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My topic is related to this post found here, although it is quite different.

I am comparing a product of three variables to a threshold value. To be more specific, let's say we have: $$ F = A*B*C $$ The range of values $A, B, C$ can take is $[0,1]$.
$F$ is below a threshold of interest $T$, and I want to determine the relative contribution of each variable ($A, B$ and $C$) to the difference $T-F$.

In two dimensions, I thought I could compute the distance between ($A,B$) and the closest point on the curve $F=T$, and determine which variable contributes the most to this distance. But it does not look like the easiest solution in 3 or more dimensions.

Thank you for your help!

carambarbe
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1 Answers1

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Let $a,b,c$ be small perturbations in the values of $A,B,C$. Then $$ (A+a)(B+b)(C+c) \approx ABC + aBC + AbC + ABc $$ since the omitted terms in the expansion contain products of two small factors. To see that geometrically, imagine a box with sides $A,B,C$.

So a small perturbation in $a$ is magnified by $BC$, etc.

You can interpret that as saying that the "contribution" of $A$ to $ABC$ is $BC$.

That assertion is independent of the threshold value $T$, which may or may not make sense in your application.

Ethan Bolker
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  • Thank you ! I still have to think whether it makes sense to not take into account the threshold. To avoid problematic cases (such as A >0 , B = 0 and C = 0), do you think of any strategy ? Would it be ok to add a very small number (<<1) to the three variables ? – carambarbe Feb 07 '24 at 18:55
  • You can do anything that makes sense in your application. – Ethan Bolker Feb 07 '24 at 19:04