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Least Squares as Random Walks (2503.20965v1)
Published 26 Mar 2025 in stat.ME, cond-mat.stat-mech, math.PR, physics.data-an, and stat.AP
Abstract: Linear least squares (LLS) is perhaps the most common method of data analysis, dating back to Legendre, Gauss and Laplace. Framed as linear regression, LLS is also a backbone of mathematical statistics. Here we report on an unexpected new connection between LLS and random walks. To that end, we introduce the notion of a random walk based on a discrete sequence of data samples (data walk). We show that the slope of a straight line which annuls the net area under a residual data walk equals the one found by LLS. For equidistant data samples this result is exact and holds for an arbitrary distribution of steps.