Re: does fuzzy bound probability?


Subject: Re: does fuzzy bound probability?
From: Scott Ferson (scott@ramas.com)
Date: Fri Nov 24 2000 - 13:31:45 MET


Vladik Kreinovich wrote:

> It looks like when you talk about a "fuzzy number", what you really have in
> mind is an interval of possible values of a certain quantity and a number in it
> (like "the most probable value" in this interval.)

Almost. I have in mind Kaufmann and Gupta's notion of a
stack of intervals. If the fuzzy number A has the membership
function mu_A(x) then, by definition, max[mu_A(x)] is one,
and every level set A_alpha = {x | mu_A(x) >= alpha} is a
closed interval of R, for any alpha in [0,1]. The three-point
notation [a,b,c] is just a shorthand for when the sides of the
fuzzy number are piecewise linear.

> Of course, if you know an interval for X and an interval for Y, then the sum
> belongs to the interval X+Y irrespective of the probability distribution that
> we have on both intervals. This is a known fact: that interval computations
> provide guaranteed estimate.

Yes, but the question is whether the guarantee extends beyond
intervals to *distributional* forms such as fuzzy numbers. Here
is the question stated more formally:

Let us say that a fuzzy number A "encloses" a probability
distribution F if

    mu_A(x) >= F(x) for all x <= max(x | mu_A(x) = 1)

and

    mu_A(x) >= 1 - F(x) for all x >= min(x | mu_A(x) = 1).

Now, suppose the random variables X and Y have F and G
respectively for their distribution functions. If the fuzzy number
A encloses F, and the fuzzy number B encloses G, is it
guaranteed that their fuzzy sum A+B encloses the distribution
of the sum X+Y?

The naive expectation is, of course, that the conjecture is true.

Scott Ferson
Applied Biomathematics

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