Lasso Regression Made Simple: A Beginners Guide

Lasso Regression

Lasso regression is a type of regularized linear regression model. The main purpose of using lasso regression is to tackle the issue of overfitting of a model.

The concept behind lasso regression is to minimize the sum of the absolute errors ( MAE ) by adding an extra penalty term to the error function, known as a regularization penalty. The regularization penalty is used to limit the number of parameters that can be used. This helps to reduce overfitting by forcing the model to consider only the most important features.

The two main benefits of using lasso regression are feature selection and shrinkage. The lasso penalty shrinks the less important feature’s coefficient to zero. This effectively does feature selection. It also reduces the variance by shrinking the unnecessary variables and makes the selected features more interpretable.

The limitations of lasso regression are that it cannot handle large feature sets, as it can only select a few features and can be computationally expensive. It is also very sensitive to outliers and can suffer from high bias due to regularization.

A few examples of where lasso regression can be used are in spam filtering, credit scoring, medical diagnosis, and stock market predictions.

A case study of lasso regression is an Assignment from the ML class of the Coursera Specialization on Machine Learning from the University of Washington. In this assignment, the task is to use lasso regression to predict house prices. The lasso regression was able to select important features from the data set and produced a good model to predict house prices.

Mathematical Formulation:

The objective of LASSO regression is to minimize a penalized version of Least Squares linear regression so that the resulting fit is more robust to outliers and more meaningful to interpret. It is expressed as:

Minimize:

W(l) = SUM[ (yi — SUM[XiWj])² + λ*|w|]

Where l is the regularization parameter that controls the sparsity of the model.

Algorithm Steps:

• Standardize predictor variables by subtracting the means of each predictor variable and dividing by the standard deviation for all predictors in the dataset.

• Initialize the parameter vector W with all 0s.

• Loop through all predictors in the dataset and fit a linear regression model for each predictor while keeping all other predictors constant.

• Calculate the coefficient of determination (R2) for each predictor and select the predictor with the highest R2 value.

• Increase the parameter value (Wj) associated with the predictor selected in step 4 and then recalculate the R2 value for all predictors.

• Continue to loop through these two steps until the R2 value for all predictors does not increase anymore.

• Calculate the penalized sum of squared errors (SSE) as the objective function to be minimized.

Parameter Tuning and Cross-Validation Techniques:

Cross-validation is a method to measure how well the LASSO regression model is doing in predicting data instances that it hasn’t seen during training. It is a model selection method that splits the dataset into a training set and an independent test set. For parameter tuning, Grid Search or Random Search is used to search for the best combination of parameters, which is defined in terms of the best R2 score.

Practical Tips:

• Start with a standard linear regression model to get an initial understanding of the datasets. Use its Coefficient of Determination (R2) value as a baseline measure for model performance.

• Use the least correlated predictors to build the model, to reduce the number of correlated predictors that might be throwing off the LASSO model.

• Set an appropriate regularization penalty (lambda value) and implement a grid search to find the best lambda value.

• Scores from the cross-validation should be used to tune the model.

• Evaluate the model performance against the baseline R2 value.

Code Snippets for Python:

First import the necessary packages — sklearn, numpy, and pandas.

#Import packages
import numpy as np
from sklearn.linear_model import Lasso
import pandas as pd

#Import data
data = pd.read_csv('data.csv')
X = data.iloc[:, :-1]
y = data.iloc[:, -1]

#Initialize Lasso
lasso_reg = Lasso(alpha=0.5)

#Fit the model
lasso_reg.fit(X, y)

#Calculate the coefficients
coef = lasso_reg.coef_

#Predict
preds = lasso_reg.predict(X)

#Calculate R2 Score
r2_score = lasso_reg.score(X, y)