Fuzzy nominal scales. Eric BENOIT, Laurent FOULLOY LAMII, ESIA, Université de Savoie,

Transcription

1 Fuzzy nominal scales Eric BENOIT, Laurent FOULLOY LAMII, ESIA, Université de Savoie,

2 Problematic 2 Measurements approaches Scale formalism Fuzzy description Isafuzzydescriptionbeascale? If yes, what kind of scale? A nominal one?

3 Scales Definition of a scale: C =<X, S, M, R X, R S, F> X Representation M S R X F RS Object States Information Entities Relations on X Relations on S Nominal Scales: F = {( ~=, )} ( x i X), ( x j X), x i x j Mx ( i ) = Mx ( j )

4 Scales Used to perform calculus on information entities Induce limitation on the set of allowed relations: F(R X ) is the set of significant relations on S Duality precision/complexity in nominal scales large set of information entities => precision small set of information entities => easy to use

5 Fuzzy description idea: perform a precise representation into a small set solution: Fuzzy description Introduce a Fuzzy relation R between X and a set of words W x Fuzzy description d X S( W) X dx ( ) = { w W xrw} w Fuzzy meaning m W S( X) W mw ( ) = { x X xrw}

6 Fuzzy description Fuzzy relation caracterized by the membership function: µ R (x,w) R x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 w w w m(w 3 ) w d(x 5 ) Fuzzy partition constraint: ( x X), () = 1 w µ mw ( ) x W

7 Fuzzy description is a scale? Considering fuzzy subsets of words as information entities X x 1 x 2 M=d PF(W) {w 3 } {0.1/w 2,0.1/w 3,0.8/w 4 } set of fuzzy subsets of words x 3 {0.9/w 1,0.1/w 4 } C = X, FS( W), d,{ ~ }, { }, {( ~, ) } this scale is precise Lot of information entities => not easy to use

8 Fuzzy description is a scale? Considering the fuzzy relation as the representation X x1 x2 x M W w 1 w 2 w 3 C = X, W, R, { ~ }, { = }, {( ~=, )} ( w, z) W 2 w=z { w} { z} this scale stay precise small set of information entities «Fuzzy nominal scale»

9 What kind of scale? Fuzzy nominal scale defines at least a nominal scale Fuzzy nominal scale induces a proximity relation on W w 1 is near w 2 if at least one d(x) of d(x) verifies: d(x) includes both w 1 and w 2 µ ( near wz, ) = sup A d X ( ) ( min ( µ A ( w), µ A ( z) )) Fuzzy nominal scale is more than a simple nominal scale Proximity scale?

10 «Recommendations» Case 1, a proximity relation on object states is present fuzzy nominal scales can take this relation into account. Case 2, no proximity relation on object states is present using a fuzzy description introduces an artificial proximity relation on information entities. => operations that use this artificial relation can perform wrong conclusions

11 Conclusion Fuzzy description is at least as a nominal scale Fuzzy description is between nominal and ordinal scales Using fuzzy description can be a bad idea next studies: fuzzy nominal scales based on different kind of fuzzy relations ordinal scales, interval scales, linear scales,... operators that use proximity relation (fuzzy rules, topology) application to formalisms based on nominal scales

12 Red Example Red Yellow Neutral Magenta Yellow Yellow Red Neutral Magenta µ m(red) (x) 1 0 Green Blue Green Green Blue Blue Cyan Cyan Red Mag. Yell. Neut. Green Cyan Blue Red Mag Yell Neut Green Cyan Blue