Least squares as random walks: The general case of arbitrary spacing

Recently, we introduced the notion of a random walk based on a discrete sequence of data samples (data walk) and discovered a surprising link between ordinary least squares (OLS) fits to evenly sampled data and random walks. Here we generalize earlier results by showing that the slope of a linear fit to data which annuls the net area under a residual data walk equals that found by OLS for irregularly spaced data sequence. We also discover a deep connection with the orthogonality principle of estimation theory, leading to interpretation of suitably defined scalar products of data vectors as areas under data walks. The results are extended to weighted and generalized least squares (GLS). The new approach is illustrated on cosmic ray arrival time series.