Iterative Least Squares Algorithm Based on M-Estimation

This text elaborates on the Iterative Least Squares Algorithm based on M-Estimation. This algorithm employs an iterative process that utilizes robust estimators such as Huber, Andrews, Hampel, and Ramsay to effectively handle data with potential outliers. The Huber estimator mitigates outlier influence by introducing a small adjustment factor in the data distribution, typically implemented through a piecewise function that transitions between quadratic and linear loss. The Andrews estimator reduces outlier weights during estimation using a sine-tapered weighting function that dampens extreme values. The Hampel estimator employs a three-part redescending function that applies progressive weighting to gradually diminish outlier impacts. The Ramsay estimator incorporates a smoothing factor through its Ea loss function to provide continuous downweighting of outliers. These diverse estimation approaches can be adaptively applied to different data distributions, enabling more accurate recovery of true underlying values. The iterative implementation typically involves: 1) Initial least squares fit 2) Residual calculation 3) Weight assignment based on chosen M-estimator function 4) Re-estimation using weighted least squares. Consequently, the M-estimation based iterative least squares algorithm serves as a crucial tool in data analysis and modeling, particularly valuable for robust regression applications where data contamination is a concern.