# Research on Prediction Method of Hydraulic Pump Remaining Useful Life Based on KPCA and JITL
## 2 Wavelet Packet Denoising Method and KPCA Method
### Wavelet Packets Denoising Method  
The wavelet packet transform analysis method is a technical improvement of the wavelet transform analysis method. It is a joint signal analysis method based on time and frequency domains. Compared with the signal analysis method of wavelet transform, the wavelet packet transform analysis method can decompose the low- and high-frequency parts of the upper layer simultaneously. Thus, it overcomes the deficiency of the wavelet transform analysis method, which only analyzes the low-frequency part of a signal. The method of wavelet packet decomposition and denoising can perform orthogonal decomposition of the collected signals in all frequency bands. An orthonormal basis that can reflect the original characteristics of the signals can be obtained by reasonably selecting the optimal wavelet packet basis function and the decomposition layer number. As shown in Figure 2a, after the original signal is decomposed by wavelet decomposition, the low-frequency coefficients will be decomposed again. As shown in Figure 2b, wavelet packet decomposition not only re-decomposes the low-frequency decomposition coefficients after each layer decomposition, but also re-decomposes the high-frequency coefficients. So, the wavelet packet denoising method is more refined and has better localization ability in the frequency domain than the wavelet denoising method. In multiresolution analysis, \(L^{2}(\mathbf{R})=\underset{j\in Z}{\oplus}W_{j}\) indicates that it is based on different scale factors \(j\). The Hilbert space, \(L^{2}(\mathbf{R})\), is decomposed into the orthogonal sums of all subspaces \(\mathbf{W}_{j}(j\in Z)\). Following signal filtering, the signals are expanded based on the wavelet packet basis. Specifically, the signal, \(f(n)\), to be analyzed is filtered through a low-pass filter H and a high-pass filter G, and thus, a group of low- and high-frequency signals is obtained. Each signal decomposition further decomposes the \(l\)-th frequency band of the upper layer, \(j+1\), into two sub-bands \(2l\)-th and \((2l+1)\)-th of the lower layer, \(j\). Wavelet packet decomposition method: \(\left\{d_{j,k}^{2l}\right\}_{k\in Z}\) and \(\left\{d_{j,k}^{2l+1}\right\}_{k\in Z}\) are calculated from \(\left\{d_{j+1,k}^{l}\right\}_{k\in Z}\), and the wavelet packet decomposition formula is  
\[\left\{ \begin{array}{c}d_{j,k}^{2l}=\underset{m\in Z}{\sum}\overline{h}_{m-2 k}d_{j+1,m}^{l}\\ d_{j,k}^{2l+1}=\underset{m\in Z}{\sum}\overline{g}_{m-2k}d_{j+1,m}^{l}\end{array}  \right., \tag{1}\]  
where \(\left\{h_{k}\right\}_{k\in Z}\) denotes the low-pass filter coefficients, \(\left\{g_{k}\right\}_{k\in Z}\) denotes the high-pass filter coefficients, and \(g_{k}=(-1)^{k}\overline{h}_{1-k}\). Figure 2: Comparison of wavelet decomposition and wavelet packet decomposition: (**a**) Wavelet decomposition and (**b**) Wavelet packet decomposition. Wavelet packet reconstruction method: \(\left\{d_{j+1,k}^{l}\right\}_{k\in\mathbb{Z}}\) is calculated from \(\left\{d_{j,k}^{2l}\right\}_{k\in\mathbb{Z}}\) and \(\left\{d_{j,k}^{2l+1}\right\}_{k\in\mathbb{Z}^{\prime}}\), and the wavelet packet reconstruction formula is  
\[d_{j+1,m}^{l}=\sum\limits_{k\in\mathbb{Z}}\left[h_{m-2k}d_{j,k}^{2l}+g_{m-2k}d_ {j,k}^{2l+1}\right], \tag{2}\]  
where Daubechies 4 wavelet is selected for wavelet packet decomposition. The steps of the wavelet packet denoising process are as follows:  
1. Optimal wavelet packet decomposition basis determination. The optimal wavelet packet decomposition basis is calculated based on the entropy standard. 2. Wavelet packet decomposition of the signal. The number of decomposition layers, \(M\), is determined, and the signal, \(f(n)\), is decomposed by an \(M\)-layer wavelet packet. In this study, the number of decomposition layers, \(M\), is 3. 3. Threshold quantization of the wavelet packet decomposition coefficients. An appropriate threshold is selected to quantify each set of wavelet packet decomposition coefficients. In this study, the default threshold is used to denoise the signals. 4.