ordinary_least_squares

Ordinary Least Squares regression with comprehensive diagnostics, robust alternatives, and best practices.

Overview

This Jupyter notebook provides an in-depth exploration of Ordinary Least Squares (OLS) regression, covering theoretical foundations, practical implementation, model diagnostics, and advanced techniques. The notebook serves as both an educational resource and a practical toolkit for regression analysis.

Features

1. Theoretical Foundation

• Mathematical formulation of OLS with proper LaTeX formatting
• Gauss-Markov theorem and assumptions
• Geometric interpretation as orthogonal projection
• Statistical inference (t-tests, F-tests, confidence intervals)

2. Comprehensive Model Diagnostics

• Residual Analysis: Six diagnostic plots including:

  • Residuals vs Fitted
  • QQ plot for normality
  • Scale-Location plot
  • Residuals vs Order
  • ACF plot for autocorrelation
  • Residual distribution histogram

• Statistical Tests:

  • Durbin-Watson for autocorrelation
  • Jarque-Bera for normality
  • Breusch-Pagan for heteroscedasticity
  • Rainbow test for linearity

3. Advanced Regression Techniques

• Non-linear relationships linear in parameters

• Categorical variables with dummy encoding

• Multicollinearity analysis:

  • Variance Inflation Factors (VIF)
  • Condition number calculation
  • Correlation matrix visualization

• Model comparison and selection:

  • Information criteria (AIC, BIC)
  • Likelihood ratio tests
  • Nested model comparison

4. Outlier and Influence Diagnostics

• Cook's Distance calculation and visualization
• DFFITS and DFBETAS analysis
• Leverage points identification
• Comprehensive influence measures

5. Robust Regression Methods

• Comparison with Huber regression
• Comparison with Bisquare (Tukey) regression
• Performance evaluation with outliers
• Scale parameter comparison

6. Practical Implementation

• Multiple approaches to OLS calculation
• Numerical stability considerations
• Cross-validation techniques
• Model validation best practices

Key Highlights

Diagnostic Tools

• VIF Calculation: Automatic detection of multicollinearity
• Influence Metrics: Cook's D, DFFITS, DFBETAS with threshold guidelines
• Residual Diagnostics: Complete visualization suite
• Statistical Tests: Comprehensive assumption checking

Robust Alternatives

• Huber Regression: M-estimation with Huber's T norm
• Bisquare Regression: Tukey's biweight estimation
• Performance Comparison: Side-by-side evaluation with OLS

Best Practices

• Model Selection: AIC, BIC, and cross-validation
• Assumption Checking: Complete diagnostic workflow
• Outlier Treatment: Identification and handling strategies
• Reporting Standards: Complete results interpretation

Requirements

Python Libraries

• numpy (1.21.0+)
• pandas (1.3.0+)
• matplotlib (3.4.0+)
• statsmodels (0.13.0+)
• scipy (1.7.0+)
• scikit-learn (0.24.0+)

Installation

pip install numpy pandas matplotlib statsmodels scipy scikit-learn

Usage

Basic OLS Estimation

import statsmodels.api as sm
import numpy as np

# Generate sample data
nsample = 100
x = np.linspace(0, 10, 100)
X = np.column_stack((x, x**2))
X = sm.add_constant(X)
beta = np.array([1, 0.1, 10])
y = np.dot(X, beta) + np.random.normal(size=nsample)

# Fit model
model = sm.OLS(y, X)
results = model.fit()
print(results.summary())

Running Diagnostics

The notebook includes a comprehensive diagnostic section that automatically:

1. Generates six diagnostic plots
2. Performs statistical tests for assumptions
3. Identifies influential observations
4. Provides interpretation guidelines

Model Comparison

Compare different model specifications using:

• Information criteria (AIC/BIC)
• Likelihood ratio tests
• Cross-validation performance

Notebook Structure

1. Introduction to OLS

   • Theoretical foundations
   • Mathematical formulation
   • Assumptions and properties

2. Basic OLS Implementation

   • Simple regression example
   • Results interpretation
   • Parameter extraction

3. Comprehensive Diagnostics

   • Visual diagnostics (6 plots)
   • Statistical tests
   • Assumption validation

4. Advanced Topics

   • Non-linear relationships
   • Categorical variables
   • Interaction terms

5. Model Comparison

   • Multiple specifications
   • Information criteria
   • Hypothesis testing

6. Outlier Analysis

   • Influence measures
   • Detection methods
   • Treatment strategies

7. Robust Regression

   • Alternative estimators
   • Performance comparison
   • Use case recommendations

8. Best Practices

   • Model selection guidelines
   • Diagnostic workflows
   • Reporting standards

Key Features for Practitioners

For Data Scientists

• Complete diagnostic workflow
• Model comparison framework
• Robust regression alternatives
• Production-ready code patterns

For Researchers

• Theoretical foundations
• Statistical inference methods
• Assumption validation techniques
• Publication-quality visualizations

For Students

• Step-by-step explanations
• Practical examples
• Common pitfalls and solutions
• Best practice guidelines

Visualizations Included

1. Diagnostic Plots:

   • Residual analysis (6 plots)
   • Multicollinearity diagnostics
   • Influence measures
   • Model comparison charts

2. Regression Results:

   • Fitted vs actual values
   • Confidence intervals
   • Prediction intervals
   • Component contributions

3. Statistical Tests:

   • QQ plots for normality
   • ACF plots for autocorrelation
   • Leverage vs residuals
   • Cook's Distance visualization

Performance Considerations

• Numerical Stability: Uses QR decomposition for better numerical stability
• Memory Efficiency: Optimized for large datasets
• Computational Speed: Efficient algorithms for common operations
• Scalability: Suitable for datasets up to millions of observations

Contributing

This notebook is designed as an educational resource. Contributions for:

• Additional diagnostic tools
• More regression techniques
• Performance optimizations
• Documentation improvements

are welcome through pull requests.

License

This project is licensed under the MIT License - see the LICENSE file for details.

Contact

For questions, suggestions, or collaboration opportunities, please open an issue in the repository.

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Note: This notebook is designed to be self-contained and educational. All code examples include explanations and interpretations to facilitate learning and practical application.