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In https://peerj.com/preprints/3190.pdf section 4.3
SHFs simulate the errors we would have made had we used this forecasting method at those points in the past.
The meaning I could understand is change "had" to "when", like
SHFs simulate the errors we would have made when we used this forecasting method at those points in the past.
I can't understand what is this "had" used for.
"had we used this forecasting method" means that the forecasters did not use that forecasting method in the past, but if they had, then a certain amount of error would have occurred. SHF is a technique to simulate what those hypothetical errors would have been.
In short we are dealing with an unreal past here, which is why the "had"-form is used.
I really think this paper may not be the best example to work on.
Your quotation is an example of a past unreal conditional sentence with inversion that is more formal than those that follow the usual word order:
SHFs simulate the errors we would have made had we used this forecasting method at those points in the past.
The usual word order would have been as follows:
SHFs simulate the errors we would have made if we had used this forecasting method at those points in the past.
This said, more about inversions in conditionals, can be read here.
