## 2 Method  
### General Regression Neural Network  
A GRNN is a type of neural network based on nonlinear regression. It is established using a one-pass learning algorithm with a highly parallel structure, which was proposed by [PERSON] in 1991 [10].  
As illustrated in Figure 1, the GRNN structure consists of four layers: the input layer, pattern layer, summation layer, and output layer. The number of inputs for the network is equal to the number of independent features (n) in the vector X = [x_1, x_2,..., x_n], which denotes a single input observation. The number of neurons (k) in the pattern layer is equal to the number of data sets used for training, and each neuron contains an input X. Equation (1) represents the Euclidean distance between the input and each neuron in the pattern layer. The symbols \(\sigma\), b and \(W_{i}\) in Equations (2) and (3) stand for spread parameter, bias, and the weight of each neuron in the pattern layer, respectively. The value of the spread parameter behaves like a regression action during network modeling, and will influence the predictive result greatly. The larger the value of \(\sigma\), the smoother output curve of the forecast outcome. By adjusting parameters \(\sigma\) and b, the output of each neuron in the pattern layer can be obtained. The summation layer of the GRNN in this study consists of only two neurons: the S- and D-summation neurons. The number of S-summation layers is equal to the number of the neuron of the output layer. It is packaged by the predicted CV from the pattern layer and manipulated by Equation (3). The number of the D-summation is a single value and stands for one. Only one \(y_{i}\) is the desired output (which is the confidence value (CV) of the machine's health status [11]) in this study. A GA is used to determine the optimal spread parameters \(\sigma\) and b in the GRNN. The relationship between the two is, in a way, a reciprocal one, as indicated in the exponential term of Equation (2). The final output value obtained as the weighted average is given in Equation (3).  
\[D_{i}^{2}=(X-P_{i})^{\text{T}}(X-P_{i}) \tag{1}\]  
\[W_{i}=\text{ exp}\,(\frac{-D_{i}^{2}b}{2\sigma^{2}}),\,0\,<\,\sigma,\,\text{ b}\,<\,1 \tag{2}\]  
\[\text{Y}=\frac{\text{S}}{D}=\frac{\sum_{i=1}^{n}y_{i}\exp{(\frac{-D_{i}^{2}}{2 \sigma^{2}})}}{\sum_{i=1}^{n}\exp{(\frac{-D_{i}^{2}}{2\sigma^{2}})}} \tag{3}\]  
The k-fold cross-validation method is used to segment the training data into modeling data and validation data. The training modeling and validation data are split into k groups, and each GRNN group model is trained at a different time. Further, by adjusting the values of the spread parameter \(\sigma\) and parameter b using the GA, the root mean square error (RMSE) value between the predicted output of the GRNN model and the corresponding output value of the training data are minimized; that is, the optimal \(\sigma\) and b of the GRNN model are obtained. Finally, the output of the entire GRNN model is obtained from the average of the output of the k GRNN models. Figure 2 presents a flow chart for determining the CV of a RUL using the GRNN model with k-fold cross-validation and GA optimization.