# Research on Prediction Method of Hydraulic Pump Remaining Useful Life Based on KPCA and JITL
## 2 Wavelet Packet Denoising Method and KPCA Method
### KPCA Method  
In traditional methods, as a gear pump performance degradation characteristic index, a single time- or frequency domain-characteristic index is frequently used as the basis. Owing to the variability and complexity of the gear pump operation process, a single index cannot accurately and comprehensively represent the faults and performance degradation information during its operation. Therefore, it is necessary to consider and extract multiple appropriate features to achieve comprehensive characterization of a gear pump performance degradation trend. For obtaining the performance degradation characteristic index of a gear pump, in this study, multiple characteristic parameters from the time and frequency domains are selected, including variance, root-mean-square value, peak value, root amplitude, peak-to-peak value, absolute mean value, standard deviation, shape factor, clearance factor, and average power. Subsequently, the dimensionality reduction calculation is performed for the selected multiple characteristic indicators. Finally, the characteristics after the dimensionality reduction are used as the performance degradation evaluation index of the gear pump. PCA [43] is a traditional mathematical linear transformation analysis method. The basic principle of the PCA method is the linear transformation of the original variables with some correlation into a set of uncorrelated variables, during which process the total variance remains unchanged. The KPCA method extracts the principal components from the original data information by a nonlinear transformation by some technical improvements of the PCA method. First, the input space is transformed into a high-dimensional space by a nonlinear transformation. Subsequently, the principal components in the information are nonlinearly analyzed in the high-dimensional space. The specific method is as follows:  
Let the training sample set be \(\mathbf{X}=\{\mathbf{x}_{i}|i=1,2,\ldots,P\}\), where \(\mathbf{x}_{i}\in R^{Q}\) is a column vector and \(P\) is the number of training samples. The KPCA method nonlinearly maps a \(Q\)-dimensional original vector from the original input space, \(R^{Q}\), to a high-dimensional space, \(\mathbf{F}\), represented as \(\mathbf{\varphi}:R^{Q}\rightarrow\mathbf{F}\). The sample in \(\mathbf{F}\) is represented as \(\varphi(\mathbf{x}_{i})\), and it satisfies \(\sum_{i=1}^{P}\varphi(\mathbf{x}_{i})=0\). Therefore, the sample covariance matrix in the \(\mathbf{F}\) space can be expressed as  
\[\mathbf{C}=\frac{1}{P}\sum_{i=1}^{P}\varphi(\mathbf{x}_{i})\varphi(\mathbf{x}_{i})^{T}. \tag{3}\]  
The eigen decomposition of the covariance matrix, \(\mathbf{C}\), can be obtained as follows:  
\[\lambda_{i}\mathbf{v}_{i}=\mathbf{C}\mathbf{v}_{i},i=1,2,\cdots,P, \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Characteristic Parameter** & **Formula** & **Characteristic Parameter** & **Formula** \\ \hline Gravity frequency & \(CF=\frac{\sum\limits_{f=0}^{f_{\text{max}}}\left[f\times S(f)\right]}{\sum \limits_{f=0}^{f_{\text{max}}}S(f)}\) & Frequency variance & \(FV=MSF-\left(CF\right)^{2}\) \\ Mean square frequency & \(MSF=\frac{\frac{f_{\text{max}}}{f-0}\left[f^{2}\times S(f)\right]}{\sum \limits_{f=0}^{f_{\text{max}}}S(f)}\) & Average power & \(f_{1}=\frac{\sum\limits_{f=0}^{f_{\text{max}}}S(f)}{f_{\text{max}}}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Frequency-domain characteristic parameters. where \(\mathbf{v}_{i}\) is the eigenvector corresponding to the eigenvalues, \(\lambda_{i}\), in the \(F\) space, and all eigenvalues of \(C\) are non-negative. Without the loss of generality, \(0\leq\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{P}\) are set, and the corresponding eigenvectors, \(\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{P}\), can be expanded by the sample, \(\varphi(\mathbf{x}_{i})\), in the \(F\) space as follows:  
\[\mathbf{v}_{r}=\sum_{i=1}^{P}\mathbf{\alpha}_{i}^{r}\varphi(\mathbf{x}_{i}), \tag{5}\]  
where \(\mathbf{\alpha}_{i}(i=1,2,\ldots,P)\) are the correlation coefficients.