## 2 Theory and Method
### Multiple Linear Regression Model  
Multiple linear regression (MLR) is a statistical method that predicts the distribution of the dependent variable by using multiple independent factors [15]. The goal of the MLR model is to establish linear links between independent and dependent characteristics that influence a given event, and it is an extension of classical least squares regression because it employs multiple explanatory factors.  
\[y=\alpha_{0}+\alpha_{1}x_{1}+\cdots\alpha_{i}x_{i}+\cdots+\alpha_{n}x_{n}+\mu_ {i} \tag{33}\]  
where \(y\) is the dependent variable, \(x_{1}\cdots x_{n}\) are the independent variables, \(\alpha_{0}\) is the \(y\) intercept, \(\alpha_{i}\) is the regression coefficient of the ith independent variable, and \(\mu_{i}\) is the model error, also known as the residual. The magnitude of the coefficient of determination(R\({}^{2}\))and the squared error(MSE)can be used to assess the predictive performance of the MLR model [15]:  
\[MSE=\frac{\sum\limits_{j=1}^{n}\left(y_{j}-\hat{y}_{j}\right)^{2}}{n} \tag{34}\]  
\[R^{2}=1-\frac{\sum\limits_{j=1}^{n}\left(y_{j}-\hat{y}_{j}\right)^{2}}{\sum \limits_{j=1}^{n}\left(y_{j}-\overline{y}_{j}\right)^{2}} \tag{35}\]  
\(y_{j}\) is the \(j\)th parameter after normalization,\(\hat{y}_{j}\) is the \(j\)th parameter predicted, \(\overline{y}_{j}\) is the mean of the predicted parameters, and \(n\) is the number of samples.  
We performed the BP test (Breusch and Pagan test) and the White test on the perturbation term \(\mu_{i}\) to see if there was heteroskedasticity. If the perturbation term is correlated with the independent variable, it may make the regression coefficients of the model inaccurate, thus leading to large errors in the results.  
In the BP test, it is assumed that the regression model is \(y_{i}=\beta_{1}+\beta_{2}x_{i2}+\cdots+\beta_{K}x_{iK}+\epsilon_{i}\), test the following original hypothesis:  
\[H_{0}:\text{E}\left(\epsilon_{i}^{2}\middle|x_{2},\cdots,x_{k}\right)=\sigma^{2}\]  
If \(H_{0}\) is not true, then the conditional variance \(\text{E}\left(\epsilon_{i}^{2}\middle|x_{2},\cdots,x_{k}\right)\) is a function of \((x_{2},\cdots,x_{k})\) and is called the conditional variance function. The BP test assumes that the conditional variance function is linear:  
\[\epsilon_{i}^{2}=\delta_{1}+\delta_{2}x_{i2}+\cdots+\delta_{K}x_{iK}+u_{i} \tag{36}\]The original hypothesis can be simplified to:  
\[H_{0}:\delta_{2}=\cdots=\delta_{K}=0\]  
If we assume that \(H_{0}\) is true, we can show that \(\varepsilon_{i}\) has no correlation with the independent variable \(x_{iK}\); that is, there is no autocorrelation, and the perturbation term has no heteroscedasticity. Since the perturbation term \(\varepsilon_{i}\) is not observable, the residual squared \(\varepsilon_{i}^{2}\) is used for auxiliary regression of the explanatory variable:  
\[e_{i}^{2}=\delta_{1}+\delta_{2}x_{i2}+\cdots+\delta_{K}x_{iK}+error_{i} \tag{37}\]  
\(nR^{2}\) statistics were used:  
\[nR^{2}\overset{d}{\rightarrow}x^{2}(K-1)\]  
\(R^{2}\) is the \(R^{2}\) of auxiliary regression. The difference between the White test and the BP test lies in that when the White test carries out auxiliary regression, there are \(x_{iK}\) square terms and cross terms in Equation (37), so the BP test can be regarded as a special case of the White test.  
In addition, we tested the model for multicollinearity, and the variance inflation factor VIF was used to eliminate the influence factors with multicollinearity, which improved the accuracy of the model:  
Assuming that there are \(k\) independent variables, then the variance inflation factor \(VIF_{n}=\frac{1}{1-R_{1+i/n}^{2}}\), \(R_{1-k/n}^{2}\) is the goodness of fit obtained by regressing the \(n\)-th independent variable as the dependent variable on the remaining \(k-1\) independent variables; the larger the \(VIF_{n}\), the greater the correlation between the \(n\)-th variable and the other variables [53]. If \(VIF_{n}\) is greater than 10, there is strict multicollinearity between the variables.