Contents

# Elastic Net

Elastic Net is a method for modeling relationship between a dependent variable (which may be a vector) and one or more explanatory variables by fitting regularized least squares model. Elastic Net regression model has the special penalty, a sum of
L
1
and
L
2
regularizations, hat takes advantage of both Ridge Regression and LASSO algorithms. This penalty is particularly useful in a situation with many correlated predictor variables [Friedman2010].

## Details

Let (
x
1
, ...,
x
p
) be a vector of input variables and
y
= (
y
1
, ...,
y
k
) be the response. For each
j
= 1, ...,
k
, the Elastic Net model has the form similar to linear and ridge regression models [Hoerl70] with one exception: the coefficients are estimated by minimizing mean squared error (MSE) objective function that is regularized by
L
1
and
L
2
penalties.
Here
x
i
,
i
= 1, ...,
p
are referred to as independent variables,
y
j
,
is referred to as dependent variable or response.
Training Stage
Let (
x
11
, ...,
x
1
p
,
y
11
, ...,
y
1
k
), ..., (
x
n
1
, ...,
x
np
,
y
n
1
, ...,
y
nk
) be a set of training data (for regression task,
n
>>
p
, and for feature selection
p
could be greater than
n
). The matrix
X
of size
n
x
p
contains observations
x
ij
,
i
= 1, ...,
n
,
j
= 1, ...,
p
of independent variables.
For each
y
j
,
j
= 1, ...,
k
, the Elastic Net regression estimates
by minimizing the objective function:
In the equation above, the first term is a mean squared error function, the second and the third are regularization terms that penalize the
L
1
and
L
2
norms of vector
, where
For more details, see [Hastie2009] and [Friedman2010].
By default, Coordinate Descent iterative algorithm is used for minimization of the objective function. SAGA solver is also applicable for minimization. See Analysis > Optimization Solvers > Iterative Solvers.
Prediction Stage
Prediction based on Elastic Net regression is done for input vector (
x
1
, ...,
x
p
) using the equation
where

#### Product and Performance Information

1

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Notice revision #20110804