When we are doing weighted least squares how do we find the weights? Where ever I see tutorials are just using $w_i = \frac{1}{(sigma)i^2}$ and doing it with basic data. But I want to know how to find the weights for real data. Is it always the inverse of the square of variance?


1 Answer 1


No, the weight does not always has to be equal to the inverse of the square of variance. There is no universally accepted scheme. Different weighting schemes are used, such as $1/x$, $1/x^2$, $1/\sqrt{x}$, where $x$ are your data. The weighting scheme depends on many things: the distribution of your data, their meaning, and in which regions you want to fit them more closely (whether small data values should have lower error), etc.

This is a paper that compares different weighting schemes for data in chemistry: [1]. The introduction in that paper has references to many other papers discussing different weighting schemes.

[1] Ram B. Jain, Comparison of three weighting schemes in weighted regression analysis for use in a chemistry laboratory, Clinica Chimica Acta, Volume 411, Issues 3–4, 2010, Pages 270-279, ISSN 0009-8981, https://doi.org/10.1016/j.cca.2009.11.021.


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