Lasso and Elastic Net Regression

Ordinary least squares chooses coefficients to minimize squared prediction errors. Lasso modifies that objective by adding a penalty proportional to the sum of absolute coefficient magnitudes. The penalty encourages shrinkage, and because the absolute-value penalty has a corner at zero, some fitted coefficients can become exactly zero. This creates a sparse model that may be easier to store or inspect. Sparsity is not proof that the retained predictors are causal or uniquely important. Elastic net combines the L1 penalty with a squared L2 penalty. Its mixing parameter controls the balance: the pure L1 endpoint is lasso, while the pure L2 endpoint is ridge in common parameterizations. The L2 part can help share weight across correlated predictors instead of arbitrarily retaining one and discarding the rest. The overall strength parameter controls how much shrinkage is applied. Exact software parameter conventions differ, so read the library documentation. Consider a hypothetical model with two nearly interchangeable measures of outreach. Lasso may retain one and zero the other; small changes to the training sample could reverse that choice. Elastic net may retain both with reduced coefficients. Either behavior can be useful depending on whether compactness, prediction stability, or interpretability is the goal. If the outcome is measured on very different scales or the features use different units, the penalty's effect is uneven unless preprocessing is appropriate. Choose penalty settings using cross-validation within the training data, keeping the final test set untouched until model selection is complete. Scaling, imputation and feature selection must be fitted separately inside each training fold to avoid leakage. For time-dependent data, use splits that respect chronology. Evaluate predictive performance and inspect coefficient stability across reasonable resamples. Lasso and elastic net are tools for regularized linear prediction; they do not correct a wrong functional form, biased data, or a mismatch between the training population and intended use.