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I want to construct a function that have these properties:

f( x, y) =  f(y,x) f(-x,-y) = -f(x,y)  f( 1,x) =  1 (if 1≥x>-1) f(-1,x) = -1 (if 1>x≥-1) f(1,-1) should be undefined f(x,-x) = 0  f(x,a) should be growing for all -1 < x < 1 

And as suspected x, y should be in [-1,1].

I want to use it to create some sort of mean-like value, to use when I have two probabilities (ranging from certainly true (1) to unknown (0) to certainly false (-1)).

How do I construct this function?

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    @jokiri, the derivative. I've rewritten that.2011-07-18

1 Answers 1

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Using some geometric considerations, I came up with:

$f(x,y) = (x+y)/(2-|x-y|)$

To construct this function, consider the square in the plane bounded by $-1 \leq x \leq1$ and $-1 \leq y \leq1$. Now given a point $(x_0,y_0)$ in the interior of the square, calculate a "signed distance" from the point to the line $y=-x$, which will be $(x_0+y_0)/\sqrt2$. Now we need to scale the distance in order that we get the appropriate value of $+1$ or $-1$ when the point is on the boundary of the square. This function gives all the symmetry properties you are looking for, and also gives the correct values on the boundary.

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    That formula works wonderfully! Thank you!2011-07-18