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

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

**Date: **Wed Nov 22 2000 - 19:20:38 MET

**sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Conference Secretary: "CFP: The International Conference on Computational Intelligence for Modelling, Control and Automation - CIMCA'2001"**Previous message:**Jaime B. Zamora S.: "Fuzzy logic and EMG pattern recognition."**Maybe in reply to:**Scott Ferson: "does fuzzy bound probability?"**Next in thread:**Vladik Kreinovich: "Re: does fuzzy bound probability?"**Maybe reply:**Scott Ferson: "Re: does fuzzy bound probability?"

Actually, fuzzy *arithmetic* (as distinct from fuzzy logic) does NOT

assume positive association between the operands. We're talking

here about (say) adding together or subtracting fuzzy numbers. As

an example, suppose

X = [3, 6, 8]

Y = [1, 2, 3]

where [a,b,c] is a fuzzy number with peak b and base from a to c.

Then X + Y is [4, 8, 11], and X - Y is [0, 4, 7]. Neither answer

makes any assumption about association or correlation of X and Y,

as they are simply level-wise generalizations of interval analysis.

This is (part of) the reason that analysts have thought that Kaufmann

and Gupta's fuzzy arithmetic might produce bounds on the answer

obtained in a probabilistic analysis.

Scott Ferson

Applied Biomathematics

WSiler@aol.com wrote:

*> In a message dated 11/21/00 5:10:46 AM Central Standard Time, scott@ramas.com
*

*> writes:
*

*>
*

*> << 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 makes
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*> 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 negative
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*> association. Jim Buckley and I have a paper or two in FS&S which discuss this
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*> point and present a family of multivalued logic parameterized in terms of the
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*> 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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*>
*

*> William Siler
*

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**Next message:**Conference Secretary: "CFP: The International Conference on Computational Intelligence for Modelling, Control and Automation - CIMCA'2001"**Previous message:**Jaime B. Zamora S.: "Fuzzy logic and EMG pattern recognition."**Maybe in reply to:**Scott Ferson: "does fuzzy bound probability?"**Next in thread:**Vladik Kreinovich: "Re: does fuzzy bound probability?"**Maybe reply:**Scott Ferson: "Re: does fuzzy bound probability?"

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