# `SparseLasso`: Sparse Solutions for the Lasso

## Introduction

SparseLasso provides a Scikit-Learn based estimation of the Lasso with cross-validation tuning for the penalty choice using the ‘one standard error’ rule to yield sparse solutions. The ‘one standard error’ rule recognizes the fact that the cross-validation path is estimated with error and selects the more parsimonious model (see Hastie, Tibshirani and Friedman, 2009). This rule thus chooses the largest possible penalty which is still within the one standard error of the cross-validation optimal value. Given that the Lasso often selects too many variables in practice, the one standard error rule provides a practical solution to yield sparser models. The software implementation of this rule is readily available in the R-package ‘glmnet’ (Friedman, Hastie and Tibshirani, 2010), however, it is absent from the Scikit-Learn module (Pedregosa et al., 2011). SparseLasso provides estimation of the penalized linear and logistic model based on Scikit-Learn’s LassoCV and LogisticRegressionCV, respectively and thus accepts the standard Scikit-Learn arguments.

## Installation

`SparseLasso` module relies on Python 3 and is based on the `scikit-learn` module. The required modules can be installed by navigating to the root of this project and executing the following command: `pip install -r requirements.txt`.

## Example

The example below demonstrates the basic usage of the `SparseLasso` module.

```# import modules
import pandas as pd
import numpy as np
from sklearn.datasets import make_regression
from sklearn.linear_model import LassoCV

# import SparseLasso
from sparse_lasso import SparseLassoCV

# simulate some example data for the linear model
X, y, coef = make_regression(n_samples=1000,
n_features=100,
n_informative=10,
noise=10,
coef=True,
random_state=0)

# estimate standard LassoCV with optimal lambda minimizing error
lasso_min = LassoCV(n_alphas=100, cv=10).fit(X=X, y=y)

# estimate SparseLassoCV with lambda using 1 standard error rule
lasso_1se = SparseLassoCV(n_alphas=100, cv=10).fit(X=X, y=y)

# compare the penalty values
print('Lasso Min Penalty: ', round(lasso_min.alpha_, 2), '\n',
'Lasso 1se Penalty: ', round(lasso_1se.alpha, 2), '\n')

# compare the number of selected features
print('Lasso Min Number of Selected Variables:     ',
np.sum((lasso_min.coef_ != 0) * 1), '\n',
'Lasso 1se Number of Selected Variables:     ',
np.sum((lasso_1se.coef_ != 0) * 1), '\n')```

For a more detailed example see the `sparse_lasso_example.py` as well as the `sparse_lasso_simulation.py` for a simulation exercise comparing the optimal cross-validation penalty choice with the one standard error rule for variable selection.

## References

• Hastie, Trevor, Robert Tibshirani, and J H. Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. , 2009. Print.
• Friedman, Jerome, Trevor Hastie, and Rob Tibshirani. “Regularization paths for generalized linear models via coordinate descent.” Journal of statistical software 33.1 (2010): 1.
• Pedregosa, Fabian, et al. “Scikit-learn: Machine learning in Python.” the Journal of machine Learning research 12 (2011): 2825-2830.

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