[Math] How to compute similarity between two numbers

arithmetic

Is there some metric for computing similarity between two numbers which takes values in the interval $[0, 1]$?

I want something like:
$sim(1,1) = 1$
$sim(1,2) = 0.5$
…
…
$sim(50,47) = 0.78$
$sim(100,99) = 0.99$

If two numbers are small, then similarity is sensitive.
But, if two numbers are big, then similarity is less sensitive.

Is there some metric like that?

Best Answer

Did you try s(x,y) = 2min(x,y)/(x+y)?

This measure can be based on the following measure for the (absolute) proportional difference: r(x,y) =|x/(x+y) - y/(x+y)| = |x-y|/(x+y). Viz. s(x,y) = 1 - r(x,y).

Does anyone know whether these measures have been elaborated?

r(x,y) has very nice properties, e.g.

  • 0 </= r(x,y) </= 1
  • r(x,x)=0 r(x,y) =1 iff either x or y =0
  • symmetric
  • scale invariant: r(ax,ay)=r(x,y)
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