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I am running a multivariate linear regression on noisy data, where the amount of error for each measurement is known (or at least estimated). It works reasonably well with weighted linear regression if I weight my rows by uncertainty, e.g. rows with high confidence have high weights and rows with high uncertainty have low weights. However I think I can do better.

Suppose instead of a single prediction for each row, I have an upper and lower bound of an 80% confidence interval. Then I can calculate my cost function as the square of the difference from the interval instead of the square of the difference from the mean.

I can code this pretty easily from scratch, but I wonder if there is an existing library to do this. I am using sklearn now, but don't mind switching from something else. This is ideally for publication, so a standard library, or at least a standard well know equation or formula, would be a big help.

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You can refer to the mapie library that provides a solution to your requirements. I am providing a link to the official documentation of mapie library, that will give you code examples of implementation along with the theoretical description.

https://mapie.readthedocs.io/en/latest/theoretical_description_regression.html

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