How Does Lasso Regression works?
Lasso regression is fundamentally an extension of linear regression. The goal of traditional linear regression is to minimize the sum of squared differences between the observed and predicted values in order to determine the line that best fits the data points. But the complexity of real-world data is not taken into account by linear regression, particularly when there are many factors.
1. Ordinary Least Squares (OLS) Regression: Lasso regression is very useful in this situation because adding a penalty term In lasso regression, minimize the sum of squared differences. The predictors’ coefficients’ absolute values serve as the basis for this penalty. The formula for OLS is:
min [Tex]RSS = Σ(yᵢ – ŷᵢ)²[/Tex]
Where,
- [Tex]y_i[/Tex] is the observed value.
- and [Tex]ŷᵢ[/Tex] is the predicted value for each data point i.
2. Penalty Term for Lasso Regression: The OLS equation is supplemented with a penalty term. This penalty term is the sum of the absolute values of the coefficients (also known as L1 regularization). The goal is now to minimize the penalty term plus the sum of squared differences:
[Tex]RSS + \lambda \times \sum |\beta_i|[/Tex]
Where,
- [Tex]\beta_i[/Tex] represents the coefficients of the predictors
- and [Tex]\lambda[/Tex] is the tuning parameter that controls the strength of the penalty. As lambda increases, more coefficients are pushed towards zero
3. Shrinking Coefficients: The penalty term in lasso regression have a unique characteristic with is its ability to reduce the coefficients of less significant variables to zero. As a result, features with zero coefficients are eliminated from the model, essentially performing variable selection. When working with high-dimensional data where there are many predictors relative to the number of observations this is especially helpful.
Lasso regression makes the model simpler and less prone to overfitting by reducing or deleting the coefficients of unimportant predictors. This improves the model’s readability and ability to be applied to fresh sets of data.
4. Selecting the optimal [Tex]\lambda[/Tex]: In lasso regression, selecting the tuning parameter lambda is essential. Frequently, cross-validation methods are employed to determine the ideal value of lambda that strikes a balance between predicted accuracy and model complexity.
The primary objective of Lasso regression is to minimize the residual sum of squares (RSS) along with a penalty term multiplied by the sum of the absolute values of the coefficients.
In the plot, the equation for the Lasso Regression of cost function, combines the residual sum of squares (RSS) and an L1 penalty on the coefficients [Tex]β_j[/Tex].
- RSS measures: The squared difference between the expected and actual values is measured.
- L1 penalty: Penalizes the coefficients’ absolute values, bringing some of them to zero and simplifying the model. The L1 penalty’s strength is managed via the lambda term. Stronger penalties result from greater lambdas, which may both increase the RSS and make the model sparser (having more coefficients equal to zero).
The graph itself shows the relationship between the value of lambda (x-axis) and the cost function (y-axis).
- y-axis: represents the value of the cost function, which Lasso Regression tries to minimize.
- Bottom axis (x-axis): represents the value of the lambda (λ) parameter, which controls the strength of the L1 penalty in the cost function.
- Green to orange curve: This curve depicts how the cost function (y-axis) changes with increasing lambda (x-axis). As lambda increases (moving to the right on the x-axis), the curve transitions from green to orange. This represents the cost function value going up (potentially due to a higher RSS term) as the L1 penalty becomes stronger (forcing more coefficients to zero).
What is lasso regression?
The Lasso Regression, a regression method based on Least Absolute Shrinkage and Selection Operator is quite an important technique in regression analysis for selecting the variables and regularization. It gets rid of irrelevant data features that help to prevent overfitting and features with weak influence become more cleanly identifiable because of shrinking the coefficients toward zero.
In this guide, we will understand core concepts of lasso regression as well as how it works to mitigate overfitting.
What is lasso regression?
- Understanding Lasso Regression
- Bias-Variance Tradeoff in Lasso Regression
- How Does Linear Regression works?
- When to use Lasso Regression
- Implementation of Lasso Regression
- Best Practices for Implementing Lasso Regression
- Advantages of Lasso Regression
- Disadvantages of Lasso Regression