## 4 Process and Result
### Identifying Unknown Artifact Types Based on Multiple Linear Regression Model  
Through the results of Sections 4.1 and 4.2, we conducted chemical content testing and analysis on a batch of newly excavated glass artifacts, as shown in Table 4, and judged the categories to which they belonged by the correlation of related elements and weathering effects. Firstly, the chemical elements with significant correlation in each category were initially screened out by the statistical law of chemical element content, and the multiple linear regression equation between elements and categories was established to find out the experimental values of categories and make errors and fits with the actual values of categories, and to verify the accuracy of the model. Figure 3: (**a**) ACF and PACF patterns of SiO\({}_{2}\) content in high potassium; (**b**) ACF and PACF patterns of SiO\({}_{2}\) content in lead-barium glasses. According to the classification rules of chemical content and surface weathering, it can be initially concluded that K\({}_{2}\)O, CaO, MgO, Al\({}_{2}\)O\({}_{3}\), FeO, PbO, BaO, and P\({}_{2}\)O\({}_{5}\) have strong correlations with surface weathering and categories, while the remaining elements have weak correlations, so the remaining elements can be deleted. In addition, because the chemical element contents of the three heavily weathered glass artifacts are very different from other contents, which will have a large impact on the analysis of the model, they are treated as outliers. In the multiple linear regression equation, the qualitative data should be set as dummy variables, so the qualitative variables (unweathered and weathered) in the surface weathering independent variable \(Suw\) can be set as quantitative variables  
Figure 4: Correlation curve of chemical element content before and after weathering: (**a**) (High potassium) SiO\({}_{2}\); (**b**) (lead barium) SiO\({}_{2}\); (**c**) (High potassium) K\({}_{2}\)O; (**d**) (lead barium) P\({}_{2}\)O\({}_{5}\); (**e**) (lead barium) PbO. (0 and 1), and the qualitative variables (high-potassium and lead-barium) in the discriminatory category dependent variable \(y_{i}\) can be set as quantitative variables (\(A\) and \(B\)), and the following multiple linear regression equation can be established as:  
\[ \begin{array}{ll}y_{i}=&a_{0}+\alpha_{1}x_{Sii}+\alpha_{2}x_{Ki}+\alpha_{3}x_{ Cai}+\alpha_{4}x_{Mgi}+\alpha_{5}x_{Ali}\\ &+\alpha_{6}x_{Fi}+\alpha_{7}x_{Pbi}+\alpha_{8}x_{Bai}+\alpha_{9}x_{Pi}+\beta Suw _{i}+\mu_{i}\\ \\ Suw_{i}=1\text{ denotes the }i-\text{th weathering sample}\\ Suw_{i}=0\text{ denotes the }i-\text{th unweathered sample}\\ E(y\mid Suw=1\text{ and other independent variables})=\beta\times 1+m(Constants)\\ E(y\mid Suw=0\text{ and other independent variables})=\beta\times 0+m(Constants)\\ \end{array}  \tag{39}\]  
The joint significance test indicators for the F-statistic [59; 60] for the above model results are as follows:  
F(10,55) represents the F joint statistic test value of 51.41, the confidence interval is 95%, and the original hypothesis \(H_{0}\) is: \(a1=a2=a3=\cdots=a9=\beta=0\). From Table 5, we can see that the \(p\)-value is 0, \(p\) is less than 0.05, and at this time the original hypothesis is rejected. We have reason to believe that the correlation coefficient is significantly different from 0, so we can consider this model to be useful. The regression coefficients and corresponding \(p\)-values for the variables of interest can be derived as in Table 6. Only when the p-value is less than 0.05, we consider it significant, and the regression coefficient is credible at this point, so we can use the regression coefficients corresponding to K\({}_{2}\)O, Al\({}_{2}\)O\({}_{3}\), PbO, BaO, and \(Suw\) (the dummy variable \"weathering\"), and the larger the absolute value of the regression coefficients, the greater the effect on the dependent variable.