# statsmodels.regression.linear_model.OLS.fit_regularized¶

OLS.fit_regularized(method='elastic_net', alpha=0.0, L1_wt=1.0, start_params=None, profile_scale=False, refit=False, **kwargs)[source]

Return a regularized fit to a linear regression model.

Parameters: method (string) – ‘elastic_net’ and ‘sqrt_lasso’ are currently implemented. alpha (scalar or array-like) – The penalty weight. If a scalar, the same penalty weight applies to all variables in the model. If a vector, it must have the same length as params, and contains a penalty weight for each coefficient. L1_wt (scalar) – The fraction of the penalty given to the L1 penalty term. Must be between 0 and 1 (inclusive). If 0, the fit is a ridge fit, if 1 it is a lasso fit. start_params (array-like) – Starting values for params. profile_scale (bool) – If True the penalized fit is computed using the profile (concentrated) log-likelihood for the Gaussian model. Otherwise the fit uses the residual sum of squares. refit (bool) – If True, the model is refit using only the variables that have non-zero coefficients in the regularized fit. The refitted model is not regularized. distributed (bool) – If True, the model uses distributed methods for fitting, will raise an error if True and partitions is None. generator (function) – generator used to partition the model, allows for handling of out of memory/parallel computing. partitions (scalar) – The number of partitions desired for the distributed estimation. threshold (scalar or array-like) – The threshold below which coefficients are zeroed out, only used for distributed estimation A RegularizedResults instance.

Notes

The elastic net uses a combination of L1 and L2 penalties. The implementation closely follows the glmnet package in R.

The function that is minimized is:

$0.5*RSS/n + alpha*((1-L1\_wt)*|params|_2^2/2 + L1\_wt*|params|_1)$

where RSS is the usual regression sum of squares, n is the sample size, and $$|*|_1$$ and $$|*|_2$$ are the L1 and L2 norms.

For WLS and GLS, the RSS is calculated using the whitened endog and exog data.

Post-estimation results are based on the same data used to select variables, hence may be subject to overfitting biases.

The elastic_net method uses the following keyword arguments:

maxiter : int
Maximum number of iterations
cnvrg_tol : float
Convergence threshold for line searches
zero_tol : float
Coefficients below this threshold are treated as zero.

The square root lasso approach is a variation of the Lasso that is largely self-tuning (the optimal tuning parameter does not depend on the standard deviation of the regression errors). If the errors are Gaussian, the tuning parameter can be taken to be

alpha = 1.1 * np.sqrt(n) * norm.ppf(1 - 0.05 / (2 * p))

where n is the sample size and p is the number of predictors.

The square root lasso uses the following keyword arguments:

zero_tol : float
Coefficients below this threshold are treated as zero.

References

Friedman, Hastie, Tibshirani (2008). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software 33(1), 1-22 Feb 2010.

A Belloni, V Chernozhukov, L Wang (2011). Square-root Lasso: pivotal recovery of sparse signals via conic programming. Biometrika 98(4), 791-806. https://arxiv.org/pdf/1009.5689.pdf