A New Grey-rough Set Model Based on Interval-Valued Grey Sets

Transcription

1 Proceedings of the 009 IEEE Interntionl Conference on Systems Mn nd Cybernetics Sn ntonio TX US - October 009 New Grey-rough Set Model sed on Intervl-Vlued Grey Sets Wu Shunxing Deprtment of utomtion Ximen University Ximen P R Chin E-mil: sxwu@xmu edu cn Lin LihuGo Qinqun Deprtment of utomtion Ximen University Ximen P R Chin E-mil: 00@qq com bstrct In this pper novel grey rough set model for intervl-vlued grey sets informtion systems nmed intervlvlued grey-rough set model is proposed nd the bsic theory nd propositions of the intervl-vlued grey-rough sets is studied sed on the proposed intervl-vlued rough grey model rough similrity degree is defined nd the clustering of the intervl-vlued grey informtion systems is lso exmined in the menwhile some exmples re presented respectively Keywords intervl-vlued grey sets grey-rough set grey cluster I INTRODUCTION Rough set theory primrily proposed by Pwlk [] is useful mthemticl frmework to del with imprecision vgueness nd uncertinty in informtion systems This theory hs been mply demonstrted to hve usefulness nd verstility by successful ppliction in vriety of problems such s in pttern recognition mchine lerning dt mining clssifiction clustering concurrent systems decision nlysis [] imge processing informtion retrievl knsei engineering signl processing system modeling nd voice recognition Pwlk proposed miniml decision rule induction from n indiscernibility reltion nd lso proposed two rough pproximtions n upper pproximtion tht dels with possibility nd lower pproximtion tht dels with certinty of informtion tbles In prticulr rough pproximtion is bsic mthemticl model for hndling informtion tbles However Clssicl rough sets minly del with nominl dt or multi-vlued discrete dt such s smll lrge short long true or flse nd the clssicl rough set theory hs difficulty in hndling intervl-vlued of ttributes which exist in the rel world Informtion tbles in the rel world however re quite complicted Mny mesured vlues given s grey dt which hve rnge for exmple ID mss No temperture nd so on Such vlue rnges cn be described by intervl sets in this pper Grey system theory proposed by Deng [] covers grey clssifiction grey control grey decision-mking grey prediction grey structurl modeling grey reltionl nlysis [] etc It dels with the uncertinty over how systems with incomplete or lck of informtion should be controlled One of the importnt concepts is grey number It is number whose exct vlue is unknown but rnge is known Thus grey system theory dels with uncertinty unlike those of fuzzy set theory or rough set theory The grey opertion is one of the opertions for grey numbers tht modifies rnge of given intervls of grey sets It is more suitble to hndling informtion tbles contining intervl set Theories of rough sets nd grey sets re generliztions of clssicl set theories for modeling vgueness nd fuzziness respectively [] It is generlly ccepted tht these two theories re relted but distinct nd complementry to ech other [90] s generliztion of grey sets the notion of intervl-vlued grey sets ws suggested by which we describe the incomplete informtion system in this pper [] In this pper we first proposed novel grey rough set model for intervl-vlued grey sets dt nmed intervl-vlued grey-rough set nd its upper pproximtion nd lower pproximtion is defined then we investigte its propositions bsed on it rough similrity degree is defined which is pointed out resonble in the menwhile some exmples re presented respectively II PRELIMINRIES Intervl-vlued grey sets Let U denote the universl set x denote n element of U Definition : n intervl-vlued grey sets is set whose exct elements is uncertin but rnge is known Let G be n intervl-vlued grey mpping set of U defined by two crisp sets of the upper bound nd the lower bound s follows: P Q where PQ re two crisp sets stisfying P Q In other words denote the set of vlue rnge tht my hold Q Throughout this pper we mrk it s = P Especilly when P = Q the intervl-vlued grey sets G becomes crisp set which mens tht intervl-vlued grey system theory dels with flexibly sets sitution Tolernce reltion bsed on intervl-vlued grey sets R Definition : Let U be non-empty finite universe be the tolernce reltion on the U with respect to ll ttributes /09/$ IEEE 0

2 in [ ] R denotes the tolernt clss which including the then ( U R is the Pwlk pproximtion spce For ny intervl-vlued grey set the tolernce reltion defined s follows: R cn be R = {( ( x ( y U U f ( ( x f ( ( y φ The tolernt clss induced by the tolernce reltion R is the set of objects ( x i e [ ( x] = { ( y U ( ( x ( y R = { ( y U f ( ( x f ( ( y φ Where [ ( x] describes the set of objects tht my tolernt with ( x in terms of in n intervl-vlued grey informtion system From the definition of [ ( x] the following properties cn be esily obtined Property : Let ( U G be n intervl-vlued grey informtion system nd then R is reflexive symmetric If then R R R [ ( x] [ ( x] [ ( x] For ny then R = R { From the definition we know P = {[ ( x] ( x U constitute covering of U Definition Given two intervl-vlued grey sets ( ( x + x x y+ = x nd ( y ( y = y where objects x y U The intervl-vlued grey sets opertion is defined s follows: ( x + ( y + ( x + x y z y + x y = x y ( x + ( y + ( x + x y z y + x y = x y ( x ( y x y x y ndx y Exmple : n intervl-vlued grey informtion system is presented in Tble Compute the clssifiction induced by the tolernce reltion R in Tble I TLE I INTERVL-VLUED GREY INFORMTION SYSTEM U ( x ( x ( x ( x ( x ( x ( x ( x { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { From tble I we cn get: [ ( x ] { ( ( ( ( = x [ ( x ] { ( ( ( ( ( = [ ( x ] { ( ( ( ( = x [ ( x ] { ( ( ( ( = [ ( x ] = { ( ( ( ( ( x [ ( x ] { ( ( ( ( ( = So [ ( x ] { ( ( ( ( = [ ( x ] { ( ( ( ( = nlogously we cn get tolernt clss of ll the objects in the universe III THE INTERVL-VLUED GREY-ROUGH SET MODELS Definition : For ny intervl-vlued grey set ( x we define: 0

3 R( ( x = { ( s U f ( ( s f ( ( x ndf ( ( s f ( ( x R( ( x = { ( s U f ( ( s f ( ( x φ ( s ( x f ( ( s f ( ( x ndf ( ( s f ( ( x Where ( s ( x denote grey objects f ( ( s f ( ( x denote the vlue of grey objects ( s ( x in terms of ttribute R( ( x R( ( x is clled the lower pproximtion nd the upper pproximtion of the intervl-vlued grey set ( x bout the pproximtion spce ( U R with respect to the ttribute respectively Property : Let ( U F be n intervl-vlued grey informtion system from the definition bove we hve tht: R( ( x = R( ( x R ( ( x = R( ( x nd if R( ( x = φ then R( ( x = φ especilly Lemm: Given two intervl-vlued grey sets x+ R( ( x y+ x nd R( ( y then: y If x = y nd x y then R( ( x R( ( y x y R( ( x R( ( y x = y If x y nd x = y then R( ( x R( ( y x y R( ( x R( ( y x y If x y nd x y then R( ( x R( ( y x y R( ( x R( ( y x y If x y nd x y then R( ( x R( ( y x y Proof: It is esy to prove this conclusion by the definition Property : Let ( U G be n intervl-vlued grey informtion system we hve tht: R( ( x R( ( x If (x = Ω then R( ( x = R( ( x = U if (x = φ then R( ( x = R( ( x = φ If ( x ( y then R( ( x R( ( y R( ( x R( ( y If then R( ( x R( ( x If ( x ( y ( z then R( ( x R( ( y R( ( z If y R( ( x then R( ( y R( ( x R( ( y R( ( x Proof: They re strightforwrd Property : Let ( U R be Pwlk pproximtion spce then the following properties hold for ny intervlvlued grey sets ( x nd ( y of the universeu : R(( ( x ( ( y R(( ( x R(( ( y R(( ( x ( ( y R(( ( x R(( ( y R(( ( x ( ( y = R(( ( x R(( ( y R(( ( x ( ( y = R(( ( x R(( ( y Where R( ( x R( ( x denote the lower pproximtion nd the upper pproximtion of the intervlvlued grey set ( x with respect to the ttribute nd f + ( ( x f ( ( x denote the upper bound nd the lower bound of the intervl-vlued grey set ( x in terms of the ttribute respectively Proof: R(( ( x ( ( y = { ( s f + ( ( s f + ( ( x f + ( ( y nd f ( ( s f ( ( x f ( ( y 0

4 Note tht { ( s f + ( ( s f + ( ( y f + ( ( x nd f ( ( s f ( ( y f ( ( x { ( s f ( ( s f ( ( x nd f ( ( s f ( ( x { ( s f ( ( s f ( ( y nd f ( ( s f ( ( y = R(( ( x R(( ( y Tht is R(( ( x ( ( y R(( ( x R(( ( y R( f ( ( x f ( ( y = { ( s f + ( ( s f + ( ( y f + ( ( x nd f ( ( s f ( ( y f ( ( x Note tht { ( s f + ( ( s f + ( ( y f + ( ( x nd f ( ( s f ( ( y f ( ( x { { ( s f ( ( s f ( ( y nd f ( ( s f ( ( y { { ( s f ( ( s f ( ( x nd f ( ( s f ( ( x = R( f ( ( x R( f ( ( y Tht is R(( ( x ( ( y R(( ( x R(( ( y R(( ( x ( ( y = { ( s f ( ( s [ f ( ( x f ( ( y ] φ + = { ( s f ( ( s f ( ( x f ( ( y φ = { ( s f ( s f ( ( x φ nd f + ( ( s f + ( ( y φ = { { ( s f + ( ( s f + ( ( x φ { { ( s f + ( ( s f + ( ( y φ = R(( ( x R(( ( y R(( ( x ( ( y = { ( s f ( ( s [ f ( ( x f ( ( y ] φ + = { ( s f ( ( s [ f ( ( x f ( ( y] φ = { ( s f ( ( s f ( ( x φ f + ( ( s f + ( ( y φ = { { ( s f + ( ( s f + ( ( x φ { { ( s f + ( ( s f + ( ( y φ = R(( ( x R(( ( y The proof is complete Exmple :( continue exmple Compute the lower pproximtion nd the upper pproximtion of the intervlvlued grey objects : R( ( x = { ( x ( x R( ( x { ( = R( ( x { ( ( ( = R ( ( x { ( ( ( ( = R ( ( x { ( ( ( ( = x R( ( x { ( ( ( = R ( ( x { ( ( ( = R( ( x { ( ( ( = R( ( x { ( ( ( = R( ( x { ( ( ( ( ( = R( ( x { ( ( ( ( = R( ( x { ( ( ( ( ( = 0

5 Then R( ( x { ( = R( ( x { ( ( ( = R( ( x { ( ( = R( ( x { ( ( ( ( = Similrly we cn get ll the objects s follows: R( ( x { ( = R( ( x { ( ( ( ( = R( ( x { ( = R( ( x { ( ( ( ( = R( ( x { ( = R( ( x = { U R( ( x { ( ( = R( ( x { ( ( ( ( = R( ( x { ( = R( ( x { ( ( = R( ( x { ( ( ( ( = IV SIMILRITY DEGREE ND PROPERTIES IN THE INTERVL-VLUED GREY SET INFORMTION SYSTEM Inspired by the wy of fuzzy clustering in fuzzy set we introduce novel similrity degree to mesure the similrity of two objects Further more we point out tht the new similrity degree we propose in this pper is meeting with the xiomtic definition of similrity degree We first give the xiomtic definition of similrity degree s follows Definition : Let γ : ( x y γ ( x y be the binry function on the U nd it stisfy tht: γ ( xx = γ( x φ = 0 Definition Given intervl-vlued grey objects ( x ( x of the U nd ( R( ( x R( ( x ( R( ( y R( ( y then we define similrity degree s follows: R( ( x R( ( y S( ( x ( y = α + R( ( x R( ( y R( ( x R( ( y β R( ( x R( ( y Where α + β = nd * denotes the crdinlity of set R( ( x R( ( y Especilly S = is clled the R( ( x R( ( y similrity degree of lower pproximtion nd R( ( x R( ( y S = is clled the similrity degree of R( ( x R( ( y upper pproximtion From the definition bove we hve tht: S( ( x ( x = S( ( x φ = 0 S( ( x ( y = S( ( y ( x I f ( x ( y ( z then S( ( x (z min{ S( ( x ( y S( (y (z Proof: is strightforwrd If ( x ( y ( z then R( ( x R( ( y R( ( z nd R( ( x R( ( y R( ( z R( ( x R( ( y R( ( z Further more R( ( x R( ( y S( ( x ( y = α R( ( x R( ( y R( ( x R( ( y + β R( ( x R( ( y R( ( x R( ( x R( ( y R( ( y γ( x y = γ( y x x y z γ( x z min{ γ( x y γ( y z Then we cll it similrity degree S( ( y ( z = α R( ( y R( ( z R( ( y R( ( z 0

6 S ( ( x ( z = α R( ( y R( ( z + β R( ( y R( ( z R( ( y R( ( y R( ( z R( ( z R ( ( x R( ( z R ( ( x R( ( z R( ( x R( ( z + β R( ( x R( ( z R( ( x R( ( x R( ( z R( ( z So we hve S( ( x (z S( ( x ( y nd S( ( x (z S( ( y ( z Tht is S( ( x (z min{ S( ( x ( y S( (y (z From bove we cn conclude sfely tht the definition of similrity degree is resonble Property :If R( ( x = R( ( y nd R( ( x = R( ( y then S( ( x ( y = Especilly when y R( ( x R ( ( y R( ( y S( ( x ( y R ( ( x R( ( x Exmple : (continue exmple Let α = 0 then similrity degree of ech object cn be clculted ccording to the definition s follows: S ( ( x ( = R( ( x ( ( R R( ( x ( ( R R( ( x ( ( R R( ( x ( ( R { ( x ( x ( x ( x { ( x ( x ( x ( x = 0 { ( x ( x ( x ( x { ( x ( x ( x ( x = / In similr wy we cn lso get ll the objects nd construct similrity degree mtrix (see tble II TLE II SIMILRITY DEGREE MTRIX S ( x ( x ( x ( x ( x ( x ( x ( x ( x / / / / / /0 / ( x / / / /0 / ( x / / / /0 / ( x / / /0 / ( x / / / ( x /0 / ( x /0 ( x From the tble bove if let λ = 0 then the universe cn V CONCLUSION be clustered into four clsses: Rough sets nd grey sets theory re two mthemticl tools { ( x ( ( { ( x ( ( to del with uncertinty Combining them together is of both theoreticl nd prcticl importnce This pper combines the { ( x { ( x intervl-vlued grey sets with the rough sets proposes novel grey rough set model for intervl-vlued grey sets dt nmed intervl-vlued grey-rough set nd studies the bsic theory of the intervl-vlued rough grey sets sed on the generlized intervl-vlued rough grey sets the clustering of the intervlvlued grey informtion systems is lso studied Our future 0

7 work will concentrte on the intervl-vlued grey rough set model in the intervl-vlued grey informtion systems n ppliction of the model which is presented in this pper will lso be reserched in the future CKNOWLEDGMENT The uthors of this pper would like to pprecite the nonymous reviewers of this pper for their crefully reding of the mnuscript s well s their mny helpfully suggestions nd corrections REFERENCES [] Z Pwlk Rough sets: Theoreticl spects of Resoning bout Dt oston: Kluwer cdemic Publishers 99 [] W X Zhng Y Ling W Z Wu Informtion system nd knowledge discovery eijing Publish House of Science 00 [] W X Zhng G F Qiu Uncertin Decision Mking sed on Rough Sets Science Press eijing Chin 00 [] S X Wu Grey Rough Set Model nd Its ppliction eijing Publish House of Science 009 [] J L Deng Foundtion of grey theory Wuhn Publish House of Huzhong University of Science nd Technology 00 [] Q Y Wng Foundtion of grey mthemtics Wuhn Huzhong University of Science nd Technology 99 [] D Ymguchi G D Li M Ngi grey-bsed rough pproximtion model for intervl dt processing Informtion Sciences vol 00 pp [] S F Liu Y Lin n Introduction to Grey Systems Methodology nd pplictions Slippery Rock: IIGSS cdemic Publisher 99 [9] S X Wu S F Liu M Q Li Study of integrte models of rough sets nd grey systems Fuzzy Systems nd Knowledge Discovery Proceedings Lecture Notes in rtificil Intelligence no ug 00 pp - [0] D Luo S F Liu S X Wu: Reserch on the Grey Rough Combined Decision-mking Model Journl of Ximen University Vol (00-0 [] Turken Intervl vlued fuzzy sets bsed on norml forms Fuzzy Sets nd Systemsvol0 9 pp9 0 [] DDubois nd HPrde Rough fuzzy sets nd fuzzy rough sets Int J Gen Syst vol no 990pp 9 09 [] ZT Gong Z Sun DG Chen Rough set theory for the intervlvlued fuzzy informtion systems Informtion Sciences vol00 pp9 9 [] S Greco Mtrzzo R Slowinski Rough sets theory for multicriteri decision nlysis Europen Journl of Opertionl Reserch