## 2 The Proposed Combined Method and Principles
### Principle of GPR  
GPR is a non-parametric regression method based on Bayesian theory, and its core idea is to view data points as a whole Gaussian process. To apply Gaussian processes to regression analysis, the regression function \(\{f(x)|x\in D\}\) can be regarded as a Gaussian process [20]. According to the properties of Gaussian processes, a Gaussian process can be fully determined by its mean function \(m(x)\) and kernel function \(k(x,x^{\prime})\). Regarding \(f(x)\) as a Gaussian process means that for any selected number of points from D, the probability distribution of \(f(x)\) follows a Gaussian distribution, as represented by formula (7).  
\[f(x)\,\sim\,GP(m(x),k(x,x^{\prime})) \tag{7}\]  
The mean function and the covariance function can be expressed as represented by formula (8).  
\[\begin{cases}\begin{aligned} m(x)=\mathbb{E}(f(x))\\ k(x,x^{\prime})=\mathbb{E}[(m(x)-f(x))(m(x^{\prime})-f(x^{\prime}))]\end{aligned}\end{cases} \tag{8}\]  
Although Gaussian process regression does not require specifying the form of the regression function, it is necessary to determine the forms of the mean function and the kernel function. In GPR, selecting an appropriate kernel function is crucial as it determines how the model interprets the relationships between data points. In this article, the mean function a is set to 0, and the Gaussian kernel is chosen as the kernel function for GPR, as shown in formula (9).  
\[k(x,x^{\prime})=\sigma^{2}\exp\left(-\frac{\left\|x-x^{\prime}\right\|^{2}}{2l ^{2}}\right) \tag{9}\]  
where \(\sigma^{2}\) is the variance parameter, determining the overall range of function values; \(l\) is the length scale parameter, controlling the rate at which function values change with the distance between input points.  
For practical modeling, the observed values should be the sum of the regression function \(f(x)\) and an independently and identically distributed Gaussian white noise \(\varepsilon\), as shown in formula (10).  
\[y=f(x)+\varepsilon,\varepsilon\sim N(0,\sigma_{n}^{2}) \tag{10}\]  
Therefore, the prior distribution of the noisy observed values is  
\[\mathbf{y}\sim N(0,\mathbf{K}(\mathbf{x},\mathbf{x})+\sigma_{n}^{2}\mathbf{I }_{n}). \tag{11}\]  
where vector \(\mathbf{y}\) refers to the observed values \([y_{1},y_{2},y_{3},\ldots,y_{n}]\); vector \(\mathbf{x}\) refers to the input vector \([x_{1},x_{2},x_{3},\ldots,x_{n}]\); and \(\mathbf{K}(\mathbf{x},\mathbf{x})=\big{(}k(x_{i},x_{j})\big{)}_{n\times p}\) is a covariance matrix, where each matrix element is computed through the kernel function, which is specifically represented by formula (8). \(\mathbf{I}_{n}\) is an n-dimensional identity matrix, while \(\sigma_{n}^{2}\mathbf{I}_{n}\) represents the covariance matrix of the noise.  
In this GPR model, \(\sigma_{n}^{2}\) is a parameter, and in formula (8), there are also parameters \(\sigma^{2}\) and \(l\). These parameters can all be obtained by maximizing the marginal likelihood function. Denoting all the parameters in the kernel function as \(\theta=[\sigma^{2},l]\), the logarithm of the marginal likelihood (LML) is given by formula (12).  
\[LML=\log p(\mathbf{y}\mid\mathbf{x},\theta,\sigma_{n}^{2})=\frac{1}{2}\log( \det(\mathbf{K}(\mathbf{x},\mathbf{x})+\sigma_{n}^{2}\mathbf{I}_{n}))-\frac{ 1}{2}\mathbf{y}^{T}[\mathbf{K}(\mathbf{x},\mathbf{x})+\sigma_{n}^{2}\mathbf{I }_{n}]^{-1}\mathbf{y}-\frac{n}{2}\log 2\pi \tag{12}\]  
By applying maximum likelihood estimation to the training dataset \(\mathbf{x}\) and \(\mathbf{y}\), the parameters in the GPR model can be determined. Once the GPR model has been established, predictions can be made on the test dataset \(\mathbf{x}^{*}\).