**Subject: **Re: does fuzzy bound probability?

**From: **Scott Ferson (*scott@ramas.com*)

**Date: **Fri Nov 24 2000 - 13:31:45 MET

**sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Neural Networks for Signal Processing 2001: "NNSP 2001, CALL FOR PAPERS"**Previous message:**albert@massivbau.tu-darmstadt.de: "fuzzy mapping rules and fuzzy implication rules"**Maybe in reply to:**Scott Ferson: "does fuzzy bound probability?"**Maybe reply:**Scott Ferson: "Re: does fuzzy bound probability?"

Vladik Kreinovich wrote:

*> It looks like when you talk about a "fuzzy number", what you really have in
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*> mind is an interval of possible values of a certain quantity and a number in it
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*> (like "the most probable value" in this interval.)
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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
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*> belongs to the interval X+Y irrespective of the probability distribution that
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*> we have on both intervals. This is a known fact: that interval computations
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*> 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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**Next message:**Neural Networks for Signal Processing 2001: "NNSP 2001, CALL FOR PAPERS"**Previous message:**albert@massivbau.tu-darmstadt.de: "fuzzy mapping rules and fuzzy implication rules"**Maybe in reply to:**Scott Ferson: "does fuzzy bound probability?"**Maybe reply:**Scott Ferson: "Re: does fuzzy bound probability?"

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