# 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
The inner product of each sample \(\varphi(\mathbf{x}_{i})\) with Equation (4) is obtained as follows:  
\[\lambda_{r}(\varphi(\mathbf{x}_{i}),\mathbf{v}_{r})=\langle\varphi(\mathbf{x}_{i}),\mathbf{C }\mathbf{v}_{r}\rangle,i,r=1,2,\ldots,P. \tag{6}\]  
Based on Equations (5) and (6), the following can be obtained:  
\[\lambda_{r}\sum_{i=1}^{P}\mathbf{\alpha}_{i}^{r}\langle\varphi(\mathbf{x}_{r}),\varphi (\mathbf{x}_{i})\rangle=\frac{1}{P}\sum_{i=1}^{P}\mathbf{\alpha}_{i}^{r}\bigg{\langle} \varphi(\mathbf{x}_{r}),\sum_{q=1}^{P}\varphi(\mathbf{x}_{q})\bigg{\rangle}\langle \varphi(\mathbf{x}_{q}),\varphi(\mathbf{x}_{i})\rangle, \tag{7}\]  
where \(r,q=1,2,\ldots,P\). A \(P\times P\)-dimensional matrix \(K\) is defined, where \(k_{iq}=\big{\langle}\varphi(\mathbf{x}_{i}),\varphi(\mathbf{x}_{q})\big{\rangle}\) and \(i,q=1,2,\ldots,P\). Because \(K\) is a symmetric matrix, \(k_{iq}=k_{qi}\), Equation (7) can be expressed as  
\[P\lambda_{r}\mathbf{\alpha}^{r}=\mathbf{K}\mathbf{\alpha}^{r}, \tag{8}\]  
where \(P\lambda_{r}\) is the eigenvalue of \(K\) and \(\mathbf{\alpha}^{r}=[\alpha_{1}^{r},\alpha_{2}^{r},\ldots,\alpha_{p}^{r}]^{T}\) is the corresponding eigenvector. Let the eigenvectors corresponding to the eigenvalues greater than zero be \(\mathbf{\alpha}^{a},\mathbf{\alpha}^{a+1},\ldots,\mathbf{\alpha}^{P}\), respectively. \(\mathbf{v}_{r}(r=a,a+1,\ldots,P)\) is normalized to obtain \(P\lambda_{r}\langle\mathbf{\alpha}^{r},\mathbf{\alpha}^{r}\rangle=1\); therefore, the projection of sample \(\varphi(\mathbf{x})\) on \(\mathbf{v}_{r}\) is  
\[t_{r}(\mathbf{x})=\langle\mathbf{v}_{r},\varphi(\mathbf{x})\rangle=\sum_{i=1}^{P}\alpha_{ i}^{r}\langle\varphi(\mathbf{x}_{i}),\varphi(\mathbf{x})\rangle, \tag{9}\]  
where \(t_{r}(\mathbf{x})\) is the \(r\)-th nonlinear principal component corresponding to \(\varphi\). All principal components are combined into vector \(\mathbf{t}(\mathbf{x})=[t_{a}(\mathbf{x}),t_{a+1}(\mathbf{x}),\ldots,t_{P}(\mathbf{x})]^{T}\) as the sample feature. To overcome the problem that the high-dimensional \(F\) space makes solving \(\mathbf{t}_{r}(\mathbf{x})\) difficult, based on the Mercer theorem [44], a kernel function \(K_{1}(\mathbf{x}_{i},\mathbf{x})=\langle\varphi(\mathbf{x}_{i}),\varphi(\mathbf{x})\rangle\) is used to replace the dot product operation of the \(F\) space. Therefore,  
\[\mathbf{t}_{r}(\mathbf{x})=\langle\mathbf{v}_{r},\varphi(\mathbf{x})\rangle=\sum_{i=1}^{P} \alpha_{i}^{r}K_{1}(\mathbf{x}_{i},\mathbf{x}). \tag{10}\]  
To solve the problem that the sampled data do not satisfy \(\sum_{i=1}^{P}\varphi(\mathbf{x}_{i})=0\), sample \(\varphi(\mathbf{x}_{i})\) in the \(F\) space can be replaced by Equation (11). \[\widetilde{\varphi}(\mathbf{x}_{i})= \varphi(\mathbf{x}_{i})-\frac{1}{P}\sum_{i=1}^{P}\varphi(\mathbf{x}_{i}). \tag{11}\]  
The kernel matrix, \(K\), in Equation (8) is replaced by \(\overline{\textbf{K}}\). \[\overline{\textbf{K}}=\textbf{K}-\mathbf{I}_{P}\textbf{K}-\textbf{K}\mathbf{I}_{P}+\bm {I}_{P}\textbf{K}\mathbf{I}_{P}, \tag{12}\]  
where \(\mathbf{I}_{P}\) is a \(P\times P\)-dimensional identity matrix with coefficient \(1/P\). The radial basis function (RBF) kernel function has many advantages, such as wide convergence range, good positive definiteness of the transformation matrix, few parameters, and simple calculation. In data processing, the accuracy and speed of the RBF kernel function are better than those of other kernel functions. Therefore, the RBF kernel function is selected as the kernel function in the KPCA method as  
\[K_{1}(x,x_{i})=\exp[-\frac{\|x-x_{i}\|^{2}}{2\sigma^{2}}]. \tag{13}\]  
The cumulative percent variances (CPVs) of the principal components are calculated to select the components that meet the conditions. Finally, the number of principal components is determined based on the actual requirements. The method and process using the _CPVs_ as the basis for selecting the principal components are as follows: First, the corresponding initial data samples are calculated and processed to obtain the covariance matrix.