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## Lasso and Elastic Net

### What Are Lasso and Elastic Net?

Lasso is a regularization technique. Use `lasso` to:

• Reduce the number of predictors in a regression model.

• Identify important predictors.

• Select among redundant predictors.

• Produce shrinkage estimates with potentially lower predictive errors than ordinary least squares.

Elastic net is a related technique. Use elastic net when you have several highly correlated variables. `lasso` provides elastic net regularization when you set the `Alpha` name-value pair to a number strictly between `0` and `1`.

For lasso regularization of regression ensembles, see `regularize`.

### Lasso and Elastic Net Details

#### Overview of Lasso and Elastic Net

Lasso is a regularization technique for performing linear regression. Lasso includes a penalty term that constrains the size of the estimated coefficients. Therefore, it resembles ridge regression. Lasso is a shrinkage estimator: it generates coefficient estimates that are biased to be small. Nevertheless, a lasso estimator can have smaller mean squared error than an ordinary least-squares estimator when you apply it to new data.

Unlike ridge regression, as the penalty term increases, lasso sets more coefficients to zero. This means that the lasso estimator is a smaller model, with fewer predictors. As such, lasso is an alternative to stepwise regression and other model selection and dimensionality reduction techniques.

Elastic net is a related technique. Elastic net is a hybrid of ridge regression and lasso regularization. Like lasso, elastic net can generate reduced models by generating zero-valued coefficients. Empirical studies have suggested that the elastic net technique can outperform lasso on data with highly correlated predictors.

#### Definition of Lasso

The lasso technique solves this regularization problem. For a given value of λ, a nonnegative parameter, `lasso` solves the problem

`$\underset{{\beta }_{0},\beta }{\mathrm{min}}\left(\frac{1}{2N}\sum _{i=1}^{N}{\left({y}_{i}-{\beta }_{0}-{x}_{i}^{T}\beta \right)}^{2}+\lambda \sum _{j=1}^{p}|{\beta }_{j}|\right).$`
• N is the number of observations.

• yi is the response at observation i.

• xi is data, a vector of p values at observation i.

• λ is a positive regularization parameter corresponding to one value of `Lambda`.

• The parameters β0 and β are scalar and p-vector respectively.

As λ increases, the number of nonzero components of β decreases.

The lasso problem involves the L1 norm of β, as contrasted with the elastic net algorithm.

#### Definition of Elastic Net

The elastic net technique solves this regularization problem. For an α strictly between 0 and 1, and a nonnegative λ, elastic net solves the problem

`$\underset{{\beta }_{0},\beta }{\mathrm{min}}\left(\frac{1}{2N}\sum _{i=1}^{N}{\left({y}_{i}-{\beta }_{0}-{x}_{i}^{T}\beta \right)}^{2}+\lambda {P}_{\alpha }\left(\beta \right)\right),$`

where

`${P}_{\alpha }\left(\beta \right)=\frac{\left(1-\alpha \right)}{2}{‖\beta ‖}_{2}^{2}+\alpha {‖\beta ‖}_{1}=\sum _{j=1}^{p}\left(\frac{\left(1-\alpha \right)}{2}{\beta }_{j}^{2}+\alpha |{\beta }_{j}|\right).$`

Elastic net is the same as lasso when α = 1. As α shrinks toward 0, elastic net approaches `ridge` regression. For other values of α, the penalty term Pα(β) interpolates between the L1 norm of β and the squared L2 norm of β.

### References

 Tibshirani, R. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, Vol 58, No. 1, pp. 267–288, 1996.

 Zou, H. and T. Hastie. Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society, Series B, Vol. 67, No. 2, pp. 301–320, 2005.

 Friedman, J., R. Tibshirani, and T. Hastie. Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, Vol 33, No. 1, 2010. `https://www.jstatsoft.org/v33/i01`

 Hastie, T., R. Tibshirani, and J. Friedman. The Elements of Statistical Learning, 2nd edition. Springer, New York, 2008.

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