2 PRELIMINARY. 2.1 Variable Precision Rough Set Theory

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1 ough Fuzzy Set sed on Logicl Disjunct Opertion of Vrible Precision nd Grde in Ordered Informtion System Ynting Guo 1 Weihu u 12 1 School of mthemtics nd sttistics Chongqing University of Technology Chongqing E-mil: @qqcom 2 The Key Lbortory of Intelligent Perception nd Systems for High-Dimensionl Informtion Ministry of Eduction Nnjing University of Science nd Technology Nnjing E-mil: chuwh@gmilcom bstrct: Combining the respective dvntges of reltive nd bsolute quntittive informtion this pper proposes the model of the rough fuzzy set which is bsed on logicl disjunct opertion of vrible precision nd grde in ordered informtion system Moreover the bsic structure nd precise description of rough set regions of the model re studied nd some importnt properties of the model re investigted crefully Further the significnce of this study is interpreted by combining the nlysis of n illustrtive cse study bout the medicl dignosis Key Words: Grded ough Fuzzy Set Logicl Disjunct Opertion Ordered Informtion System Vrible Precision ough Fuzzy Set 1 INTODUCTION ough set theory [1] is n effective tool to del with ll inds of incomplete informtion It hs been widely pplied in mny fields such s mchine lerning nowledge discovery dt mining decision support nd nlysis informtion security networing cloud computing nd biologicl informtion processing Ordered informtion systems bsed on dominnce reltions [2] s bsic tool of nowledge representtion in the rough set theory cn solve mny prcticl problems The min ide of rough set theory [3-7] is tht undefinble sets cn be pproimted by definble sets nmely the upper nd lower pproimtion set The lower pproimtion set is the union of equivlence clsses completely included in the undefinble set nd the upper pproimtion set is the union of equivlence clsses prtilly included in the undefinble set The clssicl rough set is model which not llows ny uncertinty so it lcs fult tolernce cpbility In rel life however lot of problems to solve re llowed to hve certin fult tolernce Therefore people proposed vrious generlized rough set modes [8-11] such s the decision rough set model [12]proposed by Yo Ziro's vrible precision rough set model [13] prmeterized rough set model [14] Herbert nd Yo's gme rough set model [15] grded rough set model [16] The vrible precision rough set model nd grded rough set model reflect bsolute quntittive informtion nd reltive quntittive informtion of the pproimtion spce respectively The two models re both independent nd complementry Through the combintion of vrible This wor is supported by Nturl Science Foundtion of Chin No No No ) Ntionl Nturl Science Foundtion of CQ CSTC No cstc13jcyj40051 cstc15jcyj40053) nd Grdute Innovtion Foundtion of Chongqing University of Technology No YC15227 YC14236) nd Grdute Innovtion Foundtion of CQ NoCYS15223) precision nd grded rough set theory [16-21] some higher level etension models re estblished These models cn provide more detiled description for the pproimtion spce For emple by considering the combintion of bsolute nd reltive quntittive informtion in the upper nd lower pproimtion set two inds of double-quntittive rough set model cn be obtined [18] From the perspective of logic generlized model [-21] obtined by considering the bsolute nd reltive informtion Quntittive composite models with fult tolernce bility cn be used for the qulittive nlysis of nowledge which is of gret significnce nd is very good reserch direction Therefore the rough fuzzy set model bsed on logicl disjunct opertion of vrible precision nd grde re proposed in the ordered informtion system This model provides more ccurte pproimtion of fuzzy concepts Menwhile the reserch significnce of the model is nlyzed through specific emple bout medicl dignosis 2 PELIMINY 21 Vrible Precision ough Set Theory Let C denote the crdinl number of the set C nd c[ ] ) denote error clssifiction rte of the equivlence clss [ ] with respect to c[ ] ) cn be clculted by c[ ] ) 1 [ ] # [ ] where is rel number between 0 nd 1 is clled djustble error clssifiction level nd 1 is clled precision Suppose nd denote the upper nd lower pproimtion sets of when the precision is1 where! {[ ] : c[ ] ) 1 } nd!{[ ] : c[ ] ) % } If then is /16/$3100 c 16 IEEE 5508

2 rough when the precision is1 ;otherwise is precise The upper pproimtion set is the union of equivlence clsses stisfying c[ ] ) 1 ndthe lower pproimtion set is the union of equivlence clsses stisfying c[ ] ) % It should be noted tht the rnge of prmeter is generlly [005) in the vrible precision rough set model [11] Of course the rnge cn lso be is limited to [051] which cn be regrded s promotionl rough set theory [2] Without loss of generlity the rnge of the prmeter is limited to [01] in this pper It is evident tht the former & 0 05] nd the ltter & 051] re symmetricl Therefore in generl we only need to study the former cse & 0 05] nd the ltter cse cn be obtined similrly 22 Grded ough Set Theory Let nd denote the upper nd lower pproimtion sets of when the grde is ny nturl number nd cn be defined s! {[ ] [ ] # / } nd! {[ ] [ ] [ ] # % } If then is rough when the grde is ;otherwise is precise The upper pproimtion is the union of some equivlence clsses nd ll these equivlence clsses must stisfy [ ] # / nd the lower pproimtion is the union of some equivlence clsses nd ll these equivlence clsses must stisfy [ ] [ ] # % 23 The ough Set of Ordered Informtion Systems Let I U V F) be n informtion system where the universe of discourse U { 1 2 n } is nonempty finite set of objects is nonempty finite set of ttributes V! V nd F { f U 0 V & } is the & reltionship set from U to ndv is the finite domin of For given informtion systems if there is prtil ordered reltion onv then the ttribute is clled criterion y & U y represents tht with respect to the criterion is superior to y nd y 1 f ) f y ) For 2 y represents tht with respect to the criterion set is superior to y nd y 1 it is true tht f ) f y ) for ny & If ech ttribute is criterion in I then I is clled ordered informtion system which cn be denoted by I U V F) Sometimes for convenience it cn lso be denoted by I U ) Let I U V F) informtion system For 2 the dominnce reltion of the criterion set cn be defined s { y) & U ) U : & f ) % f y )} The dominnce clss of with respect to is defined s [ ] { y & U : y) & } { y & U & f ) % f y )} In the ordered informtion system for 2 U the upper nd lower pproimtion sets of with respect to re defined s )! {[ ] [ ] # 3} nd )! {[ ] [ ] 2 } nd the positive negtive nd boundry region of with respect to re defined s pos ) ) neg ) ) nd bnd ) ) )If then with respect to with respect to is rough; if is precise or describble[2] 3 OUGH FUZZY SET THEOY then In this section new rough set model is proposed nmely the rough fuzzy set model bsed on logicl disjunct opertion of vrible precision nd grde in the ordered informtion system Let I U V F) informtion system For & F U) is the dominnce reltion of I In the ordered informtion system I generlized models of clssicl vrible precision rough sets nd grded rough sets re proposed through the promotion from clssic sets to fuzzy sets Definitions of vrible precision rough fuzzy set the grded rough fuzzy set the rough fuzzy set bsed on the dominnce reltion nd the rough fuzzy set bsed on logicl disjunct opertion of vrible precision nd grde re proposed Definition 31 The resolution rtio c[ ] ) of the dominnce clss with respect to the fuzzy set is defined s [ ] ) 1 c y) [ ] where the [ ] rel number &[01] is clled djustble resolution level nd 1 is clled precision For ny object ofu the precision is1 the upper nd lower pproimtion sets of with respect to the dominnce reltion re defined s follows: ) ) { ): [ ] ) 1 } [ ] ) ) { y ): c [ ] ) % } [ ] If then with respect to is rough when the precision is 1 ;otherwise with respect to is precise It is true tht nd re both 16 28th Chinese Control nd Decision Conference CCDC) 5509

3 fuzzy sets on the universe of discourse U The membership degree of with respect to is the supremum of with respect to dominnce clsses whose the resolution rtio is less thn 1 ; the membership degree of with respect to is the infimum of with respect to dominnce clsses whose the resolution rtio is not more thn Definition 32 Let N 05 {05 n n& N} & N 05 where N is the nturl numbers set For & U when the grde is the upper nd lower pproimtion set of with respect to re defined s follows: ) ) { ) : y y) / } [ ] + ) ) { ) : 1 y y) % } [ ] + If then with respect to is rough when the grde is ;otherwise is precise It is evident tht nd both re fuzzy sets on U The membership degree of with respect to is the supremum of with respect to dominnce clsses nd ll these dominnce clsses must stisfy y) / ;the + membership degree of with respect to is the infimum of with respect to dominnce clsses nd ll these dominnce clsses must stisfy 1 y) % Definition 33 For & F U) & U in the ordered informtion system the upper nd lower pproimtion the positive negtive nd boundry region of with respect to re defined s follows: ) ) { y): y) / 0} [ ] { & ) ) y): y } pos ) ) ) neg ) ) bn ) ) If then with respect to is rough; otherwise is precise or definble Definition 34 Let I U V F) informtion system nd is the dominnce reltion of I In the system I & F U) the upper nd lower pproimtion sets of with respect to bsed on logicl disjunct opertion of precision 1 nd grde cn be defined s follows: ) ) { y): c[ ] ) 1 or [ ] y )/ } ) ) { y): y &[ ] c[ ] ) or 1 % y ) % } y & ccording to the positive negtive nd boundry definition of clssicl rough set the definition of this model in other regions cn be similrly defined Detils re s follows: pos # neg! ) Ubn Lbn bn Ubn! Lbn where pos neg Ubn Lbn nd bn re clled the positive negtive upper boundry lower boundry nd boundry region of with respect to If then with respect to is rough otherwise is precise or definble The rough fuzzy set model bsed on logicl disjunct opertion of vrible precision nd grde is stted s bove y eploring the content the following properties cn be obtined Theorem 31 Let I U V F) informtion system is ny fuzzy set of U nd is the dominnce reltion of I The following conclusion is true nmely 1)! 2)! where &[01] nd & N 05 Proof 1) For & U itistruetht ) ) { y): c[ ] ) 1 or y) / } Itis [ ] evident tht ) ) { y): c[ ] ) [ ] 1 }) { y): y) / }) ccording to [ ] the definition 31 nd 32 there re ) ) { ): [ ] y c ) 1 } ) [ ] y &[ ] { y): y) / } ) Therefore it is true tht! 2) The method nlogous to tht used bove th Chinese Control nd Decision Conference CCDC)

4 Theorem 32 Let I U V F) informtion system is ny fuzzy set of U nd is the dominnce reltion of I It is true tht 1) pos! Ubn 2) pos! Lbn where &[01] nd & N 05 Proof ccording to the definition 34 it follows Theorem 33 Let I U V F) informtion system is ny fuzzy set of U nd is the dominnce reltion of I It is true tht 1) ) ) { ): y y) / [ ] min [ ] )} 2) ) ) { ): y y) [ ] min[ ] [ ] [ ] )} where &[01] nd & N 05 Proof 1) ccording to the definition 34 it is true tht ) ) { y): c[ ] ) 1 or [ ] y )/ } for ny object of U y considering the condition c[ ] ) 1 y) [ ] thereis [ ] ) ) { y): y) / [ ] [ ] + or y) / } Therefore there re ) ) { y): y) / [ ] or y )/ } y &[ ] 2) The property cn be proved in similr 1) Theorem 34 Let I U V F) informtion system is ny fuzzy set of U nd is the dominnce reltion of I There re the following ssertions set up 1) When 0 05nd 0 therere ) ) { pos y ):[ ] [ ] y) 1 ) [ ] }) { y) : 1 ) [ ] [ ] + ) }) y [ ] [ ] { y):[ ] 1 ) % y )/ [ ] }) ; [ ] y & neg ) ) 1 { { y ) [ ] y) / }) { y) [ ] [ ] y) / [ ] }) { y) [ ] % [ ] y & 1 )[ ] % [ ] })} ; ) ) { Ubn y ) [ ] [ ] y ) 1 ) [ ] }) { y) ) [ ] 1 y)) [ ] y) [ ] 1 )[ ] }) { y)) 1 y)) [ ] [ ] 1 ) [ ] y) [ ] } ) + { y) 1 ) [ ] [ ] [ ] + y ) [ ] }) { y)) 1 [ ] [ ] )):[ ] 1 ) y % y )/ [ ] ); y & ) ) { Lbn y ) [ ] % 1 [ ] y &[ ] % y) % [ ] }) { y)) [ ] + 1 y)) [ ] y) [ ] 1 ) [ ] ) { y)) 1 y)) y &[ ] [ ] 1 ) [ ] y) [ ] ) + { )) 1 y y)) [ ] % [ ] [ ] 1 ) y) / [ ] ) 2) When 05 % 1 nd 0 there re ) ) { pos y ):[ ] 1 ) + [ ] y) 1 ) [ ] }) { y): [ ] [ ] 1 ) }) { y / y) [ ] [ ] % y) / [ ] }) ; + ) ) 1 { { neg y ):[ ] / [ ] }) { y) / y) [ ] [ ] y & 1 ) 1 ) [ ] % y) % }) y & [ ] { ) [ ] y % y) / [ ] }) th Chinese Control nd Decision Conference CCDC) 5511

5 { y) [ ] [ ] % % y) [ ] % [ ] })} ; + ) ) { Ubn y ) [ ] 1 ) [ ] y ) 1 ) [ ] }) { y) ) [ ] 1 y)) [ ] 1 ) y) [ ] 1 )[ ] }) { y)) 1 y)) [ ] [ ] [ ] 1 ) y) / }) { + [ ] y)) 1 y)) [ ] % y) / [ ] }) ; [ ] Lbn ) ) { y ): [ ] 1 ) [ ] 1 ) [ ] % y) % }) { y)) [ ] y & 1 y)) : [ ] 1 ) y) [ ] + 1 )[ ] }) { y)) 1 y)) [ ] [ ] [ ] 1 ) y) / }) { + [ ] y)) 1 y)) [ ] % y) [ ] + [ ] }) { / y) [ ] % [ ] y &[ ] % y) % [ ] }) Proof ecuse the first cse 0 05 nd the second cse 05 % 1 re symmetry so the following only provide the resoning process of the cse 1) 1) y preconditions 0 05 nd 0 itistrue tht 1 nd 1 ) In order to obtin concise results the comprison between [ ] nd [ ] is mde When [ ] ccording to the theorem 33 there re ) ) { y) : y &[ ] y [ ] y) / } ) ) { ): [ ] [ ] } Then by pos # it is obvious tht ) ) { pos y ) [ ] [ ] y )1 )[ ] } When 1 ) [ ] it is evident tht pos ) ) y &[ ] { ) 1 ) [ ] y y) [ ] } y & When [ ] % /1 ) there re pos ) ) { ) [ ] 1 ) y % y) / [ ] } [ ] So the conclusion bout pos ) ) is true nd other conclusions cn be similrly obtined Theorem 35 Let I U V F) informtion system nd Ỹ re fuzzy sets of U nd is the dominnce reltion of I The following ssertions re vlid where& [01] l & N 05 1) 333 U U 2)If 2 Y then 2 Y 2 Y 3) )! Y 4! Y! Y ) 4! Y 4) ) # Y 2 # Y Y ) # 2 # Y 5)When l therere 2 l 2 2 l 4 l 4 4 l Proof The conclusions re obvious 4 CSE NLYSIS Let I U V F) informtion system where U { 1 2 } is composed of ptients with influenz { } is the ttribute set tht represents hedche fever cough nd muscle pin nd V {1 2 3} is the set tht represents the etent of the disese Let 1 indicte lighter degree of the disese; 2 indicte moderte degree of the disese nd 3 indicte hevier degree of the disese Detiled sttisticl dt is shown in Tble 41 t first prt of the ptients were rndomly selected s the reserch object nmely { } The etent of the ptient with influenz cn be determined by bove symptoms It is fuzzy concept tht the etent of the ptient with influenz ccording to the result of medicl emintion nd doctor dignosis the etent of seven ptients with influenz is followed by nd 064 The fuzzy set formed by the bove evlution cn be denoted by à nmely { 095) 082) 076) 088) 054) ) 064)} Then the object clssifiction bsed on dominnce reltions cn be clculted th Chinese Control nd Decision Conference CCDC)

6 Tble 41 Medicl dignosis sttistics U hedche fever cough Muscle pin Might s well choose 03 nd 1 the vlues of [ i ] y) [ ] c ) nd [ ] i y) re shown in Tble 42 Cse 1 The required vlues cn be clculted by the theorem 31 The detiled clcultion process is s follows: 1) When 03 therere { 095) 095) 095) 095) ) 11088) 13095) 16095) 17095) 18095) 19095) 064)} { 1095) 7076) 11088)} When 1therere { 095) 095) 095) 095) ) 19095)} { 095) 076) 088)} ) ccording to the results obtined from 1) nd theorem 31 the following conclusions cn be obtined nmely 0 31 { 1095) 95) 4095) 5095) 076) 095) 088) 095) ) 095) 095) 095) ) 095) 064)} { 095) 076) 088)} ) Combined with the results obtined by step 2)the following conclusions cn be obtined by the definition 34 pos 031 { 1095) 7076) 11088)} neg 031 { 1005) 05) 31) 4005) 005) 1) 024) 005) 1) 1) ) 005) 005) 005) 1) ) 005) 005) 005) 036)} Ubn 031 { 1005) 95) 4095) 5095) 024) 095) 012) 095) 095) 095) 095) 095) 095) 095) 064)} Lbn 031 { 1005) 7024) )} b n 031 { 1005) 95) 4095) 50 95) 024) 095) 012) 095) 095) 095) 095) 095) 095) 095) 0 64)} Cse 2 ccording to the theorem 34 the required vlues cn be directly clculted When 03 1therere pos 031 { 1095) 7076) ) } neg 031 { 1005) 05) 31) 4005) 005) 1) 024) 005) 1) ) 012) 005) 005) 005) ) 036)} 005) 005) 005) 005) Ubn 031 { 1005) 95) 4095) 5095) 024) 095) 012) 095) 095) 095) 095) 095) 095) 095) 064)} Lbn 031 { 10 05) 7024) )} bn 031 { 1005) 95) 4095) 5095) 024) 095) 012) 095) 095) 095) 095) 095) 095) 095) 0 64)} Obviously clcultion processes of two methods re different The step of the first method is s follows: nd re firstly clculted; then nd re obtined; finlly the other regions re clculted The second method is to directly use the result of the theorem to clculte The results of the bove two methods re consistent In prcticl ppliction it is necessry to select pproprite methods to solve problems ccording to their own needs 16 28th Chinese Control nd Decision Conference CCDC) 5513

7 Tble 42 The sttisticl dt bout dominnt clsses y) [ c[ ] ) i [ i ] i ] y) CONCLUSIONS In rel life the fuzzy phenomenon cn be seen everywhere How to ccurtely describe fuzzy concept is very meningful reserch topic From the perspective of logic reltionships this pper considers the reltive quntiztion nd bsolute quntiztion informtion of pproimtion spces It is more importnt tht the rough fuzzy set model bsed on logicl disjunct opertion of the vrible precision nd grde re proposed in the ordered informtion system Through in-depth study the bsic structure of the model nd its importnt properties re obtined Finlly n illustrtive emple is introduced to further deepen the understnding of the model EFEENCES [1] Z Pwl ough sets Interntionl Journl of Computer nd Informtion Sciences Vol11 No [2] W H u Ordered Informtion Systems nd ough Sets Science Press eijing Chin 13 [3] K Dembczyńsi Pindur Susmg Dominncebsed rough set clssifier without induction of decision rules Electronic Notes in Theoreticl Computer Science Vol82 No [4] K Dembczyńsi Pindur Susmg Genertion of ehustive Set of ules within Dominnce-bsed ough Set pproch Electronic Notes in Theoreticl Computer Science Vol82 No [5] Z Pwl ough SetsTheoreticl spects of esoning bout Dt Kluwer cdemic Publishers US 1992 [6] W ZhngYLing W Z Wu Informtion Systems nd Knowledge Discovery Science Press eijing Chin 04 [7] W Zhng WZWuJYLing DYLiough Set Theory nd Method Science Press eijing Chin 01 [8] C Liu D Mio N Zhng Grded rough set model bsed on two universes nd its properties Knowledge-sed Systems Vol33 No [9] Y Yo T Lin Grded rough setpproimtions bsed on nested neighborhood systems Proceedings of 5th Europen Congress on intelligent techniques nd Soft computing Vol [10] WuSLiuQWng TheFirstType of Grde oughset sed on ough Membership Function 10 Seventh Interntionl Conference System nd Knowledge DiscoveryFSKD10) Vol [11] Y Zhng Z W Mo Vrible precision rough sets Pttern ecognition nd rtificil Intelligence Vol17 No in Chinese) [12] Y Yo Decision-theoretic rough set models In: Proceeding of ough Sets nd Knowledge Technology Lecture Notes in rtificil Intelligence Vol4481 No [13] W Ziro Vrible precision rough set model Journl of Computer nd System Sciences Vol46 No [14] S Greco Mtrzzo Slowinsi Prmeterized rough set model using rough membership nd yesin confirmtion mesures Interntionl Journl of pproimte esoning Vol49 No [15] J Herbert J Yo Gme-theoretic rough sets Fundment Informtice Vol108 No [16] J Shen Y J Lv Promotion for rough set of vrible precision nd etent of rough set Computer Engineering nd pplictions Vol44 No in Chinese) [17] Y Yo T Lin Generliztion of rough sets using modl logics Intelligent utomtic nd Soft Computing Vol2 No [18] W u W Li Double-quntittive decision-theoretic rough set Informtion Sciences Vol [19] Zhng Z Mo F iong W Cheng Comprtive study of vrible precision rough set model nd grded rough set model Interntionl Journl of pproimte esoning Vol53 No [] Y Zhng F iong Z W Mo rough set model bsed on logicl or opertion of precision nd grde Pttern ecognition nd rtificil Intelligence Vol22 No in Chinese) [21] J H Yu W H u ough set bsed on logicl disjunct opertion of vrible precision nd grde in ordered informtion system Computer Science nd Technology Vol9 No in Chinese) th Chinese Control nd Decision Conference CCDC)