Draft:Elastic-net distribution

Elastic-net distribution

In statistics , particularly in fitting linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L₁ and L₂ penalties of the lasso and ridge methods. Hui Zou and Trevor Hastie introduce this penalty. Then Hassan M. Aljohani and his supervisor Dr. Robert G. Aykroyd wrote the penalty as distribution.

Definitions

The elastic net distribution is combined between LASSO and RIDGE, which can be written as

$P(\theta )={\frac {1}{Z(\kappa ,\gamma )}}\exp\{-\kappa (\gamma {\theta }^{2}+(1-\gamma )|\theta |)\},\;\;\theta \in \mathbb {R} ;\;\;\kappa >0,\;\;0\leq \gamma \leq 1,$

where

$Z(\kappa ,\gamma )=\left\{{\begin{array}{lr}{2}/{\kappa },&\gamma =0\\{\sqrt {\frac {4\pi }{\kappa \gamma }}}\exp {\bigg \{}{\frac {1}{4\gamma }}\kappa (1-\gamma )^{2}{\bigg \}}{\bigg (}1-\Phi ({\frac {\kappa (1-\gamma )}{\sqrt {2\kappa \gamma }}}){\bigg )},&0<\gamma <1\\{\sqrt {{\pi }/{\kappa }}},&\gamma =1.\end{array}}\right.$

The elastic net method includes LASSO and ridge regression; in other words, each is a special case where, $\gamma =0$, $0<\gamma <1$ or$\gamma =0$.

Examples of where the elastic net method has been applied are:

• Processing image
• Vibration