Change/Replace R2 Metric ?

[daniel-caichac-DHI]

Hi, I am wondering wether we should change the R2 metric. Let me explain myself.
Here I have some data. The regression does not look too bad yet R2 is negative

Which, given its definition, its ok. However, the most common or most known definition of R2 is the square of R, with R being the R=Pearson correlation coefficient. In this case for my data R_pearson=0.84 and therefore R2=0.7198.
If I plot my data in excel and do a linear regression and click show R2 I get this indeed.

Also, if I run the Mike Zero Time Series comparison (™) I get the same value (ie, Pearson R2)

I am not saying that one is better than the other, but shouldn't we stick to the most common one? Otherwise we could be potentially adding a source of error.

@jsmariegaard / @ecomodeller

[ecomodeller]

Read more about this topic in https://scikit-learn.org/stable/modules/model_evaluation.html#explained-variance-score

One of the motivations behind having modelskill as an open-source library is the that the formulation of metrics can be verified, but we should strive to use standardized nomenclature if possible, to avoid confusion.

[ecomodeller]

In the case of negative correlation, e.g. a Pearson correlation coefficient of -1, by squaring it you end up with a score of 1.0. Rather difficult metric to interpret.

[daniel-caichac-DHI]

Ok I found this very illustrative website where they show r2 and R2 and they agree that the current implementation (as is now) is indeed better
https://towardsdatascience.com/r%C2%B2-or-r%C2%B2-when-to-use-what-4968eee68ed3#:~:text=The%20Pearson%20correlation%20coefficient%20(r,the%20strength%20of%20a%20model.

but it is annoying that most software use the other one (person R square), include Mike Zero Time Series Comparison