Multi-granulation Fuzzy Rough Sets in a Fuzzy Tolerance Approximation Space

Transcription

1 46 Internatonal Journal of Fuzzy Systes Vol No 4 Deceber 0 Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space Wehua Xu Qaorong Wang and Xantao Zhang bstract ased on the analyss of the rough set odel on a tolerance relaton and the fuzzy rough set two types of fuzzy rough sets odels on tolerance relatons are constructed and researched Then we propose the optstc and pessstc ult-granulaton fuzzy rough sets odels n a fuzzy tolerance approxaton space wth the pont vew of granular coputng In these odels the fuzzy lower and upper approxatons of a fuzzy set are defned on ultple fuzzy tolerance relatons It follows the research on the propertes of the fuzzy lower and upper approxatons of the new odels The fuzzy rough set odel rough set odel on a tolerance relaton and ult-granulaton rough sets odels are specal cases of the new ones fro the perspectve of the consdered concepts related relatons and granular coputng The new odels are eanngful generalzatons of the classcal rough sets Keywords: pproxaton operators Fuzzy rough set Fuzzy tolerance relaton Mult-granulaton Introducton Rough set theory proposed by awlak [-] s a theory for the research of uncertanty anageent n a wde varety of applcatons related to artfcal ntellgence [ 5 8] The theory has been appled successfully n the felds of pattern recognton edcal dagnoss data nng conflct analyss algebra [ 4 0] whch relate an aount of precse vague and uncertan nforaton In recent years the rough set theory generated a great deal of nterest aong ore and ore researchers The generalzaton of the classcal rough set odel fro the perspectve of granular coputng s one of the study spotlghts Inforaton granules refer to peces classes and groups dvded n accordance wth characterstcs and perforances of coplex nforaton n the process of Correspondng uthor: Wehua Xu s wth the school of Matheatcs and Statstcs Chongqng Unversty of Technology Chongqng R Chna E-al: chxuwh@galco huan understandng reasonng and decson-akng Zadeh frstly proposed the concept of granular coputng and dscussed ssues of fuzzy nforaton granulaton n 979 [9] Then the basc dea of nforaton granulaton had been appled to any felds ncludng rough set [ ] In 985 Hobbs proposed the concept of granularty [4] Granular coputng played a ore and ore portant role gradually n soft coputng knowledge dscovery data nng and any excellent results were acheved [ ] In the pont vew of granular coputng the classcal awlak rough set and soe expanded rough sets lke tolerant odels [6 0 ] are based on a sngle granulaton nd an equvalence relaton or a tolerance relaton on the unverse can be regarded as a granulaton Ths approach to descrbng a concept s anly based on the followng assupton [6]: If R and R are two relatons nduced by the attrbutes subsets and and X U s a target concept then the rough set of X s derved fro the quotent set U ( R R ) = { x x [ ] [ ] R R [ x] U R [ x] U R [ x] [ x] R R R R whch suggests that we can perfor an ntersecton operaton between [ x] and [ x] and the target concept s R R approxately descrbed by usng the quotent set U ( R R) Then the target concept s descrbed by a fner granulaton (parttons or coverngs) fored through cobnng two known granulatons (parttons or coverngs) nduced fro two attrbute subsets However the cobnaton that generates a uch fner granulaton and ore knowledge destroys the orgnal granulaton structure In fact the above assupton cannot always be satsfed or requred generally The followng cases llustrate the restrctons Case In soe data analyss ssues for the sae obect there s a contradcton or nconsstent relatonshp between ts values under one attrbute set R and those under another attrbute set R In other words we cannot perfor the ntersecton operatons between ther quotent sets and the target concept cannot be approxated by usng U ( R R ) 0 TFS

2 Wehua Xu et al: Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space 47 Case In the process of soe decson akng the decson or the vew of each of decson akers ay be ndependent for the sae proect Case To extract decson rules fro dstrbutve nforaton systes and groups of ntellgent agents for the reducton of the te coplexty of rule extractons t s unnecessary for us to perfor the ntersecton operatons n between all the stes n the context of dstrbutve nforaton systes To overcoe these ltatons Qan and Xu et al extended the awlak rough set to ult-granulaton rough set odels where the approxaton operators are defned by ultple equvalence relatons on the unverse [ ] However the knd of classfcaton s stll restrctve for t pays too uch attenton to the dfferences between obects and gnores the slartes between obects So we wll extend the odels by relaxng the equvalent relatons to tolerance relatons or fuzzy tolerance relatons and by cobnng wth other theores that deal wth uncertanty knowledge such as fuzzy sets ssocated tolerant rough set wth the theory of fuzzy rough set wth granular coputng pont vew we constructed two types of ult-granulaton fuzzy rough set odels on tolerance relatons [6] Moreover there exst ssues whch have to be solved by fuzzy relatons Therefore we propose the optstc and pessstc ult-granulaton fuzzy rough sets odels n a fuzzy tolerance approxaton space n whch the needed relatons are fuzzy ones to extend Skowron s [] and J Järnen s [6] tolerance rough set odel deterned by sngle tolerance relaton nd these odels are fuzzy cases fro the perspectve of both relatons and concepts The rest of ths paper s organzed as follows Soe prelnary concepts of tolerance rough set theory fuzzy rough sets theory and ult-granulaton rough sets are showed n Secton In Secton we represent our acheveents on two types of ult-granulaton fuzzy rough approxaton operators of a fuzzy concept based on ultple ordnary tolerance relatons Then we defne the fuzzy approxatons of the fuzzy rough sets n a fuzzy tolerance approxaton space and analyze the propertes of the n Secton 4 Especally one can fnd that the defnton of lower and upper approxaton operators proposed n ths paper are the generalzed odel of other forats n Secton not only fro the aspects of the consdered concepts but also fro the perspectve of granulaton nd fnally the paper s concluded by a suary n Secton 5 relnares In ths secton we wll frst revew soe basc concepts and notons n the theory of rough set on a tolerance relaton fuzzy rough sets and ult-granulaton rough sets on the bass of equvalence relatons More can be found n ref [ ] The Rough Set on a Tolerance Relaton The noton of nforaton syste provdes a convenent tool for the representaton of obects n ters of ther attrbute values tolerance nforaton syste [] s an ordered trple I = ( U T τ ) where U s the non-epty fnte set of obects known as unverse T s the non-epty fnte set of attrbutes t s the appng fro power set T nto the faly set R of tolerance relatons satsfyng reflexvty and syetry on unverse U Let I = ( U T τ ) be a tolerance nforaton syste for T X U R Rs a relaton wth respect to the attrbutes set Denote R ( x ) = x U ( x x ) R () where { U L then R ( x ) wll be called the tolerance class of x wth respect to the tolerance relaton R Note that a tolerance relaton can construct a coverng nstead of a partton of the unverse U Let I = ( U T τ ) be a tolerance nforaton syste The lower approxaton and upper approxaton of a set X U wth respect to a tolerance relaton R are respectvely defned by R = { x U R( x) X () R = { x U R( x) X The set ndr = R ( ) ( ) X R X s called the boundary of X The set R( X ) conssts of eleents whch surely belong to X n vew of the knowledge provded by R whle R( X ) conssts of eleents whch possbly belong to X The boundary s the actual area of uncertanty It conssts of eleents whose ebershp n X can not be decded when R -related obects can not be dstngushed fro each other Let I = ( U T τ ) be a tolerance nforaton syste The propertes of the lower approxaton and the upper approxaton of sets wth respect to a tolerance relaton R are as follows f X Y U and ~ X s the copleent of X () R X R () R( ) = R( ) = R( U) = R ( U) = U () ~ R = R(~ X)~ R = R(~ X) (4) ndr( X ) = ndr(~ X ) (5) X Y R R ( Y) R R ( Y)

3 48 Internatonal Journal of Fuzzy Systes Vol No 4 Deceber 0 The Fuzzy Set and Fuzzy Rough Set We wll frstly ntroduce soe basc concepts of fuzzy set [] Let U be a fnte and non-epty set called unverse fuzzy set s a appng fro U nto the unt nterval [ 0 ] : μ : [ 0] where each x U s the ebershp degree of x n ractcally we ay consder U as a set of obects of concern and crsp subset of U represents a non-vague" concept posed on obects n U Then a fuzzy set of U s thought of as a atheatcal representaton of vague" concept descrbed lngustcally The set of all the fuzzy sets defned on U s denoted by F( U ) Let and be two fuzzy sets on U the operaton between the are defned as ( )( x) = ax { ( x) ( x) ( )( x) = n { ( x) ( x) () ~ x ( ) = x ( ) ( x) ( x)( x U) Let U be the unverse R be an equvalence relaton for a fuzzy set on U f take R ( )( x ) = { y ( ) y [ x] R (4) R ( )( x) = { y ( ) y [ x] R then R ( ) and R ( ) are called the lower and upper approxaton of the fuzzy set wth respect to the relaton R where " " eans n" and " " eans ax" and [ x ] R s the equvalence class of x wth respect to equvalence relaton R [ 9] s a fuzzy defnable set f and only f satsfes R ( ) = R ( ) therwse s called a fuzzy rough set Let I = ( U T F) be an nforaton syste [9] { F = f n s a set of relatonshp between U and T where f : U V( n) V s the doan of attrbute a and n s the nuber of the attrbutes D : U [ 0 ]( r) f denote D = { D r then ( U T F D ) s a fuzzy target nforaton syste In a fuzzy target nforaton syste we can defne the approxaton operators wth respect the decson attrbute D slarly Let U be the unverse be an equvalence relaton F( U) the fuzzy lower and upper approxaton wth respect to relaton R satsfy the followng propertes () R ( ) R ( ) () R ( ) = R ( ) R ( ) R ( ) = R ( ) R ( ) () R( ) = ~ R(~ ) R( ) = ~ R(~ ) (4) R ( ) R ( ) R ( ) R ( ) R ( ) R ( ) (5) RR ( ( )) = RR ( ( )) = R ( ) (6) RR ( ( )) = RR ( ( )) = R ( ) (7) RU ( ) = U R( ) = (8) R( ) R( ) and R ( ) R ( ) C Mult-granulaton Rough Sets For splcty we ust recall the odels of ult-granulaton rough sets and detals can be seen n ref [6-8 5] Let I = ( U T F) be an nforaton syste T = L s the nuber of the consdered attrbute sets The optstc lower and upper approxatons of the set X U wth respect to T( = L ) are = t= { [ ] = { [ ] = R ( X ) = x U x X t R ( X ) = x U x X = and R s an equvalent relaton wth respect to the attrbutes set Moreover R ( X ) R ( X ) we say that X s the optstc t = = ult-granulaton rough set therwse we say that X s the optstc ult-granulaton defnable set Let I = ( U T F) be an nforaton syste T = L s the nuber of the consdered attrbute sets The pessstc lower and upper approxatons of the set X U wth respect to T( = L ) are where [ x] { x U ( x y) R Moreover = t= { [ ] = { [ ] ϕ = R ( X ) = x U x X t R ( X ) = x U x X t = t= (5) (6) R ( X ) R ( X ) we say that X s the pessstc ult-granulaton rough set therwse we say that X s the pessstc ult-granulaton defnable set Two Types of Mult-granulaton Fuzzy Rough Sets on Tolerance Relatons In ths secton we ake researches about ultgranulaton fuzzy rough set whch are on the proble of the rough approxatons of a fuzzy set on ultple tolerance relatons

4 Wehua Xu et al: Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space 49 t frst we propose a fuzzy rough set odel (n bref FRS) on a tolerance relaton n the followng Let I = ( U T τ ) be a tolerance nforaton syste T For the fuzzy set X FU ( ) denote R( x) = { X( y) y R( x) (7) R( x) = { X( y) y R( x) where " " eans n" and " " eans ax" then R( X ) and R( X ) are the lower and upper approxaton of the fuzzy set X on the tolerance relaton R wth respect to the subset of attrbutes If R R then the fuzzy set X s a fuzzy rough set on the tolerance relaton We can easly fnd that ths odel wll be the fuzzy rough set odel we have ntroduced n the part of Secton f the above relaton R s an equvalence relaton The ptstc of Mult-granulaton Fuzzy Rough Set on Tolerance Relatons Frst the optstc two-granulaton fuzzy rough set (n bref optstc TGFRS) on tolerance relatons of a fuzzy set s defned Defnton [6]: Let I = ( U T τ ) be a tolerance nforaton syste T For the fuzzy set X FU ( ) denote { { { { R ( x) = X( y) y R ( x) X( y) y R ( x) + R+ ( x) = X( y) y R( x) X( y) y R( x) (8) where" " eans ax and " " eans n then R ( ) + X and R+ ( X ) are respectvely called the optstc two-granulaton lower approxaton and upper approxaton of X on tolerance relatons R wth respect to the subsets of attrbutes and X s a two-granulaton fuzzy rough set on tolerance relatons f and only f R+ ( X ) R+ ( X ) therwse X s a two-granulaton fuzzy defnable set on two tolerance relatons The boundary of the set s X defned as F ndr ( X ) = R ( ) (~ ( )) X R X (9) It can be found that the optstc TGFRS on tolerance relatons wll be degenerated nto fuzzy rough set when = and R( X ) and R ( X ) are equvalence classes wth respect to the subsets of attrbutes and That s to say a fuzzy rough set odel s a specal nstance of the optstc TGFRS on tolerance relatons What s ore the optstc TGFRS on tolerance relatons wll be degenerated nto a rough set odel on a tolerance relaton f = and the consdered concept X s a crsp set In the followng we eploy an exaple to llustrate the above concepts Exaple : fuzzy target nforaton syste s gven n Table The unverse U = { x x L x0 the set of condton attrbutes T = { a a a the set of decson attrbute D = { d If suppose = { a a and = { a a we consder the optstc two-granulaton lower and upper approxaton of D wth respect to and where the tolerance relaton s defned as R = {( x ) ( ) ( ) x U U f x a f x a a For the fuzzy set D = ( ) the sngle-granulaton lower and upper approxatons on a tolerance relaton are R ( ) ( ) D = R ( ) ( ) D = R ( ) ( ) D = R ( ) ( ) D = R ( ) ( ) D = R ( ) ( ) D = Fro Defnton we can copute the optstc two-granulaton lower and upper approxaton of D on tolerance relatons are R ( ) ( ) + D = R ( ) ( ) + D = bvously the followng can be found R + ( D) = R ( D) R ( D) R + ( D) = R ( ) ( ) D R D R ( D) R ( D) D R ( D) R ( D) + + Just fro Defnton we can obtan soe propertes of the optstc TGFRS n a tolerance nforaton syste Table fuzzy target nforaton syste U a a a d x x x x 4 x 5 x 6 x 7 x 8 x 9 x roposton [6]: Let I = ( U T τ ) be a tolerance nforaton syste T and X FU ( ) Then

5 50 Internatonal Journal of Fuzzy Systes Vol No 4 Deceber 0 the followng propertes hold () R + ( X ) X () R+ ( X ) X () R (~ X ) = ~ R ( X ) + + (4) R (~ X ) = ~ R ( X ) + + (5) R ( U ) = R ( U ) = U + + (6) R ( ) = R ( ) = + + (7) R ( X ) R ( R ( X )) (8) R+ ( X ) R+ ( R+ ( X )) roposton [6]: Let I = ( U T τ ) be a tolerance nforaton syste T X Y F( U) Then the followng propertes hold () R ( X Y ) R ( X ) R ( Y ) () R+ ( X Y ) R+ ( X ) R+ ( Y ) () X Y R R ( Y) + + (4) X Y R+ R+ ( Y) (5) R ( X Y ) R ( X ) R ( Y ) (6) R+ ( X Y ) R+ ( X ) R+ ( Y ) Defnton [6]: Let I = ( U T τ ) be a tolerance nforaton syste T For the fuzzy set X FU ( ) denote = = { { ( )( ) = ( )( ) ( ) R X x X y y R x = { { ( )( ) = ( )( ) ( ) R X x X y y R x = (0) where '' '' eans ''ax'' and '' '' eans ''n'' then R ( X ) and R ( X ) are respectvely called the = = optstc ult-granulaton lower approxaton and upper approxaton of X on the tolerance relatons R ( = L ) X s a ult-granulaton fuzzy rough set on the tolerance relatons R ( = L ) f and only f = = R ( X ) R ( X ) therwse X s a ult-granulaton fuzzy defnable set on the tolerance relatons R ( = L ) The boundary of the set X s de- fned as F ndr ( X ) = R ( X ) (~ R ( X )) = = = () It can be found that the optstc MGFRS on the tolerance relatons R ( = L ) wll be degenerated nto fuzzy rough set when = ( ) and R ( x ) are equvalence classes wth respect to the subsets of attrbutes ( = L ) That s to say a fuzzy rough set odel s a specal nstance of the optstc MGFRS on the tolerance relatons esdes ths odel can also be turned the optstc MGRS f the relatons are equvalence relatons and the consdered set s a crsp one What's ore the optstc MGFRS on tolerance relatons wll be degenerated nto a rough set odel on tolerance relatons f = ( ) and the consdered concept X s a crsp set The propertes about optstc MGFRS on tolerance relatons are lsted n the followng whch can be extended fro the optstc TGFRS odel on tolerance relatons roposton [6]: Let I = ( U T τ ) be a tolerance nforaton syste T and X FU ( ) Then the followng propertes hold () R ( X ) X = () R ( X ) X = () R (~ X ) = ~ R ( X ) = = (4) R (~ X ) = ~ R ( X ) = = (5) R ( U ) = R ( U ) = U = = (6) R ( ) = R ( ) = = = (7) R ( X ) R ( R ( X )) (8) = = = R ( X ) R ( R ( X )) = = = roposton 4[6]: Let I = ( U T τ ) be a tolerance nforaton syste T X Y F( U) Then the followng propertes hold () R ( X Y ) R ( X ) R ( Y ) = = = () R ( X Y ) R ( X ) R ( Y ) = = = () X Y R R ( Y) = = (4) X Y R R ( Y) = = (5) R ( X Y ) R ( X ) R ( Y ) = = = (6) R ( X Y ) R ( X ) R ( Y ) = = = The essstc Mult-granulaton Fuzzy Rough Set

6 Wehua Xu et al: Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space 5 on Tolerance Relatons In ths subsecton we wll propose a pessstc MGFRS on tolerance relatons We frst defne the pessstc two-granulaton fuzzy rough set (n bref pessstc TGFRS) on tolerance relatons Defnton [6]: Let I = ( U T τ ) be a tolerance nforaton syste T For the fuzzy set X FU ( ) denote + + ( )( ) = { ( )( ) ( ) { ( y) y R ( x) ( )( ) = { { ( )( ) ( ) { ( y) y R ( x) R X x X y y R x R X x X y y R x () then R+ ( X ) and R+ ( X ) are respectvely called the pessstc two-granulaton lower approxaton and upper approxaton of X on the tolerance relatons R wth respect to the subsets of attrbutes and X s the pessstc two-granulaton fuzzy rough set on the tolerance relatons f and only f R+ ( X ) R+ ( X ) therwse X s the pessstc two-granulaton fuzzy defnable set on these tolerance relatons The boundary of the set X s defned as S ndr ( X ) = R ( ) ( ~ ( )) X R X () It can be found that the pessstc TGFRS on tolerance relatons wll be degenerated nto the fuzzy rough set odel when = and R( x ) and R ( x ) are equvalence classes wth respect to the subsets of attrbutes and That s a fuzzy rough set odel s a specal nstance of the pessstc TGFRS on tolerance relatons What's ore the pessstc TGFRS on tolerance relatons wll be degenerated nto a rough set odel on a tolerance relaton f = and the consdered concept X s a crsp set In the followng we contnue eg to llustrate the above concepts Exaple : (Contnued fro eg ) Fro Defnton we can copute the pessstc two-granulaton lower and upper approxaton of D on the tolerance relaton R and R are R + ( D) = ( ) ( D) = ( ) R bvously the followng can be found R D = R D R D ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) R ( D) R ( D) + R + D R D R D R D R D D + + roposton 5[6]: Let I = ( U T τ ) be a tolerance nforaton syste T and X FU ( ) Then the followng propertes hold () R+ ( X ) X () R+ ( X ) X () R+ (~ X ) = ~ R+ ( X ) (4) R+ (~ X ) = ~ R+ ( X ) (5) R+ ( U) = R+ ( U) = U (6) R+ ( ) = R+ ( ) = (7) R+ ( X ) R+ ( R+ ( X )) (8) R+ ( X ) R+ ( R+ ( X )) roposton 6[6]: Let I = ( U T τ ) be a tolerance nforaton syste T X Y F( U) Then the followng propertes hold () R+ ( X Y ) = R+ ( X ) R+ ( Y ) () R+ ( X Y ) = R+ ( X ) R+ ( Y ) () X Y R+ R+ ( Y) (4) X Y R+ R+ ( Y) (5) R+ ( X Y ) R+ ( X ) R+ ( Y ) (6) R+ ( X Y ) R+ ( X ) R+ ( Y ) In the followng we wll ntroduce the pessstc ult-granulaton fuzzy rough set (n bref pessstc MGFRS) on tolerance relatons and ts correspondng propertes by extendng the pessstc TGFRS on tolerance relatons Defnton 4[6]: Let I = ( U T τ ) be a tolerance nforaton syste T For the fuzzy set X FU ( ) denote = = { { ( )( ) = ( )( ) ( ) R X x X y y R x = { { ( )( ) = ( )( ) ( ) R X x X y y R x = (4) where '' '' eans ''ax'' and '' '' eans ''n'' then R ( X ) and R ( X ) are respectvely called the = = pessstc ult-granulaton lower approxaton and upper approxaton of X on tolerance relatons R wth respect to the subsets of attrbutes ( = L ) X s the pessstc ult-granulaton fuzzy rough set f and only f R ( X ) R ( X ) therwse X s = = the pessstc ult-granulaton fuzzy defnable set on any tolerance relatons The boundary of the set X s defned as

7 5 Internatonal Journal of Fuzzy Systes Vol No 4 Deceber 0 ( ) nd X = R ( X ) (~ R ( X )) F R = = = (5) It can be found that the pessstc MGFRS wll be degenerated nto fuzzy rough set when = ( ) and R ( x ) are equvalence classes wth respect to the subsets of attrbutes ( = L ) That s a fuzzy rough set odel s also a specal nstance of the pessstc MGFRS on tolerance relatons esdes ths odel can also been turned the pessstc MGRS f the relatons are equvalence relatons and the consdered set s a crsp one What's ore the pessstc MGFRS odel wll be degenerated nto a rough set odel on a tolerance relaton f = ( ) and the consdered concept X s a crsp set The propertes about the pessstc MGFRS on tolerance relatons are lsted n the followng whch can be extended fro the pessstc TGFRS odel on tolerance relatons roposton 7[6]: Let I = ( U T τ ) be a tolerance nforaton syste T and X FU ( ) Then the followng propertes hold () R ( X ) X = () R ( X ) X = () R (~ X ) = ~ R ( X ) = = (4) R (~ X) = ~ R = = (5) R ( U ) = R ( U ) = U = = (6) R ( ) = R ( ) = = = (7) R ( X ) R ( R ( X )) = = = (8) R ( X ) R ( R ( X )) = = = roposton 8[6]: Let I = ( U T τ ) be a tolerance nforaton syste T X Y F( U) Then the followng propertes hold () R ( X Y ) = R ( X ) R ( Y ) = = = () R ( X Y ) = R ( X ) R ( Y ) = = = () X Y R R ( Y) = = (4) (5) (6) X Y R R ( Y) = = R ( X Y ) R ( X ) R ( Y ) = = = R ( X Y ) R ( X ) R ( Y ) = = = n the bass of tolerance relatons we wll nvestgate the nterrelatonshps aong SGFRS the optstc MGFRS and the pessstc MGFRS after the dscusson of the propertes of the roposton 9[6]: Let I = ( U T τ ) be a tolerance nforaton syste T X FU ( ) Then the followng propertes hold () () () (4) (5) (6) (7) (8) R ( X ) = R ( X ) = = = U R ( X ) = R ( X ) = = I R ( X ) = R ( X ) = = I R ( X ) = R ( X ) = U R ( X ) R ( X ) R ( X ) U = = = R ( X ) R ( X ) R ( X ) U = = = R ( X ) R ( X ) R ( X ) = = R ( X ) R ( X ) R ( X ) = = 4 Two Type of Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space In ths secton we wll propose two types of MGFRS odels n a tolerance approxaton space n whch the relatons are fuzzy tolerance relatons and the characterzed concepts are fuzzy sets The propertes of the approxatons wll be showed Then the optstc Mult-granulaton fuzzy rough sets wll be ntroduced frstly The ptstc Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space ( U R ) s a fuzzy tolerance approxaton space where U s a non-epty fnte set of obects known as unverse R U U s a faly set of fuzzy tolerance relatons [0] satsfyng () Reflexvty: Rxx ( ) = x U

8 Wehua Xu et al: Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space 5 () Syetry: Rxy ( ) = Ryx ( ) xy U In the followng we wll frstly defne the SGFRS n a fuzzy tolerance approxaton space Defnton 4: Let ( U R ) be a fuzzy tolerance fuzzy approxaton space R be a fuzzy tolerance relaton For the fuzzy set X FU ( ) denote RX ( )( x)= X( y) ( Rxy ( )) { RX ( )( x)= X( y) Rxy ( ) (6) then RX ( ) and RX ( ) are respectvely called the sngle-granulaton fuzzy lower and fuzzy upper approxatons of X on a fuzzy tolerance relaton X s a sngle-granulaton fuzzy rough set on a fuzzy tolerance relaton f and only f RX ( ) = RX ( ) therwse X s a sngle-granulaton fuzzy defnable set on the fuzzy tolerance relaton roposton 4: Let ( U R ) be a fuzzy tolerance fuzzy approxaton space R be a fuzzy tolerance relaton For the fuzzy set X Y F( U) we have () RX ( ) X () RX ( ) X () R(~ X) = ~ R (4) R(~ X) = ~ R (5) RU ( ) = RU ( ) = U (6) R( ) = R( ) = (7) RX ( Y) = RX ( ) RY ( ) (8) RX ( Y) = RX ( ) RY ( ) roof: We only prove the odd tes the rest can be proved slarly () Snce Rxx ( )= then X ( y) ( R( x y)) X( x) ( R( x x)) = X( x) Thus RX ( ) X () Fro Defnton 4 we have R(~ X)( x)= ( X( y)) ( R( x y)) { X y R x y = ( ) ( ) = ~ RX ( )( x) that s R(~ X) = ~ R (5) x U U( x) = and Rxx ( )= we have RU ( )( x) = U( y) ( R( x y)) = U( y) = = U( x) and RU ( )( x) = U( y) R( x y) = R( x y) = = U( x) Therefore RU ( ) = RU ( ) = U (7) x U X Y F( U) RX ( Y)( x)= ( XY)( y) ( Rxy ( )) {( X( y) ( R( x y) )) ( Y( y) ( R( x y) )) { ( X( y) ( R( x y) )) ( Y( y) ( R( x y) )) = = = RX ( ) RY ( ) Now we wll gve the defnton of the optstc ult-granulaton fuzzy rough set (n bref MGFRS) n a fuzzy tolerance approxaton space Defnton 4: Let ( U R ) be a fuzzy tolerance fuzzy approxaton space R( )be fuzzy tolerance relatons For the fuzzy set X FU ( ) denote then = = = { { y U R ( x)= X( y) ( R ( x y)) = { { y U R ( x)= X( y) R ( x y) R and = = (7) R are respectvely called the optstc ult-granulaton fuzzy lower and fuzzy upper approxatons of X on fuzzy tolerance relatons X s an optstc ult-granulaton fuzzy rough set n a fuzzy tolerance approxaton space f and only f = = R R therwse X s a ult-granulaton fuzzy defnable set n a fuzzy tolerance approxaton space We can obtan soe specal cases fro Defnton 4: () If X FU ( ) e X s a fuzzy set and fuzzy tolerance relatons R = R( ) then = = R ( x)= X( y) ( R ( x y)) = R ( x) { R ( x)= X( y) R ( x y) = R ( x) Hence the approxatons can be degenerated nto the ones n Defnton 4 () If X U R U U e X s a crsp set and R( ) are ordnary tolerance relatons R = R( ) then = R ( x)= R ( x)= X( y) ( R ( x y)) = ( y X ( x y) R ( y X y R ( x)) R ( x) X = R ( x)= R ( x)= y U stx ( y) = R ( x y) = X R ( x)

9 54 Internatonal Journal of Fuzzy Systes Vol No 4 Deceber 0 That s to say they can be changed to () () If X FU ( ) R( ) are ordnary tolerance relatons and R = R( ) then = R ( x)= R ( x) { X y R x y { X y x y R { X y y R x = ( ) ( ( )) = ( ) ) = ( ) ( ) = R ( x)= R ( x) { X( y) R ( x y) { X( y) ( x y) R { X( y) y R ( x) = = = Thus they are consstent to the approxatons n (7) (4) If X U R( ) are classcal equvalence relatons = R ( x)= R X( y) ( R( x y)) = R ( y X ( x y) R) R( y X y [ x] R ) R[ x] X = R { R = R = x U [ x] X = = R ( x)= R y U stx ( y) = R( x y) = R X R ( x) { R = R = x U [ x] X So the approxatons are the sae as the ones n (5) Thus t s easy to fnd that the odel as Defnton 4 showed s a reasonable generalzaton of odels presented n Secton In the followng we wll gve an exaple to llustrate the above defnton Exaple 4: LetU = { x x x x4 x5 x6 X = ( ) and tolerance relatons R S on U U are defned as R = S = Then the sngle-granulaton fuzzy lower and fuzzy upper approxatons of X on fuzzy tolerance relaton R S are respectvely RX ( )=( ) RX ( )=( ) S=( ) S=( ) R S=( ) R S=( ) R S=( ) R S=( ) nd the optstc ult-granulaton fuzzy lower and fuzzy upper approxatons of X on fuzzy tolerance relatons RSare R+ S =( ) R+ S =( ) We fnd that R+ S = R S R+ S = R S R S R+ S X R+ S R S Just fro Defnton 4 we can prove the followng propertes about the optstc ult-granulaton fuzzy rough set n fuzzy tolerance approxaton space hold roposton 4: Let ( U R ) be a fuzzy tolerance fuzzy approxaton space R( )be fuzzy tolerance relatons For the fuzzy set X FU ( ) the followng propertes hold () () () (4) (5) (6) = = R X R X = = = R (~ X) ~ R = = = R (~ X) ~ R = = R ( U) = R ( U) = U R = = ( ) = R ( ) =

10 Wehua Xu et al: Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space 55 (7) = = = R U R = = = (8) R I R roof: They are straghtforward fro Defnton 4 and roposton 4 roposton 4: Let ( U R ) be a fuzzy tolerance fuzzy approxaton space R( )be fuzzy tolerance relatons For the fuzzy set X Y F( U) the followng propertes hold () R ( X Y) = ( R R( Y) ) U = = () R ( X Y) = ( R R( Y) ) () (4) (5) (6) (7) I = = = = = R ( X Y) R R ( Y) = = = X Y R R ( Y) = = R ( X Y) R R ( Y) = = X Y R R ( Y) = = = R ( X Y) R R ( Y) = = = (8) R ( X Y) R R ( Y) roof: We only prove the odd tes and the others can be proved slarly () X Y F( U) ( ) ( ) = = = = R ( X Y) U R ( X Y) = U R R ( Y) () x U X Y F( U) = R ( X Y) U = ( R X R Y ) = ( ) ( ) UR UR( Y) = = (5) If X Y then X Y= X It follows fro () that = = = = R R R ( Y) = = = R R ( Y) = = R ( X Y)= R R R ( Y) (7) It s clear that X X Y andy X Y It follows that and Hence = = R ( X Y) R R X Y R Y = = = = = ( ) ( ) R ( X Y) R R ( Y) The essstc Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space In ths subsecton we wll gve the defnton of the pessstc ult-granulaton fuzzy rough set (n bref MGFRS) n a fuzzy tolerance approxaton space Defnton 4: Let ( U R ) be a fuzzy tolerance fuzzy approxaton space R( )be fuzzy tolerance relatons For the fuzzy set X FU ( ) denote R ( x)= { { X( y) ( R( x y)) = = (8) R ( x)= { X( y) R( x y) then = = R ( X ) and = y U = R ( X ) are respectvely called the pessstc ult-granulaton fuzzy lower and fuzzy upper approxatons of X on fuzzy tolerance relatons X s a MGFRS n a fuzzy tolerance approxaton space f and only f = = R R therwse X s a ult-granulaton fuzzy defnable set n a fuzzy tolerance approxaton space We can obtan soe specal cases fro Defnton 4: () If X FU ( ) e X s a fuzzy set and fuzzy tolerance relatons R = R( ) then = = R ( x)= X( y) ( R ( x y)) = R ( x) { R ( x)= X( y) R ( x y) = R ( x) Hence the approxatons can also be degenerated nto the ones n Defnton 4 () If X U R U U e X s a crsp set and R( ) are ordnary tolerance relatons R = R( ) then = R ( x)= R ( x)= X( y) ( R ( x y)) = ( y X ( x y) R ( y X y R ( x)) R ( x) X

11 56 Internatonal Journal of Fuzzy Systes Vol No 4 Deceber 0 = R ( x)= R ( x)= y U stx ( y) = R ( x y) = X R ( x) That s to say they can be changed to () () If X FU ( ) R( ) are ordnary tolerance relatons and R = R( ) then = = R ( x)= R ( x) { X y R x y { X y x y R { X y y R x = ( ) ( ( )) = ( ) ) = ( ) ( ) R ( x)= R ( x) { X( y) R ( x y) { X( y) ( x y) R { X( y) y R ( x) = = = Thus they are consstent to the approxatons n (7) (4) If X U R( ) are classcal equvalence relatons = R ( x)= R X( y) ( R( x y)) = R ( y X ( x y) R) R( y X y [ x] R ) R[ x] X = R { R = R = x U [ x] X = R ( x)= = R y U stx ( y) = R( x y) = R X R ( x) { R = R = x U [ x] X So the approxatons are the sae as the ones n (6) Thus t s easy to fnd that the odel as Defnton 4 showed s also a reasonable generalzaton of odels presented n Secton Exaple 4: (Contnued fro eg 4) Fro Defnton 4 we can copute the pessstc fuzzy lower and upper approxatons of X n the sae fuzzy tolerance space as follows R+ S =( ) R+ S =( ) We fnd that R+ S = R S R+ S = R S R+ S RS X RS R+ S Fro Defnton 4 we can prove the followng propertes about the MGFRS n a fuzzy tolerance approxaton space hold roposton 44: Let ( U R ) be a fuzzy tolerance fuzzy approxaton space R( )be fuzzy tolerance relatons For the fuzzy set X FU ( ) the followng propertes hold () () () (4) (5) (6) (7) = = R X R X = = = R (~ X) ~ R = = = R (~ X) ~ R = = R ( U) = R ( U) = U R = = ( ) = R ( ) = = = = R I R = = = (8) R U R roof: They are straghtforward fro Defnton 4 and roposton 4 roposton 45: Let ( U R ) be a fuzzy tolerance fuzzy approxaton space R( )be fuzzy tolerance relatons For the fuzzy set X Y F( U) the followng propertes hold () R ( X Y) = ( R R( Y) ) I = = () R ( X Y) = ( R R( Y) ) () (4) (5) (6) U = = = = = R ( X Y)= R R ( Y) = = = X Y R R ( Y) = = R ( X Y)= R R ( Y) = = X Y R R ( Y)

12 Wehua Xu et al: Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space 57 (7) = = = R ( X Y) R R ( Y) = = = (8) R ( X Y) R R ( Y) roof: We only prove the odd tes and the others can be proved slarly () X Y F( U) ( ) ( ) = = = = R ( X Y) I R ( X Y) = I R R ( Y) () x U X Y F( U) = R ( X Y) I = ( R X R Y ) = ( ) ( ) = IR IR( Y) = = (5) If X Y then X Y= X It follows fro () that = = = = = R = R R ( Y) = = = R R ( Y) = = R ( X Y)= R R R ( Y) (7) It s clear that X X Y andy X Y It follows that and Hence = = R ( X Y) R R ( X Y) R ( Y) = = = = = R ( X Y) R R ( Y) roposton 46: Let ( U R ) be a fuzzy tolerance fuzzy approxaton space R( )be fuzzy tolerance relatons For the fuzzy set X FU ( ) the followng propertes hold () () = = R R R = = R R R () U R R = = = = (4) R U R = = R I R I R R = roof: The frst two tes are straghtforward to prove by roposton 4 and 44 We ust prove te () and = te (4) can be proved slarly by te () Snce x y U we have U I R R R = = R( x y) R( x y) R( x y) = = = = U I = = then by te () R ( x y) R ( x y) R ( x y) R ( x) R ( x) R ( x) = = = = U R R R and R R I R hold Ths proposton exposes the relatonshps aong SGFRS MGFRS and MGFRS n a fuzzy tolerance approxaton space 5 Conclusons The theory of rough sets has been sgnfcantly extended fro the pont vew of granular coputng by cobnng the theory of fuzzy sets The theory of fuzzy sets pays ore attenton to the fuzzness of knowledge whle the theory of rough sets to the roughness of knowledge n the pont vew of granular of knowledge For the copleent of the two types of theory fuzzy rough set odels are nvestgated to solve practcal proble Gven that the equvalence relaton n the fuzzy rough set theory s too rgorous for soe practcal applcaton t s necessary to weaken the equvalence relatons to tolerance relatons or fuzzy relatons The contrbuton of ths paper s havng constructed two new types of fuzzy rough set on fuzzy tolerance relatons assocated wth granular coputng called ult-granulaton fuzzy rough set odels n a fuzzy tolerance approxaton space n whch the set approxaton operators are defned on the bass of ultple fuzzy tolerance relatons What's ore we ake conclusons that fuzzy rough set odel rough set odel on a tolerance relaton and ult-granulaton rough sets odels are specal cases of the two types of ult-granulaton fuzzy rough set n a fuzzy tolerance approxaton space by analyzng the defntons of the More propertes of the two odels are dscussed and coparsons are ade wth fuzzy rough set on a tolerance relaton The constructon of the new types of fuzzy rough set odels n a fuzzy tolerance approxaton space are extensons n the pont vew of granular coputng and are eanngful n ters of the generalzaton of rough set theory

13 58 Internatonal Journal of Fuzzy Systes Vol No 4 Deceber 0 cknowledgent Ths work s supported by the Natonal Natural Scence Foundaton of Chna (No and 007) the Natonal Natural Scence Foundaton of Chongqng (Nocstc04007) and the ostdoctoral Scence Foundaton of Chna (No00048) References [] V S nanthanarayana M M Narasha and D K Subraanan Tree structure for effcent data nng usng rough sets attern Recognton Letter vol 4 pp [] I Düntsch and G Gedga Uncertanty easures of rough set predcton rtfcal Intellgence vol 06 pp [] D Dubos and H rade uttng rough sets and fuzzy sets together In: R Slownsk (Ed) Intellgent Decson Support: Handbook of pplcatons and dvances of the Sets Theory Kluwer Dordrecht pp 0-99 [4] J R Hobbs Granularty In: roceedng of Internaton Jont Conference on rtfcal Intellgence Los ngeles pp [5] G Jeon D K and J Jeong Rough sets attrbutes reducton based expert syste n nterlaced vdeo sequences IEEE Transactons on Consuer Electroncs vol 5 pp [6] J Järnen pproxatons and rough sets based on tolerances Sprnger erln/hedelberg pp [7] M Khan and M aneree Multple-Source approxaton systes: ebershp functons and ndscernblty roceedng of The Thrd Internatonal Conference on Rough Sets and Knowledge Technology Chengdu Chna vol 4 pp [8] J Y Lang C Y Dang K S Chn and C M Ya Rchard new ethod for easurng uncertanty and fuzzness n rough set theory Internatonal Journal of General Systes vol pp [9] J M Ma W X Zhang Y Leung and X X Song Granular coputng and dual Galos connecton Inforaton Scences vol 77 no pp [0] Y uyang Z D Wang and H Zhang n fuzzy rough sets based on tolerance relatons Inforaton Scences vol 80 no 4 pp [] Z awlak Rough sets Internatonal Journal of Coputer and Inforaton Scences vol no 5 pp [] Z awlak Rough Set: Theoretcal spects of Reasonng about Data Kluwer cadec ublshers Dordrecht 99 [] Z awlak and Skowron Rudents of rough sets Inforaton Scences vol [4] Z awlak Rough sets decson algorths and ayes s theore Europe Journal of peratonal Research vol 6 no pp [5] Y H Qan and J Y Lang Rough set ethod based on ult-granulatons The 5th IEEE Internatonal Conference on Cogntve Inforatcs eng Chna 006 [6] Y H Qan J Y Lang Y Y Yao and C H Dang MGRS: ult-granulaton rough set Inforaton Scences vol 80 pp [7] Y H Qan J Y Lang W edrycz and C Y Dang ostve approxaton: n accelerator for attrbute reducton n rough set theory rtfcal Intellgence vol 74 pp [8] Y H Qan J Y Lang and W We essstc rough decson In: Second Internatonal Workshop on Rough Sets Theory pp [9] T R Qu Q Lu and H K Huang Granular Coputng pproach to Knowledge Dscovery n Relatonal Databases cta utoatca Snca vol 5 no 8 pp [0] R W Swnarsk and Skowron Rough set ethod n feature selecton and recognton attern Recognton Letter vol 4 pp [] Skowron and J Stepanuk Tolerance approxaton space Fundaental Inforaton vol 7 pp [] W Z Wu Y Leung and J S M Granular Coputng and Knowledge Reducton n Foral Contexts IEEE Transactons on Knowledge and Data Engneerng vol no 0 pp [] Z Xu X G Hu and H Wang general rough set odel based on tolerance Internatonal cadec ublshers World ublshng Corporaton pp [4] W H Xu X Y Zhang and W X Zhang Knowledge granulaton knowledge entropy and knowledge uncertanty easure n ordered nforaton systes ppled soft coputng pp [5] W H Xu X Y Zhang and W X Zhang Two new types of ult-granulaton rough set Inforaton Scences (Subtted) [6] W H Xu Q R Wang and X T Zhang Mult-granulaton Fuzzy Rough Set Models on Tolerance Relatons roceedng of The Fourth Internatonal Workshop on dvanced Coputatonal Intellgence Wuhan Hube Chna pp

14 Wehua Xu et al: Mult-granulaton Fuzzy Rough Sets n a Fuzzy Tolerance pproxaton Space 59 [7] Y Y Yao Inforaton granulaton and rough set approxaton Internatonal Journal of Intellgent Systes vol 6 pp [8] Y Y Yao erspectves of granular coputng In: roceedngs of 005 IEEE Internatonal Conference on Granular Coputng vol pp [9] L Zadeh Fuzzy Sets and Inforaton Granularty dvances n fuzzy set theory and applcaton North Holland ublshng sterda 979 [0] W X Zhang Y Lang and W Z Wu Fuzzy nforaton systes and knowledge dscovery In: Inforaton systes and knowledge dscovery Scence ress pp Wehua Xu s a rofessor and Master Supervsor at Chongqng Unversty of Technology R Chna He receved a hd fro X an Jaotong Unversty R Chna n 007 He has publshed over 50 ournal papers Hs an research felds are rough sets fuzzy sets and the atheatcal foundaton of nforaton scence Qaorong Wang s a aster at Chongqng Unversty of Technology R Chna Her an research areas are rough sets fuzzy sets and the atheatcal foundaton of nforaton scence Xantao Zhang s a aster at Chongqng Unversty of Technology R Chna Hs an research areas are rough set and ntellgent coputng