Couldn't decide whether the question belongs on Mathematics or here, but I guess I'll ask it here.
I've got a discrete - valued function defined on a finite segment of a line that looks like this:
Then the task is, given a set of other functions, find a function in this set that is the closest - in some way that I'm about to describe - to the original one.
The problem is that the candidate functions may be, actually, circularily shifted with respect to one another. If the functions were not shifted, any reasonable norm of difference (l1, l2, any lp, I guess) would've been okay. I also want to notice that I do not necessarily want a function that satisfies the mathematical definition of a norm (or even a distance) - I just want the problem solved.
The obvious thing to do is to look at the maximum of circular cross - correlation. However, this is actually a bad idea because even if I normalize both compared function by their max values, the function f(x) = 1 will give the maximal value with any other function.
I want to know whether there is some standard approach for such a situation or should I just compute an l1 - norm for every possible circular shift and then take the minimum value.