## 3 Elastic Net  
Elastic net is a revised variable selection algorithm based on lasso. To overcome the deficiencies of traditional biased estimation methods, such as stepwise regression and ridge regression, in the selection of variables, such as poor anti-interference ability, [PERSON] proposed the lasso algorithm, which has had breakthrough significance [34,41,42]. There is the observed value \({\mathbf{Y}}\in R^{N\times 1}\) and an input matrix \({\mathbf{X}}\in R^{N\times m}\), and we approximate the regression function by a linear model \(\mathrm{E}(\mathrm{Y}|\mathrm{X}=x)={a_{0}}+{x^{T}}{\mathbf{a}}\). The regression model of lasso is:  
\[\left\{{\begin{array}{*{20}{l}}{{\dot{\mathbf{a}}}}=\min\limits_{{({a_{0}},{a})} \in R^{N\times n}}\left[{\frac{1}{2N}\sum\limits_{n=1}^{N}{\left({{y_{n}}-{\mathbf{ a}}_{0}}-{x_{n}^{T}}{\mathbf{a}}\right)^{2}}+\lambda{\left\|{{\mathbf{a}}}\right\|_{1}}} \right]}\\ {\left\|{{\mathbf{a}}}\right\|_{1}}=\sum\limits_{j=1}^the first term on the right is the least square term, and the second is the penalty term, where \(\mathbf{a}\) is the regression coefficient; \(\|\mathbf{a}\|_{1}\) is the L1 norm, which is the sum of the absolute values of all elements in the matrix and \(\lambda\) is a penalty parameter and has the effect of a compression variable, and its numerical value indicates the severity of punishment. By punishing the sum of the absolute value of the regression coefficient, lasso realizes the compression regression coefficient, making the partial regression coefficient strictly equal to zero. That makes the lasso sparse and avoids overfitting problems. Sparseness will make the lasso have two characteristics, one is feature selection. There are some elements in the input data that are not related to the observed values. When solving the minimum residual error, although this part of the elements will reduce the training error, this part of the unrelated elements will interfere with the prediction model when predicting new samples, resulting in overfitting. Feature selection can accurately screen out irrelevant information. The second characteristic is interpretability. Among the many input data, there are always some elements that have a decision-making influence on the observation value. The identification of this part can accurately describe the characteristics of the model and make the model principle highly transparent and clear, so that the model has a strong interpretation. However, the lasso cannot handle multicollinearity, and [PERSON] et al. and [PERSON] et al. proposed an elastic net based on the lasso to address these deficiencies. Elastic net is the first sparse model with an automatic group effect [34], and the model is:  
\[\left\{ \begin{array}{l}\overset{\wedge}{\mathbf{a}}=\min_{(a_{0},a)\in\mathbb{R }^{m}}\left[\frac{1}{2N}\sum\limits_{n=1}^{N}\left(y_{n}-\mathbf{a}_{0}-x_{n}^{T} \mathbf{a}\right)^{2}+\lambda_{1}\|\mathbf{a}\|_{1}+\lambda_{2}\|\mathbf{a}\|_{2}^{2}\right] \\ \|\mathbf{a}\|_{1}=\sum\limits_{j=1}^{m}|a_{j}|\\ \|\mathbf{a}\|_{2}^{2}=\sum\limits_{j=1}^{m}a_{j}^{2}\end{array} \right., \tag{9}\]  
where the first term on the right of the equal sign is the loss function and the second and third terms are penalty terms. \(\|\mathbf{a}\|_{2}^{2}\) is the L1 norm, which represents the sum of the squares of all elements of the matrix, which is the Euclidean distance. \(\lambda\) is a penalty parameter. Specifically, \(\lambda_{1}\) is a one-norm penalty and \(\lambda_{2}\) is a second-norm penalty. Both the L1 norm and L2 norm have the function of the compressive regression coefficient. Elastic net compresses the regression coefficient twice and leads to an increase in estimation bias. Therefore, it is necessary to improve the above formula.