Last Updated : 15 May, 2024

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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

## Understanding Lasso Regression

Lasso (Least Absolute Shrinkage and Selection Operator) regression typically belongs to regularization techniques category, which is usually applied to avoid overfitting. Lasso Regression enhance the linear regression concept by making use of a regularization process in the standard regression equation. Linear Regression operates by minimizing the sum of squared discrepancies between the observed and predicted values by fitting a line (or, in higher dimensions, a plane or hyperplane) to the data points.

However, multicollinearity a condition in which features have a strong correlation with one another occurs in real-words datasets. This is when the regularization approach of Lasso Regression comes in handy. Regularization, in simple term add penalty term to model, preventing it from overfitting.

For example, if you’re attempting with a model to forecast house prices based on features such as location, square footage, and the number of bedrooms. Lasso Regression will let us determine which feature is more important or whereas location and square footage are major determinants of price. By zeroing out the coefficient for the bedroom feature, Lasso simplifies the model in order to increase the model accuracy.

## Bias-Variance Tradeoff in Lasso Regression

The balance between bias (error resulting from oversimplified assumptions in the model) and variance (error resulting from sensitivity to little variations in the training data) is known as the bias-variance tradeoff.

While implementing lasso regression, the penalty term (L1 regularization) helps to significantly lowers the variance of the model by decreasing the coefficients of less significant features towards zero. By doing this, overfitting may be avoided. Hence, the model identifies noise in the training set rather than the underlying patterns. **However, as the model may become overly simplistic and unable to represent the true underlying relationships in the data, raising the regularization strength to reduce variance may also increase bias. **

Thus, bias and variance are traded off in lasso regression, just like in other regularization strategies. ** Achieving the ideal balance usually entails minimizing the total prediction error(MSE) by adjusting the regularization parameter using methods like **cross-validation

**.**## 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].

The squared difference between the expected and actual values is measured.**RSS measures:**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).**L1 penalty:**

The graph itself shows the relationship between the value of lambda (x-axis) and the cost function (y-axis).

represents the value of the cost function, which Lasso Regression tries to minimize.**y-axis:**represents the value of the lambda (λ) parameter, which controls the strength of the L1 penalty in the cost function.**Bottom axis (x-axis):**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).**Green to orange curve:**

## When to use Lasso Regression

When working with high-dimensional datasets that contain a large number of features some of which may be redundant or irrelevant, lasso regression is very helpful. Moreover, we can use lasso regression in following situations:

: By reducing the coefficients of less significant features to zero, Lasso regression automatically chooses a selection of features. When you have a lot of features and want to find the ones that are most significant, this is helpful.**Feature Selection**By reducing the coefficients of correlated variables and choosing one, lasso regression might be useful when there is multicollinearity—that is, when the predictor variables have a high degree of correlation with one another.**Collinearity:**: By penalizing big coefficients, Lasso regression can aid in preventing overfitting. When the number of predictors approaches or surpasses the number of observations, this becomes particularly significant.**Regularization**: Compared to conventional linear regression models that incorporate all features, lasso regression often yields sparse models with fewer non-zero coefficients. This could make the final model simpler to understand.**Interpretability**

Nevertheless, It is important to remember that lasso regression could not work effectively if the true underlying model is dense (many predictors having non-zero coefficients) and there are high correlations among the variables.

## Implementation of Lasso Regression

Lasso Regression can be implemented using various tools, like Python and R. Python offers a rich ecosystem of libraries for machine learning, commonly used is scikit-learn. Scikit-learn provides a user-friendly interface for executing Lasso Regression. Similar to Python, R also offers package for implementing Lasso Regression which is glmnet package** . **These tools offer convinient functions for data preprocessing, hyperparameter adjustment, and model evaluation for executing Lasso regression. You are free to select the tool that best suits your demands and level of programming language competence based on your preferences and unique requirements.

After Data Preprocessing, the following steps are necessary for implementing Lasso Regression:

: Lasso regression standardize the features before fitting the model to ensure no biasedness in comparison of their coefficients.**Feature Selection**: With the preprocessed data, we fit lasso regression to the model. During this step, the algorithm iteratively updates the coefficients to minimize the sum of squared differences between actual and predicted values, along with the penalty term.**Model Fitting**- Hyperparameter Tuning: Cross-validation techniques plays an role here for choosing the optimal alpha value that balances model complexity and predictive performance to controls the strength of the penalty term
: Evaluating the performance using appropriate metrics such as Mean Squared Error (MSE), R-squared, or others.**Evaluation**: Concluding the model results, paying close attention to the non-zero coefficients.**Interpretation****These coefficients indicate the importance of the corresponding features in predicting the target variable.**

For more refer to:

- Implementation of Lasso Regression From Scratch using Python
- Lasso Regression in R Programming

## Best Practices for Implementing Lasso Regression

: As Lasso struggles to handle multicollinear features, make sure all of the features are multicollinear.**Dealing with Multicollinearity**: To reduce dimensionality, group features that are highly connected or take into account methods like Principal Component Analysis (PCA).**Feature Classification**: Recognize the balance between variation and bias. Raising the alpha (stronger regularization) decreases variance but increases bias.**Balance between Variance and Bias**

## Advantages of Lasso Regression

Lasso regression eliminates the need to manually select the most relevant features, hence, the developed regression model becomes simpler and more explainable.**Feature Selection:**Lasso constrains large coefficients, so a less biased model is generated, which is robust and general in its predictions.**Regularization:**With lasso, models are often sparsity induced, therefore, they are easier to interpret and explain, which is essential in fields like health care and finance.**Interpretability:**Lasso lends itself to dealing with high-dimensional data like we have in genomic as well as imaging studies.**Handles Large Feature Spaces:**

## Disadvantages of Lasso Regression

Lasso, might arbitrarily choose one variable in a group of highly correlated variables rather than the other, thereby yielding a biased model in the end.**Selection Bias:**Lasso is demanding in the respect that features of different orders have a tendency to affect the regularization line and the model’s precision.**Sensitive to Scale:**Lasso can be easily affected by the outliers in the given data, resulting into the overfitting of the coefficients.**Impact of Outliers:**In the environment of multiple correlated variables the lasso’s selection of variable may be unstable, which results in different variable subsets each time in tiny data change.**Model Instability:**Analyzing different λ (alpha) values may be problematic and maybe solved by cross-validation.**Tuning Parameter Selection:**

## Conclusion

In conclusion, Lasso regression is one of the best splendid techniques of machine learning and statistics to build up models concisely and to handle high-dimensional data. Through introducing penalty terms to large coefficients, Lasso regulation adds the sparsity nature of the selection procedure in features. Identifying its sphere of influence and boundaries is of great value for the development of this approach in various fields of use. In the face of vast feature data, Lasso regression can be quite useful as it adds to the effectiveness of a model and its interpretability.

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