## 2 Theory and Method  
### An Improved Logistic Regression Model with Double Adaptive Elastic Net  
In this analysis of ancient glass artifacts, the relationship between glass weathering and its chemical composition was identified and statistically analyzed by using an improved logistic regression model based on a double adaptive elastic net, i.e., the Daen-Logistic regression (Daen-LR) model. Furthermore, we calculated the \(p\)-value of each correlation factor and counted whether each regression coefficient was significant at the 90% confidence level, extracted the strong correlation elements, and excluded the weak correlation elements. The characteristics of the model are as follows:  
Logistic regression is an effective method to solve classification problems in which effective estimation of parameters and selection of variables are extremely important. The regularization method [18], which considers adding a penalty term to the optimized loss function to estimate parameters, can simultaneously solve the two key points of logistic regression. Elastic net [19] is one of the representatives of this method. However, considering the inadequacy of the traditional logistic regression model for the estimation of parameters and the identification of important variables, it has two major shortcomings: First, the selected variables may not be consistent, i.e., they lack oracle properties [20]. Second, the specific effects of strongly correlated variables on the independent variables are not considered, i.e., adaptive categorical effects are missing [21; 22]. To overcome the first deficiency of Elastic net, adaptive elastic net is established by combining Adaptive lasso [23] and Ridge to achieve consistency in selected variables. However, the Adaptive coefficient vector W1, which makes an adaptive elastic net with oracle properties, is not easy to set correctly. It is generally determined by the initial estimates of parameters and the constant \(\delta\). To solve the second defect of Elastic net, [PERSON] et al. [24] proposed a Generalized ridge in which parameters are first divided into groups and then given different Ridge penalties for each group. The Generalized ridge has an adaptive grouping effect, and its Adaptive ridge also enjoys that effect. However, Generalized ridge does not have the function to select variables and is of limited application. Based on existing solutions to the Elastic net deficiency, it follows that Adaptive lasso and Adaptive ridge have oracle properties and adaptive grouping effects, respectively, so they can be combined to avoid the two existing disadvantages. This combination of penalties can be called the double adaptive elastic net. It is assumed that in the composition analysis of glass artifacts, there are \(m\) chemical composition influence factors, \(X=(x_{1},x_{2},\cdots,x_{m})\) is the characteristic variable of chemical composition content (i.e., independent variable), \(m\) is the number of variables, and the weathering status of the corresponding glass artifacts is set as \(y\) (i.e., dependent variable), where \(y\) represents the dichotomous variable of weathering or not (i.e., 0 means \"unwueh-eered\" and 1 means \"weathered\"). To assess the magnitude of the probability of whether a particular glass artifact is weathered, it is necessary to calculate the predicted outcome of the model as the probability of occurrence of \(y=1\), which can then be expressed as \(P=f(y=1\mid x_{1},x_{2},\cdots,x_{m})\), i.e., the mathematical expression of the traditional logistic regression model is:  
\[\text{Logit}(P)=\ln\frac{P(y=1)}{1-P(y=1)}=\beta_{0}+\beta_{1}x_{1}+\beta_{2 }x_{2}+\cdots+\beta_{m}x_{m} \tag{1}\]  
i.e.,  
\[P(y=1)=\frac{\exp(\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\cdots+\beta_{m}x_{ m})}{1+\exp(\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\cdots+\beta_{m}x_{m})}\]  
In the above equation, \((\beta_{0},\beta_{1},\beta_{2},\cdots,\beta_{m})\) are the regression coefficients to be determined.