# Cable Temperature Prediction Based on RF-GPR for Digital Twin Applications
## 3 RF-GPR Prediction Model of Cable Temperature
### Construction of RF-GPR Cable Temperature Prediction Model  
In the calculation of the importance score of the characteristic variables, it can be seen that among the 10-dimensional input characteristic variables, 6 input characteristic variables were more than 4% important in model training, and 4 input characteristic variables were less than 2% important in model training. Six characteristic variables with an important degree of more than 4% were selected as new input characteristic variables. The temperature prediction model of RF-GPR cable was constructed by using the Gaussian process regression algorithm in the form of the series mixing model. Gaussian process regression is a machine learning regression method developed in recent years. Its principle is based on statistical learning theory. It is a non-parametric model that takes the Gaussian process as a priori. The basic idea of Gaussian process regression is to map low-dimensional characteristic variables to high-dimensional characteristic variables and then carry out regression analysis on the processed high-dimensional characteristic variables. Gaussian process regression has good characteristics for dealing with nonlinear models and small sample data and is widely used in parameter control, model prediction and time series analysis [29]. During the construction of the cable temperature prediction model based on Gaussian process regression, the data set was first processed according to the normalization requirements, and the result after processing is shown in Formula (10). \[D=\{(x_{i},T_{i})\},i=1,2\cdots n \tag{14}\]  
where \(x_{i}\) is a group of six-dimensional input characteristic variables, and \(T_{i}\) is the output temperature value of the corresponding group. Then, the basis function, kernel function, kernel size and Sigma parameters of the Gaussian process regression model were optimized. The basis function has decisive factors on the specific form of the prior mean function of the Gaussian process regression model. Generally, zero value function, constant function and linear function can be selected. Choosing different kernel functions affects the accuracy of the Gaussian process prediction model. The mean function \(m(x)\) and covariance function \(k(x,x^{\prime})\) together constitute the kernel functions in the Gaussian process regression, and the kernel functions affect all the statistical characteristics of Gaussian process regression. Among them, the covariance function can be divided into four types: the Matern covariance function, square exponential covariance function, periodic covariance function and rational quadratic covariance function. Sigma is used to set the initial value of the standard deviation of observed noise [30]. For the characteristic variable type of the cable temperature output calculated by the finite element method in this paper, the optimizer type in Gaussian process regression model was set as Bayesian optimization, the basis function type as linear function, the mean function as zero mean function and the kernel function type as square exponential covariance function. The specific expression is shown as follows:  
\[k\big{(}x_{i},x_{j}\big{)}=\sigma_{f}^{2}exp\left[-\sum_{m=1}^{n}\frac{\big{(} x_{im}-x_{jm}\big{)}^{2}}{2\sigma_{m}^{2}}\right] \tag{15}\]  
where \(x_{im}\) and \(x_{jm}\) are the variable in vectors \(x_{i}\) and \(x_{j}\), respectively; \(\sigma_{f}\) is the variance; \(\sigma_{m}\) is the variance scale; and \(\sigma_{f}\) and \(\sigma_{m}\) constitute the hyperparameters in the covariance function. After setting the specific parameters of the Gaussian process regression prediction model, the regression expression of the prediction model is obtained, as shown in Equation (16). \[y\sim GP\big{(}0,k\big{(}x,x^{\prime}\big{)}\big{)} \tag{16}\]  
Finally, the input data set of the Gaussian process regression prediction model is trained and tested.