|f_{\text{dn}}(i) \right|}\right]^{2}\) & Kurtosis index & \(K_{\text{v}}=\frac{K}{X_{\text{rms}}}\) \\ Peak-to-peak value & \(X_{\text{pp}}=\max[f_{\text{dn}}(i)]-\min[f_{\text{dn}}(i)]\) & Clearance factor & \(CL_{\text{f}}=\frac{X_{\text{rms}}}{X_{\text{r}}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Time-domain characteristic parameters.  
quently, the frequency-domain characteristics of the time-domain signals are calculated based on the power spectral density; therefore, the performance degradation trend of the mechanical equipment can be determined more intuitively and accurately. The calculation methods of the characteristic parameters in the relevant frequency domain are summarized in Table 2.  
The \(f_{\text{max}}\) is the upper limit of the analysis frequency band and \(S(f)\) is the amplitude of the power spectrum at frequency \(f\).