# Research on Prediction Method of Hydraulic Pump Remaining Useful Life Based on KPCA and JITL
## 3 Principle of JITL Method  
The principle of the JITL method is \"similar input data produce similar output data.\" Its basic workflow is to store a large amount of sample information in the sample data memory. Based on the actual input data, data similar to the input sample data are found in the sample data storage, and subsequently the corresponding output data of the input data are obtained from the sample data information. For a nonlinear mapping of multiple inputs and single output, \(f:R^{n}\to R\), it is assumed that the input and output data of the system (i.e., \(\left\{x_{q},y_{q}\right\}_{q=1}^{V}\), where \(V\) is the number of training samples) can be measured. Moreover, the input and output data have the following functional relationship:  
\[y_{q}=f(x_{q})+\zeta_{q}, \tag{16}\]  
where \(x_{q}=\left\{\;x_{i,q}|i=1,2,\ldots,T,q=1,2,\ldots,V\right\}\) represents the input sample data and \(T\) is the dimension of the input sample data. The output sample data are \(y_{q}\in R\). \(\zeta_{q}\) is an independent random variable with a zero mean and a variance of \(\sigma^{2}\). The JITL method establishes a model in a certain space by recursion to predict the output. The input and output relationship of the established model is described as follows:  
\[f(x_{q},\mathbf{\theta})=x_{q}^{T}\mathbf{\theta}=\theta_{0}+x_{1,q}\theta_{1}+x_{2,q }\theta_{2}+\ldots+x_{T,q}\theta_{T}, \tag{17}\]where \(\mathbf{\theta}=[\theta_{0},\theta_{1},\ldots,\theta_{T}]^{T}\) represents the sub-model parameters. Combined with a historical database, mapping in the local space is established to obtain the model output value, \(\hat{y}_{q}\). Therefore, the predicted output problem can be solved using an optimal solution, and the formula is as follows:  
\[\min\sum_{(\mathbf{x}_{q},y_{q})\in\mathbf{\Omega}_{k}}\big{[}y_{q}-f(\mathbf{x}_{q},\mathbf{ \theta})\big{]}^{2}w_{q}, \tag{18}\]  
where \(\mathbf{\Omega}_{k}\) is the local space consisting of \(k\) sample data near \(\mathbf{x}_{q}\). \(f\) is the mapping relationship between input data \(\mathbf{x}_{q}\) and output data \(y_{q}\). \(w_{q}\) is the weight corresponding to the \(q\)-th historical sample data, which represents the weight of the influence of the \(q\)-th sample data contained in the local sample space on the model output. The selection of the local space has a significant influence on the accuracy of the sub-models. For the input data, \(\mathbf{x}_{q}\), the selected local space is used to select the appropriate historical data. These historical data consist of \(k\) samples that are most similar to the input data, \(\mathbf{x}_{q}\). To select historical data similar to the input vector, \(\mathbf{x}_{q}\), some scholars have proposed k-NN, k-surrounding neighbors, k-bipartite neighbors, and other methods to screen the modeling data. However, these methods mainly refer to the Euclidean distance as the basis for modeling, which is insufficient for fully exploring the internal relationships among the input and nearest data. In [45], the k-VNN method was proposed, which considers both the Euclidean distance and angle relationship between two vectors. Therefore, the k-VNN method is used as the neighborhood selection criterion in this study. As we all know, k-VNN considers both the Euclidean distance and angle relationship between two vectors, while k-NN mainly refers to the Euclidean distance as the basis for modeling, which is insufficient for fully exploring the internal relationships among the input and nearest data. Therefore, sample selection method based on k-VNN of the RUL prediction method can significantly improve the prediction accuracy of the RUL prediction model. Assuming that the data in the historical database are \(\mathbf{  
\(D(\mathbf{x}_{i},\mathbf{x}_{q})\). This indicates that the historical data, \(\mathbf{x}_{i}\), have a significant influence on the sub-model (i.e., the weight coefficient, \(w_{q}\), is large).