Case Study: Gaussian Mixture Models (GMM) Implementation

This chapter documents the complete EXTREME TDD implementation of aprender's Gaussian Mixture Model clustering algorithm using the Expectation-Maximization (EM) algorithm from Issue #16.

Background

GitHub Issue #16: Implement Gaussian Mixture Models (GMM) for Probabilistic Clustering

Requirements:

  • EM algorithm for fitting mixture of Gaussians
  • Four covariance types: Full, Tied, Diagonal, Spherical
  • Soft clustering with predict_proba() for probability distributions
  • Hard clustering with predict() for definitive assignments
  • score() method for log-likelihood evaluation
  • Integration with UnsupervisedEstimator trait

Initial State:

  • Tests: 577 passing
  • Existing clustering: K-Means, DBSCAN, Hierarchical
  • No probabilistic clustering support

Implementation Summary

RED Phase

Created 19 comprehensive tests covering:

  • All 4 covariance types (Full, Tied, Diag, Spherical)
  • Soft vs hard assignments consistency
  • Probability distributions (sum to 1, range [0,1])
  • Model parameters (means, weights)
  • Log-likelihood scoring
  • Convergence behavior
  • Reproducibility with random seeds
  • Error handling (predict/score before fit)

GREEN Phase

Implemented complete EM algorithm (334 lines):

Core Components:

  1. Initialization: K-Means for stable starting parameters
  2. E-Step: Compute responsibilities (posterior probabilities)
  3. M-Step: Update means, covariances, and mixing weights
  4. Convergence: Iterate until log-likelihood change < tolerance

Key Methods:

  • gaussian_pdf(): Multivariate Gaussian probability density
  • compute_responsibilities(): E-step implementation
  • update_parameters(): M-step implementation
  • predict_proba(): Soft cluster assignments
  • score(): Log-likelihood evaluation

Numerical Stability:

  • Regularization (1e-6) for covariance matrices
  • Minimum probability thresholds
  • Uniform fallback for degenerate cases

REFACTOR Phase

  • Added clippy allow annotations for matrix operation loops
  • Fixed manual range contains warnings
  • Exported in prelude for easy access
  • Comprehensive documentation

Final State:

  • Tests: 596 passing (577 → 596, +19)
  • Zero clippy warnings
  • All quality gates passing

Algorithm Details

Expectation-Maximization (EM):

  1. E-step: γ_ik = P(component k | point i)
  2. M-step: Update μ_k, Σ_k, π_k from weighted samples
  3. Repeat until convergence (Δ log-likelihood < tolerance)

Time Complexity: O(nkd²i)

  • n = samples, k = components, d = features, i = iterations

Space Complexity: O(nk + kd²)

Covariance Types

  • Full: Most flexible, separate covariance matrix per component
  • Tied: All components share same covariance matrix
  • Diagonal: Assumes feature independence (faster)
  • Spherical: Isotropic, similar to K-Means (fastest)

Example Highlights

The example demonstrates:

  1. Soft vs hard assignments
  2. Probability distributions
  3. Model parameters (means, weights)
  4. Covariance type comparison
  5. GMM vs K-Means advantages
  6. Reproducibility

Key Takeaways

  1. Probabilistic Framework: GMM provides uncertainty quantification unlike K-Means
  2. Soft Clustering: Points can partially belong to multiple clusters
  3. EM Convergence: Guaranteed to find local maximum of likelihood
  4. Numerical Stability: Critical for matrix operations with regularization
  5. Covariance Types: Trade-off between flexibility and computational cost