Analysis of Linear Interpolation of Fuzzy Sets with Entropy-based Distances

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1 cta Polytechnca Hungarca Vol No 3 3 nalyss of Lnear Interpolaton of Fuzzy Sets wth Entropy-base Dstances László Kovács an Joel Ratsaby Department of Informaton Technology Unversty of Mskolc 355 Mskolc- Egyetemváros Hungary kovacs@tun-mskolchu Department of Electrcal an Electroncs Engneerng rel Unversty Center of Samara rel 47 Israel ratsaby@arelacl bstract: n nterpolaton of fuzzy sets s an mportant metho n evelopment of effcent fuzzy rule systems n mportant property of the nterpolate set s the stance mnmum property s can be seen the valty of ths property epens on the apple stance metrc The authors analyse the stance relatonshp among the base an generate fuzzy sets n the case of KH lnear nterpolaton The paper presents new propertes among the entropy-base stances an proposes an approprate metho for stance optmum nterpolaton Keywors: fuzzy nterpolaton; escrptve complety; entropy; stance metrc Introucton Interpolaton s a wely use metho to etermne the values of a target functon f at a poston n a real nterval [ab] where fa an fb are gven but f s not known In a more general approach the metho can be etene for an arbtrary oman D wth a a an D to etermne f from fa fa n Our nvestgaton focuses on set D of fuzzy sets The noton of a fuzzy set was ntrouce by [4] It s a class of objects wth contnuous values of membershp an hence etens the classcal efnton of a set to stngush t from a fuzzy set we refer to t as a crsp set Formally a fuzzy set s a par E m where E s a set of objects an m s a membershp functon m : E [ ] Fuzzy set theory can be use n a we range of omans n whch nformaton s ncomplete or mprecse such as pattern recognton an ecson theory [] [3] In the area of fuzzy rule nterpolaton FRI [7] the goal s to generate new fuzzy rules from estng rules n mportant component of FRI s the generaton of anteceent an consequent fuzzy sets usng a Fuzzy Set Interpolaton FSI metho In the most wely use approaches f s generate as a weghte sum 5

2 L Kovács et al nalyss of Lnear Interpolaton of Fuzzy Sets wth Entropy-base Dstances 5 of fa where the weght value epens on the stance between an a : In the case of lnear nterpolaton the sum of weghts s equal to : n n w a f w f The KH metho evelope by Kóczy an Hrota [8] uses lnear nterpolaton as a stanar FSI metho The poston of the generate fuzzy set * s calculate wth the formula n n * * * where enotes the anteceent set an s the consequence set The symbol enotes a -cut whch s efne as H α = { E m H α} for any H fuzzy set wth membershp functon m H In aton to the KH metho several new approaches are avalable n the lterature In the mofe α-cut base nterpolaton MCI [] fuzzy sets are escrbe wth two vectors contanng the left lower an rght upper flanks The mprove verson of MCI s calle the multmensonal mofe α-cut base nterpolaton [9] an t etens MCI wth the fuzzness conservaton technque propose by [] more etale survey of FRI methos can be foun n [7] [] among others In all versons the stance value [] has a central role n the nterpolaton algorthm sem-metrc functon to measure the stance D D : meets the followng contons: y z y z y y y For the Euclean space the most wely use metrc s the Mnkowsk stance between two ponts an y n n whch s efne as / r y y r r n r For sets n Euclean space there are several varants for the metrc functon The Hausorff stance q s efne as supnf sup nf ma v u v u V U q V U v u U u V v 3

3 cta Polytechnca Hungarca Vol No 3 3 Ths can be etene to fuzzy sets as follows Let E be a fnte set an let E be the set of all fuzzy subsets of E Then for two fuzzy subsets E the stance n 3 can be etene to the followng stance between an q q fferent approach s the Hammng stance for fuzzy sets Conser two fuzzy subsets E wth membershp functons m m : E [ ] Then can be etene to the followng Hammng stance / r r r m m r E The Euclean stance has the followng nce property: conser two elements n the space then for every element C that satsfes the followng equalty hols C [] C C C 5 e the ponts of the connectng lne are etreme ponts from the vewpont of stance relatonshp Ths nce property wll not n general be met for other stances The goal of our nvestgaton s to analyze the relatonshp between the lnear nterpolaton of fuzzy sets an the stance functon n the case of a specfc metrc the entropy-base stance functon The analyss shows that the fuzzy sets generate by lnear nterpolaton wll not meet 5 an a fferent generaton metho shoul be use to fulfll ths etreme conton In Secton three basc entropy-base stance efntons for fuzzy sets are presente The frst approach correspons to a global entropy fference the secon metho s base on an element-wse entropy fference an the thr approach uses a escrptve complety wth symmetrc fference of the corresponng membershp functons In Secton 3 the property of stance optmalty s nvestgate n KH nterpolaton for the fferent stance nterpretatons It wll be shown that the KH nterpolaton algorthm s not sutable to generate a fuzzy set lyng on the stance optmum mle pont between the operan fuzzy sets To prove the estence of such an optmum fuzzy set a generaton algorthm s also presente n the secton The theoretcal conseratons are emonstrate wth numercal eamples n the paper 4 53

4 L Kovács et al nalyss of Lnear Interpolaton of Fuzzy Sets wth Entropy-base Dstances Entropy-base Dstances Dfferent applcaton areas requre fferent smlarty an stance nterpretatons In the case of fuzzy sets there are bascally three man aspects of smlarty [5]: - smlarty of the support set n E Hausorff metrc; - smlarty of the values of membershp functons Hammng metrc - smlarty of the fuzzness of membershp functons In the latter we assume a contnuous E oman The fuzzness of E s efne by De Luca an Termn [6] as where entropy S m S lg lg One approach to nclue the fuzzness nto the stance calculaton s gven by the followng formula: S entropy entropy s the entropy functon maps the fuzzy sets nto s meets the requrements of a metrc functon nother way s to efne an element-wse fference as where / entropy { } entropy { S } entropy S{ } lg lg Ths approach maps the fuzzy sets nto a mult-mensonal vector space where the apple Euclean stance s a metrc; s meets also here the requrements of a metrc functon The thr approach uses the stance functon that s base on a escrptvecomplety [6] Ths stance uses the symmetrc fference of the corresponng membershp functons an s base on the followng conseratons Gven two fuzzy subsets [N] wth membershp functons m m we enote by an m m mn m m ma m m

5 cta Polytechnca Hungarca Vol No 3 3 Defne by = \ the symmetrc fference between crsp sets For fuzzy sets [N] efne by m m m Defne a sequence of ernoull ranom varables X for [N] takng the value wth respect to m an the value wth respect to - m Defne by HX the entropy of X H X m logm m log m Defne the ranom varable X w p m w p m We efne a new stance between [N] as for screte oman an N S3 H X N S 3 H X 8 for contnuous oman In [6] we prove that the functon S3 s a sem-metrc on Φ[N]; e t s non-negatve symmetrc equals zero f = an satsfes the trangle nequalty Note that for any [N] wth a crsp membershp value e m = or m = we have m an hence n ths case H X Ths means that for a crsp set for all m {} our stance has the followng property we call ths the complement-property st From an nformaton theoretc perspectve ths property s epecte snce knowng a set automatcally means that we also know how to escrbe ts complement Hence there s no atonal escrpton necessary to escrbe gven ts complement Ths s what st means It can be seen from the efnton that the functon st may equal zero even when s an eample conser the fuzzy sets C an the complement ' wth membershp functons as shown n Fgure Note that an ts complement are crsp sets The stance matr D = [ j ] s shown below; the rows an columns correspon to C an ' so that for nstance the element 3 = S3 C = 79 55

6 L Kovács et al nalyss of Lnear Interpolaton of Fuzzy Sets wth Entropy-base Dstances 354 D Dstance matr D s can be seen C s a translate verson of an they are both the same stance from Ths s ue to H X H X an C are farther apart C than an Snce S3 ' = then each one of C s of the same stance to as to ' Fgure [6] Fuzzy sets C an c 3 Dstance-Optmal Interpolaton lgorthm ccorng to 5 a lnear nterpolaton wth Euclean metrc generates elements wth optmal stance In ths paper we obtane epermental results usng the KH metho whch was use to generate the ntermeate fuzzy set C for gven [N] In these tests the value runs from to The test results are shown n Fgure In the Fgure the -as shows the value of ; on the y-as the value ffc = C + C - s gven The top re lne s the escrptve complety stance S3 the mle blue lne s the element-wse entropy stance S an the bottom green lne refers to the entropy-fference stance S The ffc value ncates whether the generate C element s the closest element to both an If ffc s equal to zero the trangle nequalty yels an equalty an C les on the lne connectng to 56

7 cta Polytechnca Hungarca Vol No Fgure Dstance fferences for S S an S3 ase on the test results we conclue the followng: Property : For the entropy-fference stance S for elements generate by KH nterpolaton the stance fference ffc s equal to zero Proof Let us take trapezo membershp functons wth the followng parameters for a set : } sup{ } sup{ } nf{ } nf{ 4 3 where symbol α=c enotes the set of ponts wth the membershp functon equal to c The entropy ffers from zero only on the ntervals an 3 4 The entropy value entropy s calculate wth log log Wth corresponng substtutons the ntegral can be transforme nto the form 4 log log z z z z z z Thus the entropy value for the set s equal wth entropy

8 L Kovács et al nalyss of Lnear Interpolaton of Fuzzy Sets wth Entropy-base Dstances e t s equal to the length of t non-crsp parts Takng a C KH-nterpolate set wth parameter the C wll be also a trapezo fuzzy set wth the followng parameters: It follows from the lnearty that also hols Thus C entropy C entropy entropy entropy C [mn{ entrpoy entropy } ma{ entrpoy entropy }] an ffc = s met ssumng the membershp functon can be appromate wth a chan of lnear segments the ffc = conton s fulflle for fuzzy sets of arbtrary shapes Property For every E the S3 S nequalty hols Proof Conser frst the followng nequalty H X entropy { } entropy { } 9 for every E The nequalty n 9 can be converte nto the followng epresson: K entropy { } entropy { } entropy { } The entropy functon can be substtute wth ts efnton: K log log log log log log where enotes m Let us f b to a value b an smplfy notaton a to s contans two absolute value epressons four fferent subomans shoul be efne: R: b entropy entropy b R : b entropy entropy b R3: b entropy entropy b R4 : b entropy entropy b In suboman R formula can be wrtten as K b log b b log b blog b blog b log log 58

9 cta Polytechnca Hungarca Vol No 3 3 The etreme pont of K meets the followng equaton Ths yels n an K log b log b log log b b b In suboman R the etreme ponts le on the lne y = In a smlar way the etreme ponts are the followng n the other subomans: R: y R : y R3 : no soluton R4 : no soluton s can be easly verfe the followng contons are met: K K K b Thus for every b [] the K functon has the followng functon-value segments: zero ncreasng ecreasng zero ncreasng ecreasng zero From ths fact t follows that K for every an b value Thus conton 9 s met The measure K values are gven n Fgure 3 59

10 L Kovács et al nalyss of Lnear Interpolaton of Fuzzy Sets wth Entropy-base Dstances From the fact Fgure 3 The K fference functon t follows that an H X entropy { } entropy { } H X entropy { } entropy { } H X H X entropy { } entropy { } Etenng the epresson to nfnte elements we get the epecte property S3 S s can be seen the KH nterpolaton algorthm s not sutable to generate a fuzzy set C lyng on the mle pont between an e S C S3 C 3 S3 In the net step an algorthm s presente for generatng the requre C set Property 3 The requre C set can be generate from n such a way that every elements of C s ether equal to or to Proof For the requre element C the equaton ff C S3 C S3 C S3 shoul be met It follows from efnton 8 that ff C H X C H X C H X 6

11 cta Polytechnca Hungarca Vol No 3 3 In a smlar way as was shown n the proof of Property we get an H X H X H X [] C C H X H X H X C C f an only f m m C C m or m If m = or = then m C can be equal to zero or too The same s true for m also Fgure 4 shows the value for m C [] m = m = 7 Fgure 4 The fference functon ase on ths result a constructve algorthm can be gven to generate C from the sets an The algorthm assgns ponts to C from n a greey way untl t reaches the requre stance value: Gen C = = whle S3 C > S3 { C = C +N ++ } 6

12 L Kovács et al nalyss of Lnear Interpolaton of Fuzzy Sets wth Entropy-base Dstances In Fgure 5 the fuzzy sets generate by KH an the propose Gen functon are splaye The two target trapezo fuzzy sets an are shown n Fgure 5a The KH nterpolate fuzzy set C' wth =5 s gven n Fgure 5b n the mle n a sol blue lne The nterpolate fuzzy set C'' generate wth Gen s shown n Fgure 5b wth a thck brown lne In the eample the followng stance values can be measure: S3 S3 S3 S3 S3 645 C' 56 C' 687 C C '' '' Thus the Gen metho yels the requre stance relatonshp for the nterpolate C set usng the escrptve complety stance Fgure 5a The an fuzzy sets Fgure 5b The nterpolate C sets 6

13 cta Polytechnca Hungarca Vol No 3 3 Concluson Ths paper analyzes the stance relatonshp among the base an generate fuzzy sets for KH lnear nterpolaton In the case of Euclean stance the usual behavor can be seen but n the case of entropy-base stances the new generate sets o not prove the stance optmum The paper presents new propertes among the entropy-base stances an proposes an approprate metho of stance optmum nterpolaton cknowlegement Ths research was supporte by the Hungaran Natonal Scentfc Research Fun Grant OTK K7789 References [] M Deza an E Deza Encyclopea of Dstances Vol 5 of Seres n Computer Scence Sprnger-Verlag 9 [] J Ratsaby Informaton Effcency In Proc of 33 r Conference on Current Trens n Theory an Practce of Computer Scence SOFSEM 7 Vol LNCS 436 pp [3] J Ratsaby Informaton Set-Dstance Proc of the Mn-Conference on pple Theoretcal Computer Scence MTCOS pp 6-64 Unversty of Prmorska Press Koper Slovena [4] L Zaeh Fuzzy Sets Informaton Control 8: [5] R Zwck E Carlsten an D V uescu Measures of Smlarty among Fuzzy Concepts: Comparatve nalyss Internatonal Journal of ppromate Reasonng : [6] L Kovács J Ratsaby: Descrptve-Complety-base Dstance for Fuzzy Sets CoRR abs/34: [7] Zs Cs Johanyák Sz Kovács: ref Survey an Comparson on Varous Interpolaton-base Fuzzy Reasonng Methos cta Polytechnca Hungarca Vol 3 No 6 pp 9-5 [8] Kóczy L T Hrota K: Rule Interpolaton by α-level Sets n Fuzzy ppromate Reasonng In J USEFL utomne UR-CNRS Vol 46 Toulouse France 99 pp 5-3 [9] Wong K W Geeon T D Tkk D: n Improve Multmensonal α- Cut-base Fuzzy Interpolaton Technque In Proc Int Conf rtfcal Intellgence n Scence an Technology IST Hobart ustrala pp 9-3 [] Geeon T D Kóczy L T: Conservaton of Fuzzness n the Rule Interpolaton Intellgent Technologes Int Symposum on New Trens n Control of Large Scale Systems Vol Herl any 996 pp

14 L Kovács et al nalyss of Lnear Interpolaton of Fuzzy Sets wth Entropy-base Dstances [] Tkk D arany P: Comprehensve nalyss of a New Fuzzy Rule Interpolaton Metho IEEE Trans Fuzzy Syst Vol 8 June pp 8-96 [] Perflva I Wrublova M Hoakova P: Fuzzy Interpolaton ccorng to Fuzzy an Classcal Contons cta Polytechnca Hungarca Vol 7 No 4 pp