MATLAB Source Code for Iterative Reweighted Least Squares (IRLS) Algorithm

The IRLS algorithm, short for Iterative Reweighted Least Squares, is an optimization method specifically designed for regression problems, particularly effective in modeling data contaminated with outliers or non-Gaussian noise distributions. The core concept involves iteratively adjusting a weight matrix to progressively focus on well-fitting data points. The implementation follows these key computational steps: Initialization Phase: Start with ordinary least squares (OLS) to obtain initial parameter estimates using MATLAB's backslash operator (\) or pinv() function for matrix inversion. Weight Calculation: Compute weights for each data point based on current residuals, where larger errors receive smaller weights. Common weight functions include: - Huber function: Transition between quadratic and linear loss using a threshold parameter - Bisquare function: Smooth weighting that gradually reduces influence of outliers In MATLAB, this typically involves vectorized operations using element-wise arithmetic. Weighted Least Squares: Perform weighted least squares estimation using the calculated weight matrix. This can be implemented through: W = diag(weights); % Create diagonal weight matrix params = (X' * W * X) \ (X' * W * y); % Solve weighted normal equations Convergence Check: Monitor parameter changes or residual norms against tolerance thresholds. Common convergence criteria include: - Relative parameter change: norm(params_new - params_old) / norm(params_old) < tol - Maximum iteration count prevention for computational efficiency Critical MATLAB implementation considerations: - Weight update strategy significantly impacts algorithm robustness and convergence rate - Termination conditions must balance numerical precision with computational complexity - Matrix operations require careful handling of ill-conditioned systems using techniques like Tikhonov regularization - Debugging should verify weight function behavior and convergence logic through residual analysis Compared to standard least squares, IRLS offers superior robustness against outliers and finds extensive applications in statistics, signal processing, and machine learning. The algorithm's effectiveness makes it particularly valuable for real-world datasets where Gaussian assumptions may not hold.