Re: determining contributions to a function

briggs_at_encompasserve.org
Date: 06/01/04 

Date: 1 Jun 2004 10:16:06 -0600

In article <pan.2004.06.01.00.10.19.656887@presidency.com>, Rajarshi Guha <rajarshi@presidency.com> writes:
> Hello,
> (I'm not sure whether this is the right group to post this question so
> if its not sorry.)
>
> If I have a function of several variables, say f(x,y,z).
> For given values of x,y and z is there a way to determine the
> contribution to the value of f(x,y,z)? And similarly for y and z?

Before you can determine the contribution of x, y and z to f(x,y,z)
you must _define_ the contribution of x, y and z to f(x,y,z)

(Maybe that is what you meant to ask)

In that case, the answer is yes and no. Yes, you could pick any old
formula and define "contribution" accordingly. No, no such formula
is the one true underlying meaning of "contribution" - there is no
such underlying meaning.

What properties would you like "contribution" to satisfy? Should
the three contributions add up to 100%? Should they reflect the
incremental sensitivity of f(x,y,z) to small changes in x, y and z?
Should they reflect the cumulative change from the origin? Should
they be insensitive to a coordinate system transformation? Should they
be invariant if a constant is added or subtracted from f? I think
we can take it as a given that "contribution" should behave symmetrically
with respect to x, y and z and symmetrically with respect to positive
and negative numbers.

Not all these conditions can hold at once. You need to choose.
Perhaps, if you choose carefully, there will be a unique formula
that will satisfy your conditions.

maybe c_x(x,y,z) = (f(x,y,z) - f(0,0,0)) * |x| / (|x| + |y| + |z|)

That one adds to 100%, is invariant with respect to the addition
or subtraction of a constant and possesses the requisite symmetries.

        John Briggs

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