Ridge Regression vs Lasso Regression: A Practical Guide for House Price Prediction

Introduction

Linear regression is a fundamental technique in data science that models relationships between variables. In house price prediction, we have features like house size, number of bedrooms, and location to estimate prices. While basic linear regression works well in simple scenarios, it often struggles with real-world complexities like noisy data, correlated features, and overfitting. This article explores two powerful solutions to these problems: Ridge and Lasso regression.

1. Ordinary Least Squares (OLS) - The Foundation

What is OLS?

Ordinary Least Squares (OLS) is the standard method for training linear regression models. It works by finding the line (or hyperplane in higher dimensions) that minimizes the sum of squared differences between predicted and actual values.

Objective: Minimize the sum of squared residuals:

Loss = Σ(y_actual - y_predicted)²

The Overfitting Problem

Imagine predicting house prices using not only relevant features (size, location) but also irrelevant ones (color of front door, street name). OLS will try to use all these features to fit the training data perfectly. This creates two problems:

1. Unstable coefficients: Small changes in data cause large coefficient swings
2. Poor generalization: The model memorizes training data noise instead of learning patterns

Example: If we include "house number" as a feature, OLS might find patterns that don't generalize to new houses.

2. The Power of Regularization

Solving the Overfitting Problem

Regularization adds a penalty term to the loss function that discourages overly complex models. Think of it as adding "training wheels" to prevent the model from overcomplicating itself.

Why Penalties Help:

• They shrink coefficients toward zero
• They reduce model variance
• They improve generalization to new data

3. Ridge Regression (L2 Regularization)

The Ridge Loss Function

Loss = Σ(y_actual - y_predicted)² + λ * Σ(coefficients²)

Where λ (lambda) controls regularization strength - higher λ means more penalty.

How L2 Penalty Works

Ridge adds the sum of squared coefficients to the loss. This:

• Shrinks all coefficients proportionally
• Never sets coefficients exactly to zero
• Works like a gentle pull toward zero

Why No Feature Selection?

Because squaring small coefficients makes them even smaller but never zero. All features remain in the model, just with reduced influence.

4. Lasso Regression (L1 Regularization)

The Lasso Loss Function

Loss = Σ(y_actual - y_predicted)² + λ * Σ|coefficients|

How L1 Differs from L2

Instead of squaring coefficients, Lasso uses absolute values. This subtle change has dramatic effects:

• Creates "corner solutions" in optimization
• Can set coefficients exactly to zero
• Performs automatic feature selection

Why Zero Coefficients Matter

When a coefficient hits zero, that feature is completely removed from the model. Lasso automatically selects only the most important features - perfect for identifying which house characteristics truly matter.

5. Ridge vs Lasso: Key Differences

Aspect	Ridge (L2)	Lasso (L1)
Feature Selection	No - keeps all features	Yes - can eliminate features
Coefficient Behavior	Shrinks evenly, never zero	Can shrink to exactly zero
Interpretability	All features remain, harder to interpret	Fewer features, simpler model
Best For	Many useful features	Few important features

6. House Price Prediction Application

Scenario A: All Features Contribute

If we believe all our features (size, bedrooms, distance, schools) genuinely affect price, Ridge regression is preferable. It will use all available information while preventing any single feature from dominating unreasonably.

Why Ridge? It preserves all features while controlling their influence.

Scenario B: Few Important Features

If many features are noisy or irrelevant (like "neighbor's car color"), Lasso regression excels. It will identify and keep only the truly important predictors while eliminating noise.

Why Lasso? It acts like a feature detective, separating signal from noise and giving us a simpler, more interpretable model.

7. Model Evaluation Strategies

Detecting Overfitting

Train-Test Split Method:

1. Split data into training (80%) and testing (20%) sets
2. Train model on training data
3. Compare performance:
   • Good: Similar performance on both sets
   • Overfit: Much better on training than testing
   • Underfit: Poor performance on both

Example: If your model predicts training houses perfectly but fails on new houses, it's overfitting.

The Role of Residuals

Residuals (errors) = Actual price - Predicted price

What Residuals Tell Us:

• Patterned residuals: Model missing something (maybe non-linear relationships)
• Random residuals: Good model fit
• Large residuals: Poor predictions

Residual analysis helps diagnose whether our regularization is working properly.

Conclusion

Choosing between Ridge and Lasso depends on your problem context:

• Use Ridge when you believe most features contribute meaningfully
• Use Lasso when you suspect many features are irrelevant
• Use OLS only with few features and plenty of clean data

For house price prediction, Lasso often works well because only certain features (size, location, bedrooms) strongly influence prices, while others (exact age in days, specific street names) add mostly noise. Regularization techniques give us the control we need to build models that generalize well from training data to real-world predictions.

The goal isn't perfect training performance, but accurate predictions on houses we haven't seen before. Regularization helps us achieve this balance between complexity and generalizability.