A 2-Parameters Weight Definition in Least-Squares Regression Towards Data Fitting with Uncertainty

Linear regression often fits a model to observations without accounting for associated uncertainties, as seen in Ordinary Least Squares (OLS). To address this limitation, a new two-parameter weight system is evaluated within the Weighted Least-Squares (WLS) method, which, unlike OLS, accounts for uncertainty in the observations. The weights are defined by a delay, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tau$</tex>, and a form factor, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{I}. \tau$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{I}$</tex> modulate the weight based on the observation uncertainty <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sigma_{i}$</tex> and the prediction error <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\varepsilon_{i}$</tex>, adapting the weight dynamically depending on their relative values. The weights are tested against known target functions, using the model-to-target distance over a selected interval for various observations errors and number of observations. Estimators using these new weights outperform the naive method (which ignores uncertainty) and are particularly effective when traditional weights, defined as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1/\sigma^{2}$</tex>, are inadequate, such as in homoskedastic scenarios.