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

**From: **Vladik Kreinovich (*vladik@cs.utep.edu*)

**Date: **Thu Nov 23 2000 - 17:22:21 MET

**sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Pim van den Broek: "Re: Fuzzy relations vs. Mamdani model"**Previous message:**Nadine Gisler: "Welcome to the First ICSC Congress on Neuro-Fuzzy NF'2002"**Maybe in reply to:**Scott Ferson: "does fuzzy bound probability?"**Next in thread:**Scott Ferson: "Re: does fuzzy bound probability?"**Maybe reply:**Vladik Kreinovich: "Re: does fuzzy bound probability?"

Dear Scott,

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.)

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.

Vladik

*> Date: Tue, 21 Nov 2000 09:57:23 -0500
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*> From: Scott Ferson <scott@ramas.com>
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*> To: NAFIPS-L <nafips-l@sphinx.Gsu.edu>, WSiler@aol.com
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*> Subject: Re: does fuzzy bound probability?
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*>
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*>
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*> Actually, fuzzy *arithmetic* (as distinct from fuzzy logic) does NOT
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*> assume positive association between the operands. We're talking
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*> here about (say) adding together or subtracting fuzzy numbers. As
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*> an example, suppose
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*>
*

*> X = [3, 6, 8]
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*> Y = [1, 2, 3]
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*>
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*> where [a,b,c] is a fuzzy number with peak b and base from a to c.
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*> Then X + Y is [4, 8, 11], and X - Y is [0, 4, 7]. Neither answer
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*> makes any assumption about association or correlation of X and Y,
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*> as they are simply level-wise generalizations of interval analysis.
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*> This is (part of) the reason that analysts have thought that Kaufmann
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*> and Gupta's fuzzy arithmetic might produce bounds on the answer
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*> obtained in a probabilistic analysis.
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*>
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*> Scott Ferson
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*> Applied Biomathematics
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*>
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*>
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*>
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*> WSiler@aol.com wrote:
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*>
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*> > In a message dated 11/21/00 5:10:46 AM Central Standard Time,
*

scott@ramas.com

*> > writes:
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*> >
*

*> > << Because (standard) fuzzy arithmetic makes no assumptions about the
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*> > > dependence or independence among quantities, it has been suggested
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*> > > that fuzzy arithmetic might be able to provide bounds on probability
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*> > > distributions in cases where the dependence among input variables
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*> > > cannot be specified empirically.
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*> > >>
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*> >
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*> > I'm not sure what basis was offered for the statement that fuzzy logic
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makes

*> > no assumptions about independence, but I'm afraid that the statement is not
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*> > true. Standard Zadehian min-max fuzzy logic assumes implicitly that the
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*> > operands of the logical operations AND and OR are positively associated as
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*> > much as possible, just as the probabilistic AND (a*b) and OR (a + b - a*b)
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*> > assumes independence (zero association) and the bounded sum and difference
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*> > logic AND (max(0, a + b - 1)) and OR (min(a +b - 1)) assume maximum
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negative

*> > association. Jim Buckley and I have a paper or two in FS&S which discuss
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this

*> > point and present a family of multivalued logic parameterized in terms of
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the

*> > correlation coefficient between the operands, based either on neccessity (a
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*> > AND NOT a) or past history.
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*> >
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*> > Consequently, any manipulations which employ fuzzy logical operations (such
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*> > as the extension principle) implicitly make an assumption about the
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*> > independence of the operands. That would seem to include fuzzy arithmetic.
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*> >
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*> > William Siler
*

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**Next message:**Pim van den Broek: "Re: Fuzzy relations vs. Mamdani model"**Previous message:**Nadine Gisler: "Welcome to the First ICSC Congress on Neuro-Fuzzy NF'2002"**Maybe in reply to:**Scott Ferson: "does fuzzy bound probability?"**Next in thread:**Scott Ferson: "Re: does fuzzy bound probability?"**Maybe reply:**Vladik Kreinovich: "Re: does fuzzy bound probability?"

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