The Anatomy of the Least Squares Method, Part Two (thepalindrome.org)

Part Two of "The Anatomy of the Least Squares Method" walks readers through learning regression by simulating data, showing how deliberately generated datasets give you ground truth for experiments and intuition that real data alone can’t provide. The post demonstrates how to translate a simple linear model into Python—choose betas, draw independent variables (integers in the punk-shows example), add normally distributed noise, clip unrealistic outcomes, then fit with least squares—so you can see how estimator accuracy changes with noise and sample size. It also previews the rest of the series (real-data examples and modeling GPT-2 activations), and stresses ethics: always label simulated data clearly. Technically useful takeaways: build a design matrix (column of ones + IVs), prefer numerically stable routines (numpy.linalg.lstsq uses QR/Gaussian-elimination instead of an unstable matrix inverse), and compare estimated betas to known ground truth to study bias and variance (small N yields high variability; large N converges to true betas). The post emphasizes diagnostic visualization—data vs. fit, predicted vs. residuals (their linear correlation is always zero by least-squares geometry, so check for nonlinear structure), and residual histograms for Gaussianity—and touches on model-fit metrics (adjusted R²). Overall, it’s a practical, code-backed primer for developing robust intuition about least-squares in ML workflows.