International audienceThis paper presents how a representation with fuzzy subsets of symbols can be used to build a scale. Then, the incidence of the fuzziness on this new kind of nominal scale, called fuzzy nominal scale, is analysed

Benoit, Eric

Foulloy, Laurent

HAL Université de Savoie

Fuzzy Nominal ScalesEric Benoit, Laurent FoulloyTo cite this version:Eric Benoit, Laurent Foulloy. Fuzzy Nominal Scales. IMEKO TC7 Symposium, Jun 2002,Cracow, Poland. IMEKO, pp 21-25, 2002. <hal-00147234>HAL Id: hal-00147234https://hal.archives-ouvertes.fr/hal-00147234Submitted on 16 May 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ABSTRACTThe main objective of a measurement process is to convertthe physical state of an entity into an information entity. Ata higher level, relations on physical states are alsoconverted into relations between the information entities.Then it’s possible to manipulate the information entitieswith these relations instead of manipulating the physicalstates. Nominal scales are scales where the only relationbeing converted is an equivalence relation. In mostapplications of nominal scales, the symbolicrepresentation of a physical state is a symbol. An otherapproach shows that a physical state can be represented bya fuzzy subset of symbols. This paper presents how arepresentation with fuzzy subsets of symbols can be usedto build a scale. Then, the incidence of the fuzziness on thisnew kind of nominal scale, called fuzzy nominal scale, isanalysed.Keywords: Measurement theory, Scales, Nominal scales,Fuzzy Scales.1. INTRODUCTION1.1. Nominal scalesLet us present the general scale formalism in order tointroduce the aim of the paper. Any scale can be definedwith a symbolism [1]:C = < X, S, M, RX, RS, F> (1)- where X refers to a set of object states, and RX is a setof relations on X.- S refers to a set of information entities [2], and RS is aset of relations on S.- M, called the representation, is a mapping from X to S.- F is a one-to-one mapping with domain RX and rangeRS.In the case of nominal scales [3], RS contains only theequality relation on S, and RX contains an equivalencerelation on X denoted ~. F is then simply defined by.(2)The equivalence relation ~ defines a partition of the set ofobject states, and M associates each item of the partition toan information entity. The general symbolism of a nominalscale is:C = < X, S, M, {~}, {=}, {(~,=)}> (3)The easiest way for the definition of a nominal scale is todefine a partition and a one-to-one mapping from thepartition to the set of information entities.Depending on the measurement aim, the nature ofinformation entities differs:- An information entity can be a symbol. This is thesimplest case used for example with colormeasurement.- An information entity can be a subset of symbols. Forexample, a subset of predicates can be used as arepresentation [4].In this paper we consider the mechanism that representseach object with a fuzzy subset of symbols. Thismechanism, called fuzzy description and introduced fewyears after the fuzzy subset theory in [5], is now usuallyapplied in fuzzy sensors [6][7].1.2. Fuzzy descriptionIf numerical values are commonly used to representmeasurement results, it is now admitted that someapplications would better manipulate symbolic values orlinguistic terms. For example, a rule-based decision systemuses linguistic term in its rules. The advantages of usinglinguistic terms instead of numeric ones is to reduce thenumber of symbols.A measurement result is represented by a fuzzy subset oflinguistic terms. The conversion from numerical tolinguistic representation is called a linguistic description.To perform a linguistic measurement, it is necessary toclearly specify the relation between linguistic terms andnumbers [8].Let X be the universe of discourse associated with themeasurement of a particular physical quantity. In order tolinguistically characterise any measurement over X, let Wbe a set of words, representative of the physicalphenomenon. Denotes R the relation defined on X×W thatformalize the link between items of the universe ofdiscourse and the words of the lexical universe. In order tosimplify the management of this link, two mapping arededuced form this relation. The first one is a one-to-onemapping m called the meaning of a linguistic terms [5]. Itassociates any word w of W with a subset of X i.e. an itemof the set S(X) of subsets of X.F ~ =( , ){ }=xi X∈( )∀ xj X∈( )∀ xi xj∼ M xi( )⇔, , M x j( )=Fuzzy nominal scalesEric BENOIT and Laurent FOULLOYLAMII, ESIA, Université de Savoie,B.P. 806, F-74016 Annecy cedex, FranceEric.Benoit@univ-savoie.fr, Laurent.Foulloy@univ-savoie.frBenoit E., Foulloy L., “Fuzzy Nominal Scales”,Proc. of IMEKO TC7 Symposium, Cracow,Poland, 2002, pp. 21-25.(4)The other one is a one-to-one mapping d called thedescription of a measurement.(5)From the linguistic description point of view, it isinteresting to define R such that each object state x ischaracterized by only one word w: d(x) = {w}. This meansthat the range of the mapping m is a partition on X i.e. eachm(w) is an equivalence class. Thus, the mapping m definesan equivalence relation ~ on X. Then the descriptionmechanism can be considered as a nominal scale where:(6)- The universe of discourse is the set of object states.- Information entities are singletons of the lexical set W.- M is the description d.If the number of information entities is small then theprecision of this symbolism is said low. When object stateschange in a same equivalence class, the associatedinformation entity does not change. Introducing thevagueness into the relation between object states andinformation entities is a solution for increasing theprecision of a symbolism.Consider now that the relation R is a fuzzy relation onX×W. It is characterized by its membership functiondenoted µR(x,w). Then the fuzzy meaning of a word andthe fuzzy description of an object state are respectivelydefined as:(7)(8)Where FS(M) is the set of fuzzy subsets of M, and FS(X) isthe set of fuzzy subsets of X. These two mappings aresimply linked by(9)As for a crisp description mechanism, it is interesting todefine R such that the range of the fuzzy mapping m be apartition on X. Then, a fuzzy equivalence relation ~ on X,also called similarity relation, can be deduced. Severalfuzzy partitions can be defined. The preferred partitionsused for the description with linguistic terms respect thefollowing constraint:. (10)This constraint reveals the need of exclusion betweenlinguistic terms for the description of states. Based on thispartition a similarity relation can be extracted:(11)This relation is reflexive, symmetric and verifies a weakversion of transitivity based on the Lukasiewicz T-norm.With this fuzzy equivalence relation it seems possible tointerpret the fuzzy description as a nominal scale.2. THE FUZZY DESCRIPTIONMECHANISM IS A SCALEIn this paper, some answers to the following questions areproposed:- Can it be possible to interpret the fuzzy description ofobjects into the general scale formalism?- If possible, what kind of scale uses a fuzzy descriptionas representation mechanism?2.1. Considering fuzzy subsets of words asinformation entitiesOne solution for the interpretation of the fuzzy descriptioninto the scale formalism is to define the representation setas the set of fuzzy subsets of words, and the mapping M asthe fuzzy description.Fig. 1. Scale interpretationThe following symbolism can then be defined:(12)- where X refers to the set of object states,- where information entities are fuzzy subsets of thelexical set W,- the fuzzy description d is the representation mapping,- the relation ~ is the fuzzy equivalence relation on Xdefined in (11),- and ≡ is the equivalence relation on fuzzy subsets ofW such that:(13)This equivalence relation is only defined on the range ofthe fuzzy description d. It can be extended to any fuzzysubset A of W that respect (14).m W S X( )→w W∈∀ m w( ) x X∈ xRw{ }=d X S W( )→x X∈∀ d x( ) w W∈ xRw{ }=C X w{ } w W∈( ){ } d ~{ } ={ } ~ =,( ){ }, , , , , =m X FS W( )→w W∈∀ x X∈∀, µm w( ) x( ) µR x w,( )=d W FS X( )→w W∈∀ x X∈∀, µd x( ) w( ) µR x w,( )=w W∈∀ x X∈∀, µd x( ) w( ) µm w( ) x( )=x X∈( )∀ µm w( ) x( )w W∈, 1=µ~x y,( ) min µm w( ) x( ) µm w( ) y( ),( )( )w W∈={0.8/w1, 0.2/w2}{0.4/w2, 0.6/w3}{w3}x1x2x3X PF(W)MC X FS W( ) d ~{ } ≡{ } ~ ≡,( ){ }, , , , , =x y,( ) X2∈∀ x y∼ d x( ) d y( )≡⇔(14)Denote MFS(W) the subset of FS(W) such that each item ofMFS(W) verifies (14). It’s interesting to see that MFS(W)is the union of ranges of all possible fuzzy descriptions onthe lexical set W.(15)In this solution the symbolisation is precise. This propertycome from the size of the set of information entities thatcan be very large. This is also contradictory with the firstinterest of fuzzy description i.e. reducing the number ofinformation entities in order to simplify the manipulationof these entities.2.2. Considering fuzzy description as therepresentation mappingIn an other solution, the information entities are symbols,and M is a fuzzy relation from X to the lexical set W.(16)- where X refers to the same set of object states,- where information entities are words of the lexical setW,- where R the fuzzy relation from X to W that defines thefuzzy meaning and the fuzzy description (7) and (8).- the relation ~ is the fuzzy equivalence relation on Xdefined in (11),- and = is an equivalence relation on items of W. Thisrelation can be simply defined such that:(17).Fig. 2. Scale interpretationIn this case, the symbolisation stays precise, because theequivalence relation on object states is the same than inpreceding case. The number of representational symbols issmall. This is useful when we need to handle theknowledge associated with object states.It can be noted that using a crisp partition instead of a fuzzyone induces a usual nominal scale.3. FUZZY NOMINAL SCALESTo the question «what kind of scale uses a fuzzydescription as representation mechanism?», the lastsymbolism presented seems to let conclude that a fuzzydescription is a nominal scale. But we consider that thisconclusion is dangerous. Indeed, in the case of a nominalscale, the only relation that can be used on the lexical set isan equivalence relation. But the introduction of fuzzynessinto the measurement mechanism induces a new proximityrelation on the lexical set. This relation comes fromrelation between the meanings of words i.e. the items ofthe fuzzy partition.As each word w belongs to an item of d(X) it is possible todefine an equivalence relation based on the generalcombination-conjunction:(18)such that(19)This relation is an equivalence relation that verify a weaktransitivity based on Lukasiewicz T-norm. This relationdoes not verify a strict definition of the transitivity i.e. thesup-min transitivity. It is then considered as a proximityrelation. This relation is not the crisp equality relation, andcan link different words. Actually it acts as a proximityrelation between words.Introducing a fuzzy description of object states instead ofa crisp one induces a proximity relation between items ofthe fuzzy partition and then induces a proximity relationbetween information entities that represent these objectsstates. As symbolisms are used in order to give tools formanipulating information entities instead of theirassociated object states, a symbolism do not have tointroduce relations that not exist between object states.Using fuzzy description can be an approach for taking intoaccount existing proximity relation between object states.And fuzzy nominal scales can be used when nominalscales are too poor and when an ordinal scale cannot bedefined.4. COLOR MEASUREMENT EXAMPLEThe color measurement is a typical measurement where itsseems to be possible to use a scale more informative thana simple nominal scale. Indeed, we can imagine that aproximity relation can exist into a color space. Red color isclose to orange color but not close to green color.Unfortunately no existing crisp scale presently includesthis proximity relation.A color measurement is performed with three photometrictransducers that recreate the effects of red, green and bluehuman cones. The set of color measurements is part of theso-called RGB cube. Then a non linear mapping is appliedµA w( )( )w W∈1=A B,( ) MFS W( )2∈∀µ≡A B,( ) min µA w( ) µB w( ),( )w W∈=C X W R ~{ } ={ } ~ =,( ){ }, , , , , =w z,( ) W2∈∀ w=z w{ }≡ z{ }⇔x1x2x3XMw1w2w30.80.20.40.61Ww z,( )∀ W2∈µ=w z,( ) supA d X( )∈ min µA w( ) µA z( ),( )( )=w∀ W∈ supA d X( )∈ µA w( )( ) 1=to the RGB cube. This mapping has been chosen in orderto obtain a colour information which do not depend on theluminosity. This information, called the chrominance, isincluded into the chrominance plane. In this example, wechoose the C1C2 chrominance plane [6]. Seven typicalcolors are localized on this plane (Fig. 3.).Fig. 3. C1C2 chrominance plane used for colormeasurementsA fuzzy partition based on an interpolation of humanknowledge can be defined on this chrominance plane (seeFig. 3.). The human knowledge is defined by a set ofmeasurement states, With each state associated to a word.The interpolation knowledge is performed with thetriangulation method. Other methods like fuzzy C-means,or K-Nearest neighbour that can also be used are notpresented in this paper.Fig. 4. Human knowledge and triangulation of thechrominance space.Fig. 5. Fuzzy meaning of the word Red.This fuzzy partition is used for the definition of the fuzzynominal scale. The proximity relation on the lexical set canthen be define (see table 1).The equivalence/proximity relation = on color wordsshows that red colors are near yellow colors. In can benoted that if the lexical set is larger, including for examplethe orange color, then the relation = can differs, and redcolors can be near orange colors but not near yellow colors.Actually, the definition of the proximity relation on thelexical set of words depends on the definition of the fuzzypartition. It depends on the fuzzy meaning of each word.5. CONCLUSIONIn this paper, the definition of a fuzzy nominal scale hadbeen presented, and the interest and the danger of usingsuch scale where exposed. Using fuzzy nominal scale for ameasurement induces a proximity relation on therepresentational set that is more informative than theequality. This is an improvement only if this proximityrelation represents a real proximity relation between objectstates. If not, the danger is to introduce a relation that doesnot have any meaning. As several family of fuzzyequivalence relations can be used, it will be useful to studythe properties and the useability of associated fuzzynominal scales.RedMagentaBlueCyanGreenYellowNeutral C2C1RedRed MagentaBlueBlueCyanGreen GreenYellowYellowNeutralTable1: = relation on color words= Red Mag. Yell. Neut. Green Cyan BlueRed 1 0.5 0.5 0.5 0 0 0Mag. 0.5 1 0 0.5 0 0 0.5Yell. 0.5 0 1 0.5 0.5 0 0Neut. 0.5 0.5 0.5 1 0.5 0 0.5Green 0 0 0.5 0.5 1 0.5 0.5Cyan 0 0 0 0 0.5 1 0.5Blue 0 0.5 0 0.5 0.5 0.5 110µm(Red)(x)C1C2REFERENCES[1] Finkelstein L. Representation by symbol systems as anextension of the concept of measurement, Kybernetes,Vol. 4, pp. 215-223, 1975.[2] Mari L., The duality of sensing and coding as afundamental topic of information science, Proc. of theWorld Conf. on Systemics, Cybernetics andInformatics (SCI 2001), Vol. XVI, Orlando, USA,2001, pp. 177-180.[3] Watanabe H., Ryumae S.,Oka Y., Shigemasu K.,Moritora Y., “A study of nominal scale”, Proc. ofIMEKO TC7 int. symp. on AIMaC’91, Kyoto, Japan,1991, pp. 295-302.[4] Watanabe H., Theory of classification to reinterpret thenominal scale for our better understanding ofmeasurement information, Proc. of the World Conf. onSystemics, Cybernetics and Informatics (SCI 2001),Vol. XVI, Orlando, USA, 2001, pp. 191-198.[5] Zadeh L.A., Quantitative fuzzy semantics, InformationSciences, Vol. 3, 1971, pp. 159-176.[6] Benoit E., Mauris G., Foulloy L., A fuzzy coloursensor, Proc. of XIII IMEKO World congress, Torino5-9 September 1994, pp. 1015-1020[7] Mauris G., Benoit E., Foulloy L., Fuzzy symbolicsensor : from concept to applications, Measurement,ISSN 0263-2241, No 12, pp 357-184, 1994.[8] Benoit, E., Foulloy, L., Symbolic sensors, Proc. theInternational Symposium on Artificial IntelligenceBased Measurement and Control, Kyoto, Japan,September 1991.

Fuzzy Nominal Scales

Hal - Université Grenoble Alpes

https://hal.archives-ouvertes.fr/hal-00147234

Fuzzy Nominal Scales

Abstract

Similar works

Full text

Available Versions

HAL Université de Savoie

Hal - Université Grenoble Alpes