## 2 Theory and Method
### An Improved Logistic Regression Model with Double Adaptive Elastic Net
Define \(D_{\lambda_{1},\lambda_{2}}(k,l)=\frac{1}{n}\left|w_{2k}\hat{\mathbf{\beta}}_{k}( \lambda_{1},\lambda_{2})-w_{2l}\hat{\mathbf{\beta}}_{l}(\lambda_{1},\lambda_{2})\right|\), then:_  
\[D_{\lambda_{1},\lambda_{2}}(k,l)\leq\frac{\sqrt{2(1-\rho_{kl})}+\frac{\lambda _{1}}{n}\left|w_{1k}-w_{1l}\right|}{2\lambda_{2}} \tag{8}\]  
_where_  
\[\rho_{kl}=\text{corr}(x_{k},x_{l})\]  
By combining the above two schemes to improve the logistic regression equation, \(\mathbf{X}_{i}=(1,x_{i1},x_{i2},\cdot\cdot\cdot,x_{im})^{T}\), \(\mathbf{\beta}=(\beta_{0},\beta_{1},\beta_{2},\cdot\cdot\cdot,\beta_{m})^{T}\),\(y_{i}\in\{0,1\},i=1,2,\cdot\cdot\cdot,n\), its estimated value of \(\beta\) is:  
\[\hat{\mathbf{\beta}}_{\text{Daen}}=\underset{\mathbf{\beta}}{\text{argmin}}\left\{ \ln(l(\beta;X))+\lambda_{1}{\sum\limits_{j=1}^{m}w_{1j}\left|\hat{\mathbf{\beta} }_{j}\right|}+\lambda_{2}{\sum\limits_{j=1}^{m}w_{2j}\beta_{j}^{2}}\right\} \tag{9}\]  
where  
\[w_{1j}=\left|\hat{\mathbf{\beta}}_{j}^{*}\right|^{-\delta},w_{2j}>0,\lambda_{1}, \lambda_{2}>0,\delta>0\]  
\[\hat{\mathbf{\beta}}^{*}=\underset{\mathbf{\beta}}{\text{argmin}}\left\{\ln(l(\beta; X))+\lambda_{2}{\sum\limits_{j=1}^{m}w_{2j}\beta_{j}^{2}}\right\}. \tag{10}\]  
Since incorporating the correlation of variables into the regression model helps to improve the accuracy of parameter estimation and variable selection [27],  
\[w_{2j}=\frac{\sum\limits_{k=1,k\
eq j}^{m}\left|\rho_{kj}\right|}{m-1}+\varepsilon _{j},\]  
where \(\rho_{kj}=corr\left(x_{k},x_{j}\right)\) is the correlation coefficient between variables \(x_{k}\) and \(x_{j}\), \(\left(\varepsilon_{1},\varepsilon_{2},\cdot\cdot\cdot,\varepsilon_{m}\right)^{T}\) is the vector that can make \(w_{21},w_{22},\cdot\cdot\cdot,w_{2m}\) unequal to each other, and \(\frac{\sum_{i=1}^{m}\varepsilon_{j}}{m}=1,0.95\leq\varepsilon_{j}\leq 1.25\). Equation(9) is equivalent to:  
\[\hat{\mathbf{\beta}}_{\text{Daen}}=\underset{\beta}{\text{argmin}}\left\{\ln(l( \beta;X))\right\}, \tag{11}\]  
where  
\[\alpha=\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}},t>0.\]Using the coordinate gradient method and the Newton method to solve \(\beta\), Equation (9) can be rewritten as:  
\[a_{n}\left(\hat{\mathbf{\beta}}^{*}-\mathbf{\beta}_{0}\right)=\mathcal{O}_{p}(1) \text{and}\lim\limits_{n\rightarrow\infty}\frac{\lambda_{1}a_{n}^{\delta}}{ \sqrt{n}}=\infty;\]0  
where  
\[a_{n}\left(\hat{\mathbf{\beta}}^{*}-\mathbf{\beta}_{0}\right)=\mathcal{O}_{p}(1) \text{and}\lim\limits_{n\rightarrow\infty}\frac{\lambda_{1}a_{n}^{\delta}}{ \sqrt{n}}=\infty;\]1  
If \(\beta(t)\) is the solution of \(\beta\) at step t, then \(I(\beta)\) can be approximated as:  
\[a_{n}\left(\hat{\mathbf{\beta}}^{*}-\mathbf{\beta}_{0}\right)=\mathcal{O}_{p}(1) \text{and}\lim\limits_{n\rightarrow\infty}\frac{\lambda_{1}a_{n}^{\delta}}{ \sqrt{n}}=\infty;\]2  
where \(g(t)\) and \(\mathbf{h}(t)\) are the gradients of \(I(\beta)\) at \(\beta=\beta(t)\), respectively, and the Hessian matrix, adding \(\lambda_{1}\sum\limits_{j=1}^{m}w_{1j}\Big{|}\beta_{j}\Big{|}\) to the Equation (12) and making \(\frac{\partial I(\beta)}{\partial\beta}=0\) yields:  
\[a_{n}\left(\hat{\mathbf{\beta}}^{*}-\mathbf{\beta}_{0}\right)=\mathcal{O}_{p}(1) \text{and}\lim\limits_{n\rightarrow\infty}\frac{\lambda_{1}a_{n}^{\delta}}{ \sqrt{n}}=\infty;\]3  
where  
\[a_{n}\left(\hat{\mathbf{\beta}}^{*}-\mathbf{\beta}_{0}\right)=\mathcal{O}_{p}(1) \text{and}\lim\limits_{n\rightarrow\infty}\frac{\lambda_{1}a_{n}^{\delta}}{ \sqrt{n}}=\infty;\]4  
\[K(Q,W)= \begin{cases}Q-W,&0\leq W<Q\\ Q+W,&0\leq W<-Q\\ 0,&|Q|\leq W\end{cases}  \tag{14}\]  
Since \(\lambda_{1}\mathbf{h}^{-1}(t)\mathbf{W}_{1}\) may have some numbers less than 0, the parameters of some irrelevant variables cannot become 0. Thus, it can be directly rewritten as \(\lambda_{1}\mathbf{W}_{1}\). By the above inference, the solution process of \(\hat{\beta}_{\text{Danen}}\) can be derived: first generate an initial value of \(\beta\), then repeat the calculation of \(g(t)\), \(\mathbf{h}(t)\), and \(\beta(t+1)\), until convergence [26].