## 2 Theory and Method
### An Improved Logistic Regression Model with Double Adaptive Elastic Net
The great likelihood estimation method is used to find these coefficients:  
\[P(y=1\mid X)=1-\frac{1}{1+\exp(\beta_{n}+\beta_{1}x_{1}+\beta_{2}x_{2}+\cdots+ \beta_{m}x_{m})}=\pi \tag{2}\]  
\[P(y=0\mid X)=1-\pi \tag{3}\]  
Combining the probability functions of \(y\) as:  
\[P(y_{i})=\pi^{y_{i}}(1-\pi)^{y_{i}},y_{i}=0,1;i=1,2,\cdots,n \tag{4}\]  
According to the Bernoulli distribution, the maximum likelihood function can be expressed as:  
\[l(\beta;X)=\prod_{i=1}^{n}P(y_{i})=\prod_{i=1}^{n}\pi_{i}^{y_{i}}(1-\pi_{i})^{ 1-y_{i}} \tag{5}\]  
The log-likelihood function is expressed as:  
\[\ln(l(\beta;X))=\sum_{i=1}^{n}\{y_{i}(\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{i2 }+\cdots+\beta_{m}x_{im})\,-\ln[1+\exp(\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{ i2}+\cdots+\beta_{m}x_{im})]\} \tag{6}\]  
Since the maximum likelihood function is a convex function, the point at which its first derivative equals zero is the point of maximum value. By calculating the first derivative of the undetermined coefficient \((\beta_{0},\beta_{1},\beta_{2},\cdots,\beta_{m})\) in Equation (6) and setting it equal to zero, all the parameters to be solved in the equation group can be solved. Considering the shortcomings and deficiencies of the traditional logistic regression model, the oracle effect and adaptive classification effect are integrated into the traditional logistic model to create a double adaptive elastic net model, which makes the identification results of glass cultural relics more relevant and persuasive [25; 26]. **Theory 1. **_Oracle property_  
_In the Logistic model, suppose the real parameters \(\beta_{0}=(\beta_{01},\beta_{02},\cdots,\beta_{0m})^{\text{T}}\), \(\mathcal{A}=\left\{j\mid\beta_{0j}\
eq 0\right\}\)_  
\(=\{1,2,\cdots,m_{0}\},m_{0}<m,\) _[PERSON] information matrix \(I(\beta_{0})= \begin{pmatrix}I_{11}&I_{12}\\ I_{21}&I_{22}\end{pmatrix} \), where \(I_{11}\) is a square matrix of order \(m_{0},\phi\left(\mathbf{X}^{T}\beta\right)=\ln\left(1+\mathbf{e}^{\mathbf{X} ^{T}\beta}\right)\), then the double adaptive elastic net logistic has oracle property according to the following conditions._  
1. \(I(\beta_{0})\) Is a positive definite matrix. 2. There exists an open set containing \(\beta_{0}\), such that for any \(\beta\in\Omega\) there exists a function \(N(\cdot)\) satisfying: \[\mid\phi^{-}\left(\mathbf{X}^{T}\beta\right)\mid\leq N(\mathbf{X})<\infty,\] (7)and for any m-dimensional vector u, we have \(E\left(N(\mathbf{X})\left(\mathbf{X}^{T}\mathbf{u}\right)^{3}\right)<\infty\);
3. \(\lambda_{1}=o(\sqrt{n})\), and there is a sequence \(\{a_{n}\}\), such that \[a_{n}\left(\hat{\mathbf{\beta}}^{*}-\mathbf{\beta}_{0}\right)=\mathcal{O}_{p}(1) \text{and}\lim\limits_{n\rightarrow\infty}\frac{\lambda_{1}a_{n}^{\delta}}{ \sqrt{n}}=\infty;\]
4. \(\lambda_{2}=o(n)\) and \(\lim\limits_{n\rightarrow\infty}\frac{\lambda_{2}}{\sqrt{n}}\sum\limits_{j=1 }^{m_{0}}\hat{\rho}_{0j}^{2}=0\). When conditions 1-4 hold, double adaptive elastic net estimate \(\hat{\mathbf{\beta}}\) has the following properties: 1. \(\sqrt{n}\left(\hat{\mathbf{\beta}}_{\mathcal{A}}-\mathbf{\beta}_{\mathcal{A}}\right) \overset{D}{\rightarrow}N\left(0,\text{I}_{11}^{-1}\right)\);
2. \(\lim\limits_{n\rightarrow\infty}P\left(\hat{\mathbf{\beta}}_{\mathcal{A}^{*}}=0 \right)=1\). **Theorem 2**. : _Adaptive Classification Effect_  
_Given the binary data \(\{(\mathbf{X}_{i},y_{i})\}_{i=1}^{n}\), where \(\mathbf{X}_{i}=\left(x_{i1},x_{i2},\cdot\cdot\cdot,x_{im}\right)^{T}\) and \(\forall j\in\{1,2,\cdot\cdot\cdot,m\}\), \(\sum\limits_{i=1}^{n}x_{ij}=0\), \(\sum\limits_{i=1}^{n}x_{ij}=1,y_{i}\in\{0,1\}\). Let \(\hat{\mathbf{\beta}}(\lambda_{1},\lambda_{2})\) be the estimate of the model and assume that \(\hat{\mathbf{\beta}}_{k}(\lambda_{1},\lambda_{2})\hat{\mathbf{\beta}}_{l}(\lambda_{1}, \lambda_{2})>0\).