Generalising Beyond Linearity: A Look at Basis Expansions and Penalised Regression

Introduction

Traditional linear regression models are powerful for understanding relationships between predictors and target variables. However, in many real-world datasets, the underlying patterns are non-linear, making basic linear models insufficient. Basis expansions and penalised regression techniques bridge this gap by allowing models to generalise beyond strict linearity while preventing overfitting.

For learners enrolled in a data scientist course in Ahmedabad, mastering these concepts is crucial. They form the foundation for building flexible, interpretable, and high-performing predictive models across diverse applications like finance, healthcare, marketing, and engineering.

The Limitations of Simple Linear Regression

Linear regression assumes that the relationship between independent variables (features) and the dependent variable (target) is strictly linear. This assumption creates challenges when:

• The underlying relationship is non-linear. 
• Predictors interact in complex ways. 
• Multicollinearity among variables leads to unstable estimates. 
• Overfitting occurs in high-dimensional datasets. 

To overcome these limitations, we need methods like basis expansions and penalised regression to balance flexibility with model control.

What Are Basis Expansions?

Basis expansions transform the original input features into new sets of variables, enabling the model to capture non-linear relationships. Instead of fitting a straight line, the model uses transformed features to approximate complex patterns.

1. Polynomial Basis Expansion

Involves creating higher-order polynomial terms for features:

y=β0​+β1​x+β2​x2+β3​x3+⋯+ϵ

• Advantage: Captures curves and interactions efficiently. 
• Limitation: High-degree polynomials risk overfitting and extreme variance. 

2. Splines and Piecewise Basis

Splines divide the input range into segments, fitting lower-degree polynomials within each.

• Natural Splines: Reduce edge variability. 
• B-splines: Offer smooth transitions between segments. 
• Use Case: Popular in time-series forecasting and demand modelling. 

3. Fourier Basis Expansion

Uses sine and cosine functions to model cyclical patterns.

• Ideal for periodic datasets, like seasonal sales or temperature data. 

4. Interaction Basis Functions

Captures relationships between variables by adding terms like x₁ × x₂.

• Critical for fields like marketing analytics, where the combined effect of promotions and pricing drives sales. 

Students of a data scientist course in Ahmedabad gain hands-on experience applying these techniques to datasets, learning to choose the right transformation for a given problem.

The Challenge: Overfitting in Expanded Models

While basis expansions improve flexibility, adding too many transformed features increases model complexity, causing:

• Poor generalisation to new data. 
• Inflated variance and unstable coefficients. 
• Computational inefficiency in large datasets. 

This is where penalised regression becomes essential.

Penalised Regression: Balancing Fit and Complexity

Penalised regression methods introduce a penalty term to the regression objective, discouraging overly complex models.

1. Ridge Regression (L2 Regularisation)

Adds the squared sum of coefficients to the loss function:

L=∑(yi​−y^​i​)2+λ∑βj

• Effect: Shrinks coefficients but doesn’t eliminate them. 
• Best for: Multicollinearity and scenarios where all features contribute somewhat. 

2. Lasso Regression (L1 Regularisation)

Adds up the absolute value of coefficients to the loss:

L=∑(yi​−y^​i​)2+λ∑∣βj​∣

• Effect: Forces some coefficients to zero, performing feature selection. 
• Best for: High-dimensional datasets with many irrelevant features. 

3. Elastic Net Regression

Combines L1 and L2 penalties, balancing feature selection and stability.

• Use Case: Works well when features are highly correlated. 

4. Generalised Additive Models (GAMs)

GAMs integrate basis expansions and penalisation, modelling each predictor as a smooth non-linear function:

y=β0​+f1​(x1​)+f2​(x2​)+…+ϵ

• Advantage: Interpretable, flexible, and avoids overfitting through smoothness penalties. 

Applications of Basis Expansions and Penalised Regression

1. Healthcare Predictive Modelling

• Predict disease progression by modelling non-linear effects of biomarkers. 
• Penalised regression ensures stable predictions in high-dimensional genomic datasets. 

2. Financial Risk Scoring

• Capture non-linear credit behaviour patterns in loan defaults. 
• Lasso regression filters out irrelevant financial indicators. 

3. Marketing Analytics

• Use interaction terms to measure the combined effect of discounts and advertising. 
• Basis expansions improve demand forecasts for seasonal products. 

4. Energy and Climate Modelling

• Fourier expansions track temperature cycles. 
• Ridge regression stabilises predictions under highly correlated weather variables. 

Best Practices for Implementing These Techniques

1. Start Simple, Scale Gradually

• Begin with basic polynomial expansions before introducing advanced splines or Fourier transformations. 

2. Cross-Validation for Hyperparameter Tuning

• Use k-fold cross-validation to select the optimal penalty term (λ). 

3. Combine Domain Knowledge

• Avoid irrelevant feature expansions by aligning transformations with real-world behaviours. 

4. Automate Feature Selection

• Use Lasso or Elastic Net for automatic elimination of redundant predictors. 

5. Monitor Model Stability

• Evaluate performance across training, validation, and test datasets to detect variance issues. 

Tools and Libraries to Use

• Python: scikit-learn, statsmodels, pyGAM 
• R: caret, glmnet, mgcv 
• Visualisation: matplotlib, seaborn for assessing basis function impacts 
• Deployment: Integrate optimised models into CI/CD pipelines for production-ready solutions.
  

Learners in a data scientist course in Ahmedabad practice these tools through real-world capstone projects, preparing them to build scalable, production-grade models.

Case Study: E-Commerce Price Optimisation

Scenario:
An e-commerce company wanted to model customer response to pricing changes.

Approach:

• Applied polynomial basis expansions to model non-linear pricing effects. 
• Used lasso regression to eliminate irrelevant variables like secondary page visits. 
• Deployed the model into production to recommend dynamic pricing strategies. 

Outcome:

• Improved conversion rates by 24%. 
• Reduced overfitting by tuning penalty parameters via cross-validation. 
• Enhanced revenue predictability across seasonal sales events. 

Future Trends

1. AI-Driven Basis Expansions

Neural networks will automate the creation of optimal basis functions for complex data.

2. Sparse High-Dimensional Modelling

Advanced penalisation methods like group lasso and fused lasso will dominate research.

3. Integration with Explainable AI

Basis functions combined with GAMs will provide interpretable insights alongside predictive power.

4. Real-Time Penalisation in Big Data Pipelines

Streaming frameworks will incorporate adaptive penalisation to update models continuously.

Conclusion

Generalising beyond linearity using basis expansions and penalised regression unlocks the ability to model real-world complexities effectively. These techniques balance flexibility, interpretability, and stability, making them indispensable for modern data scientists.

For aspiring professionals, a data scientist course in Ahmedabad provides practical experience applying these concepts to real-world projects, presenting you with the expertise needed to design high-performing, scalable, and robust predictive models.