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How to create a penalty function in two dimensions for a profile curve?
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Hello, I have a computer model that models a physical process that is represented by the black line shown in the attached figure. A traditional penalty function would compare the individual Y values at each point in time, with something like error squared. Thus, a modeled point at say 0.92 and 5.25 (the blue diamond) would provide a very large error (5.25^2), when in actuality I would be happy with any result that was within the two dotted red lines.
My question is: Is there a penalty function formulation that would compare the “relative closeness” of the computed function compared to the ideal function, taking into account “how close it is to the black line” (maybe closest distance?), or if it is within the red lines, and closest to the black line. Kind of like a visual measurement of how good the “eye fit” is in both time and amplitude. If the computed function had a sharp vertical rise at say 0.98, that would be very good, but I do not know how to compute a penalty function to account for if the computed function is “close” to the black line in time (left-right) space as well as amplitude (up-down, which I do know how to do).
Any help would be greatly appreciated.