12-19-06 - 1

You have two terms A and B and wish to combine them. They might be errors, predictions, whatever. There are several interesting possibilities.

First of all, generally the output should be "linear" in A's and B's. That is, sqrt(A*B) is good but A*B is not, because it gives you 2 powers of A or B. We think like physics and assume these things have units, the output should have the same units.

Secondly, we must be aware of scales. If A & B are scaled in exactly the same way so their numeric values have the same significance, then (A+B) is a good combiner. If they are not scaled the same, then any forms which add A & B are not okay. In that case you only really have one option :

sqrt(A*B) * G(A/B)

Often A & B have the same significance which means the output should be symmetric in swap A <-> B. In that case the G function has limitted forms. I haven't thought about exactly what they can be, but it has to be things like (A/B) + (B/A). In fact if A & B aren't on a similar scale even that form is not okay.

If we assume A & B are on the same scale, then additive forms are okay and it opens up some other options. (A+B)/2 is obviously your first guess.

2AB / (A+B) is an interesting combiner. If an A&B are in [0,1] then this takes (0,x)->0 , (.5,.5) -> .5 and (1,1) -> 1, and sort of penalizes when they're not equal. It takes (x,x) -> x which is a nice property of any combiner when you're trying to make a combiner that can stand in for (A+B)/2.

sqrt(A^2 + B^2) is another, and then you can take simple multiples of these, which gives you forms like (A^2 + B^2)/(2AB).

Anyway the point is that there really aren't very many forms to choose from which satisfy the basic properties and you can easily try them all.

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