## How to Develop Elastic Net Regression Models in Python

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Regression is a modeling job that entails predicting a numeric worth given an enter.

Linear regression is the usual algorithm for regression that assumes a linear relationship between inputs and the goal variable. An extension to linear regression entails including penalties to the loss operate throughout coaching that encourage easier fashions which have smaller coefficient values. These extensions are known as regularized linear regression or penalized linear regression.

**Elastic web** is a well-liked sort of regularized linear regression that mixes two fashionable penalties, particularly the L1 and L2 penalty capabilities.

On this tutorial, you’ll uncover the right way to develop Elastic Web regularized regression in Python.

After finishing this tutorial, you’ll know:

- Elastic Web is an extension of linear regression that provides regularization penalties to the loss operate throughout coaching.
- The right way to consider an Elastic Web mannequin and use a remaining mannequin to make predictions for brand spanking new information.
- The right way to configure the Elastic Web mannequin for a brand new dataset by way of grid search and mechanically.

Let’s get began.

## Tutorial Overview

This tutorial is split into three components; they’re:

- Elastic Web Regression
- Instance of Elastic Web Regression
- Tuning Elastic Web Hyperparameters

## Elastic Web Regression

Linear regression refers to a mannequin that assumes a linear relationship between enter variables and the goal variable.

With a single enter variable, this relationship is a line, and with increased dimensions, this relationship may be regarded as a hyperplane that connects the enter variables to the goal variable. The coefficients of the mannequin are discovered by way of an optimization course of that seeks to reduce the sum squared error between the predictions (*yhat*) and the anticipated goal values (*y*).

- loss = sum i=Zero to n (y_i – yhat_i)^2

An issue with linear regression is that estimated coefficients of the mannequin can grow to be massive, making the mannequin delicate to inputs and probably unstable. That is significantly true for issues with few observations (*samples*) or extra samples (*n*) than enter predictors (*p*) or variables (so-called *p >> n issues*).

One method to addressing the soundness of regression fashions is to vary the loss operate to incorporate extra prices for a mannequin that has massive coefficients. Linear regression fashions that use these modified loss capabilities throughout coaching are referred to collectively as penalized linear regression.

One fashionable penalty is to penalize a mannequin based mostly on the sum of the squared coefficient values. That is referred to as an L2 penalty. An L2 penalty minimizes the dimensions of all coefficients, though it prevents any coefficients from being faraway from the mannequin.

- l2_penalty = sum j=Zero to p beta_j^2

One other fashionable penalty is to penalize a mannequin based mostly on the sum of absolutely the coefficient values. That is referred to as the L1 penalty. An L1 penalty minimizes the dimensions of all coefficients and permits some coefficients to be minimized to the worth zero, which removes the predictor from the mannequin.

- l1_penalty = sum j=Zero to p abs(beta_j)

Elastic web is a penalized linear regression mannequin that features each the L1 and L2 penalties throughout coaching.

Utilizing the terminology from “The Elements of Statistical Learning,” a hyperparameter “*alpha*” is offered to assign how a lot weight is given to every of the L1 and L2 penalties. Alpha is a worth between Zero and 1 and is used to weight the contribution of the L1 penalty and one minus the alpha worth is used to weight the L2 penalty.

- elastic_net_penalty = (alpha * l1_penalty) + ((1 – alpha) * l2_penalty)

For instance, an alpha of 0.5 would offer a 50 p.c contribution of every penalty to the loss operate. An alpha worth of Zero provides all weight to the L2 penalty and a worth of 1 provides all weight to the L1 penalty.

The parameter alpha determines the combo of the penalties, and is usually pre-chosen on qualitative grounds.

— Web page 663, The Elements of Statistical Learning, 2016.

The profit is that elastic web permits a steadiness of each penalties, which may end up in higher efficiency than a mannequin with both one or the opposite penalty on some issues.

One other hyperparameter is offered referred to as “*lambda*” that controls the weighting of the sum of each penalties to the loss operate. A default worth of 1.Zero is used to make use of the absolutely weighted penalty; a worth of Zero excludes the penalty. Very small values of lambada, corresponding to 1e-Three or smaller, are frequent.

- elastic_net_loss = loss + (lambda * elastic_net_penalty)

Now that we’re conversant in elastic web penalized regression, let’s have a look at a labored instance.

## Instance of Elastic Web Regression

On this part, we are going to show the right way to use the Elastic Web regression algorithm.

First, let’s introduce an ordinary regression dataset. We’ll use the housing dataset.

The housing dataset is an ordinary machine studying dataset comprising 506 rows of information with 13 numerical enter variables and a numerical goal variable.

Utilizing a check harness of repeated stratified 10-fold cross-validation with three repeats, a naive mannequin can obtain a imply absolute error (MAE) of about 6.6. A top-performing mannequin can obtain a MAE on this identical check harness of about 1.9. This supplies the bounds of anticipated efficiency on this dataset.

The dataset entails predicting the home worth given particulars of the home’s suburb within the American metropolis of Boston.

No must obtain the dataset; we are going to obtain it mechanically as a part of our labored examples.

The instance under downloads and masses the dataset as a Pandas DataFrame and summarizes the form of the dataset and the primary 5 rows of information.

# load and summarize the housing dataset from pandas import read_csv from matplotlib import pyplot # load dataset url = ‘https://uncooked.githubusercontent.com/jbrownlee/Datasets/grasp/housing.csv’ dataframe = read_csv(url, header=None) # summarize form print(dataframe.form) # summarize first few strains print(dataframe.head()) |

Working the instance confirms the 506 rows of information and 13 enter variables and a single numeric goal variable (14 in complete).

We are able to additionally see that every one enter variables are numeric.

(506, 14) 0 1 2 3 4 5 … 8 9 10 11 12 13 0 0.00632 18.0 2.31 0 0.538 6.575 … 1 296.0 15.3 396.90 4.98 24.0 1 0.02731 0.0 7.07 0 0.469 6.421 … 2 242.0 17.8 396.90 9.14 21.6 2 0.02729 0.0 7.07 0 0.469 7.185 … 2 242.0 17.8 392.83 4.03 34.7 3 0.03237 0.0 2.18 0 0.458 6.998 … 3 222.0 18.7 394.63 2.94 33.4 4 0.06905 0.0 2.18 0 0.458 7.147 … 3 222.0 18.7 396.90 5.33 36.2
[5 rows x 14 columns] |

The scikit-learn Python machine studying library supplies an implementation of the Elastic Web penalized regression algorithm by way of the ElasticNet class.

Confusingly, the *alpha* hyperparameter may be set by way of the “*l1_ratio*” argument that controls the contribution of the L1 and L2 penalties and the *lambda* hyperparameter may be set by way of the “*alpha*” argument that controls the contribution of the sum of each penalties to the loss operate.

By default, an equal steadiness of 0.5 is used for “*l1_ratio*” and a full weighting of 1.Zero is used for alpha.

... # outline mannequin mannequin = ElasticNet(alpha=1.0, l1_ratio=0.5) |

We are able to consider the Elastic Web mannequin on the housing dataset utilizing repeated 10-fold cross-validation and report the typical imply absolute error (MAE) on the dataset.

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# consider an elastic web mannequin on the dataset from numpy import imply from numpy import std from numpy import absolute from pandas import read_csv from sklearn.model_selection import cross_val_score from sklearn.model_selection import RepeatedKFold from sklearn.linear_model import ElasticNet # load the dataset url = ‘https://uncooked.githubusercontent.com/jbrownlee/Datasets/grasp/housing.csv’ dataframe = read_csv(url, header=None) information = dataframe.values X, y = information[:, :–1], information[:, –1] # outline mannequin mannequin = ElasticNet(alpha=1.0, l1_ratio=0.5) # outline mannequin analysis technique cv = RepeatedKFold(n_splits=10, n_repeats=3, random_state=1) # consider mannequin scores = cross_val_score(mannequin, X, y, scoring=‘neg_mean_absolute_error’, cv=cv, n_jobs=–1) # power scores to be optimistic scores = absolute(scores) print(‘Imply MAE: %.3f (%.3f)’ % (imply(scores), std(scores))) |

Working the instance evaluates the Elastic Web algorithm on the housing dataset and studies the typical MAE throughout the three repeats of 10-fold cross-validation.

Your particular outcomes could range given the stochastic nature of the educational algorithm. Contemplate operating the instance a number of instances.

On this case, we are able to see that the mannequin achieved a MAE of about 3.682.

We could resolve to make use of the Elastic Web as our remaining mannequin and make predictions on new information.

This may be achieved by becoming the mannequin on all out there information and calling the *predict()* operate, passing in a brand new row of information.

We are able to show this with a whole instance, listed under.

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# make a prediction with an elastic web mannequin on the dataset from pandas import read_csv from sklearn.linear_model import ElasticNet # load the dataset url = ‘https://uncooked.githubusercontent.com/jbrownlee/Datasets/grasp/housing.csv’ dataframe = read_csv(url, header=None) information = dataframe.values X, y = information[:, :–1], information[:, –1] # outline mannequin mannequin = ElasticNet(alpha=1.0, l1_ratio=0.5) # match mannequin mannequin.match(X, y) # outline new information row = [0.00632,18.00,2.310,0,0.5380,6.5750,65.20,4.0900,1,296.0,15.30,396.90,4.98] # make a prediction yhat = mannequin.predict([row]) # summarize prediction print(‘Predicted: %.3f’ % yhat) |

Working the instance suits the mannequin and makes a prediction for the brand new rows of information.

Subsequent, we are able to have a look at configuring the mannequin hyperparameters.

## Tuning Elastic Web Hyperparameters

How do we all know that the default hyperparameters of alpha=1.Zero and l1_ratio=0.5 are any good for our dataset?

We don’t.

As a substitute, it’s good follow to check a collection of various configurations and uncover what works greatest.

One method can be to gird search *l1_ratio* values between Zero and 1 with a 0.1 or 0.01 separation and *alpha* values from maybe 1e-5 to 100 on a log-10 scale and uncover what works greatest for a dataset.

The instance under demonstrates this utilizing the GridSearchCV class with a grid of values we’ve outlined.

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# grid search hyperparameters for the elastic web from numpy import arange from pandas import read_csv from sklearn.model_selection import GridSearchCV from sklearn.model_selection import RepeatedKFold from sklearn.linear_model import ElasticNet # load the dataset url = ‘https://uncooked.githubusercontent.com/jbrownlee/Datasets/grasp/housing.csv’ dataframe = read_csv(url, header=None) information = dataframe.values X, y = information[:, :–1], information[:, –1] # outline mannequin mannequin = ElasticNet() # outline mannequin analysis technique cv = RepeatedKFold(n_splits=10, n_repeats=3, random_state=1) # outline grid grid = dict() grid[‘alpha’] = [1e–5, 1e–4, 1e–3, 1e–2, 1e–1, 0.0, 1.0, 10.0, 100.0] grid[‘l1_ratio’] = arange(0, 1, 0.01) # outline search search = GridSearchCV(mannequin, grid, scoring=‘neg_mean_absolute_error’, cv=cv, n_jobs=–1) # carry out the search outcomes = search.match(X, y) # summarize print(‘MAE: %.3f’ % outcomes.best_score_) print(‘Config: %s’ % outcomes.best_params_) |

Working the instance will consider every mixture of configurations utilizing repeated cross-validation.

You would possibly see some warnings that may be safely ignored, corresponding to:

Goal didn’t converge. You would possibly need to improve the variety of iterations. |

Your particular outcomes could range given the stochastic nature of the educational algorithm. Attempt operating the instance a number of instances.

On this case, we are able to see that we achieved barely higher outcomes than the default 3.378 vs. 3.682. Ignore the signal; the library makes the MAE destructive for optimization functions.

We are able to see that the mannequin assigned an alpha weight of 0.01 to the penalty and focuses completely on the L2 penalty.

MAE: -3.378 Config: {‘alpha’: 0.01, ‘l1_ratio’: 0.97} |

The scikit-learn library additionally supplies a built-in model of the algorithm that mechanically finds good hyperparameters by way of the ElasticNetCV class.

To make use of this class, it’s first match on the dataset, then used to make a prediction. It is going to mechanically discover applicable hyperparameters.

By default, the mannequin will check 100 alpha values and use a default ratio. We are able to specify our personal lists of values to check by way of the “*l1_ratio*” and “*alphas*” arguments, as we did with the handbook grid search.

The instance under demonstrates this.

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# use mechanically configured elastic web algorithm from numpy import arange from pandas import read_csv from sklearn.linear_model import ElasticNetCV from sklearn.model_selection import RepeatedKFold # load the dataset url = ‘https://uncooked.githubusercontent.com/jbrownlee/Datasets/grasp/housing.csv’ dataframe = read_csv(url, header=None) information = dataframe.values X, y = information[:, :–1], information[:, –1] # outline mannequin analysis technique cv = RepeatedKFold(n_splits=10, n_repeats=3, random_state=1) # outline mannequin ratios = arange(0, 1, 0.01) alphas = [1e–5, 1e–4, 1e–3, 1e–2, 1e–1, 0.0, 1.0, 10.0, 100.0] mannequin = ElasticNetCV(l1_ratio=ratios, alphas=alphas, cv=cv, n_jobs=–1) # match mannequin mannequin.match(X, y) # summarize chosen configuration print(‘alpha: %f’ % mannequin.alpha_) print(‘l1_ratio_: %f’ % mannequin.l1_ratio_) |

Your particular outcomes could range given the stochastic nature of the educational algorithm. Attempt operating the instance a number of instances.

Once more, you would possibly see some warnings that may be safely ignored, corresponding to:

Goal didn’t converge. You would possibly need to improve the variety of iterations. |

On this case, we are able to see that an alpha of 0.Zero was chosen, eradicating each penalties from the loss operate.

That is completely different from what we discovered by way of our handbook grid search, maybe because of the systematic means during which configurations had been searched or chosen.

alpha: 0.000000 l1_ratio_: 0.470000 |

## Additional Studying

This part supplies extra assets on the subject in case you are seeking to go deeper.

### Books

### APIs

### Articles

## Abstract

On this tutorial, you found the right way to develop Elastic Web regularized regression in Python.

Particularly, you realized:

- Elastic Web is an extension of linear regression that provides regularization penalties to the loss operate throughout coaching.
- The right way to consider an Elastic Web mannequin and use a remaining mannequin to make predictions for brand spanking new information.
- The right way to configure the Elastic Web mannequin for a brand new dataset by way of grid search and mechanically.

**Do you may have any questions?**

Ask your questions within the feedback under and I’ll do my greatest to reply.

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