About a Fuzzy Distance between Two Fuzzy Partitions and Application in Attribute Reduction Problem

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1 BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND IORMATION TECHNOLOGIES Volume 6, No 4 Sofa 206 Prt ISSN: ; Ole ISSN: DOI: 0.55/cat About a Fuzzy Dstace betwee Two Fuzzy Parttos ad Applcato Attrbute Reducto Problem Cao Chh Ngha, Demetrovcs Jaos 2, Nguye Log Gag 3, Vu Duc Th 4 People s Polce Academy, Vet Nam 2 Isttute for Computer ad Cotrol (SZTAKI) Hugara Academy of Sceces 3 Isttute of Iformato Techology, VAST, Vet Nam 4 Isttute of Iformato Techology, VNU, Vet Nam Emals: ccgha@gmal.com demetrovcs@sztak.mta.hu lgag@ot.ac.v vdth@vu.edu.v Abstract: Accordg to tradtoal rough set theory approach, attrbute reducto methods are performed o the decso tables wth the dscretzed value doma, whch are decso tables obtaed by dscretzed data methods. I recet years, researches have proposed methods based o fuzzy rough set approach to solve the problem of attrbute reducto decso tables wth umercal value doma. I ths paper, we propose a fuzzy dstace betwee two parttos ad a attrbute reducto method umercal decso tables based o proposed fuzzy dstace. Expermets o data sets show that the classfcato accuracy of proposed method s more effcet tha the oes based fuzzy etropy. Keywords: Fuzzy rough set, fuzzy equvalece relato, fuzzy dstace, decso table, attrbute reducto, reduct.. Itroducto Attrbute reducto s a mportat ssue of data preprocessg steps data mg ad kowledge dscovery. The am of attrbute reducto s elmated redudat attrbutes to ehace the effectveess of data mg algorthms. Rough set theory of Pawlak s a effectve tool to solve the attrbute reducto problem decso tables ad the oe that researchers has performed for a log tme. Rough set based attrbute reducto methods are performed o decso tables wth dscretzed value doma [2-4]. I fact, the attrbute value doma of decso tables ofte cotas umercal values ad cotuous values. For example, the attrbute of body weght ad blood pressure patet data tables s usually umercal value, cotuous value. Whe performg attrbute reducto methods based o rough set, data eeds to be dscretzed. However, these dscretzed methods do ot preserve the tal dfferece 3

2 betwee objects the orgal data, so t reduces the classfcato accuracy after attrbute reducto. To solve the ssue of attrbute reducto decso tables wth umercal value ad cotuous value, recet years, researches have proposed ew methods based o fuzzy rough set approach. D u b o s ad P r a d e [] proposed fuzzy rough set theory s a combato of rough set theory ad fuzzy set theory order to approxmate fuzzy sets based o fuzzy equvalece relato. The fuzzy equvalece s determed by the attrbute value doma. Tradtoal rough set based o smlarty relato to approxmate sets. I rough set theory, two objects are called equvalet o the attrbute set (the smlarty s ) f ther attrbute values are equal o all attrbutes. Coversely, they are ot equal (the smlarty s 0). The fuzzy rough set theory has used the fuzzy equvalece relato to replace the equvalece relato. The value smlarty the rage [0, ] shows the close or smlar propertes of two objects. Therefore, the fuzzy equvalece preserves the dfferece or the smlarty of objects. Attrbute reducto methods based o fuzzy rough set approach has the potetal to preserve the classfcato accuracy after mplemetg attrbute reducto methods. I recet years, the topc of the attrbute reducto based o fuzzy rough set has attracted may researchers [5-4]. Wth attrbute reducto ssue drectly o the decso table based o fuzzy rough set, these related researches are cocerg two ma drectos: fuzzy postve rego approach ad fuzzy etropy approach. Based o fuzzy postve rego, H u, Xe ad Y u [8] proposed FAR-VPFRS algorthm to fd oe fuzzy postve rego reduct whch use the fuzzy membershp fucto. The expermetal data sets show that the classfcato accuracy of FAR- VPFRS algorthm s better tha the oe of algorthm whch use the membershp fucto accordg to tradtoal rough set. Q a et al. [4] proposed FA_FPR algorthm, whch s a mprovemet of FAR-VPFRS [8] terms of executed tme. Accordg to fuzzy etropy approach, H u, Y u ad Xe [7] proposed the fuzzy etropy whch s based o etropy Shao ad troduces FSCE to fd oe reduct usg fuzzy etropy. D a ad Xu [6] proposed fuzzy ga rato based o fuzzy etropy ad troduces GAIN_RATION_AS_FRS to fd oe reduct usg fuzzy ga rato. The expermetal data sets show that the classfcato of FSCE, GAIN_RATION_AS_FRS algorthms are better tha the oes based o tradtoal rough set. Q a et al. [4] who proposed FA_FSCE algorthm, s a mprovemet of FSCE algorthm [7] terms of executed tme. I both drecto approaches, authors [4] have ever evaluated the classfcato accuracy after mplemetg mproved algorthms FA_FPR, FA_FSCE. Wth drect reducto attrbute o the decso table based o fuzzy rough set, the am of ths paper s to propose a ew method whch mproves the classfcato accuracy more tha the prevous oes. I ths paper, we propose the heurstc algorthm to fd a best reducto of the decso table wth umercal attrbute value doma usg a fuzzy dstace. The fuzzy dstace s costructed betwee two parttos. The expermetal results data sets from UCI [7] show that the classfcato accuracy of proposed algorthm s better tha FA_FSCE ad FA_FPR algorthms [4]. The structure of ths paper s as follows. Secto 2 presets some basc cocepts of fuzzy rough set theory. Secto 3 presets the method of costructg the fuzzy dstace betwee two attrbute sets. 4

3 Secto 4 presets a attrbute reducto method usg fuzzy dstace measure. Secto 5 presets expermetal results. Fally, Secto 6 gves a cocluso of ths paper ad subsequet developmets. 2. Basc cocepts I ths secto, we troduce some cocepts rough set theory, fuzzy rough set theory ad some related cocepts fuzzy partto space. DS U, C D, where U be a o-empty fte set; A decso table s a par C s called codtoal attrbute set, D s called decso attrbute set wth C D. DS s called the umercal decso table where the value doma of c C s the umercal for ay c C. Pawlak s tradtoal rough set theory [5] used a equvalece relato to approxmate sets. A subset P C determes equvalece relato o attrbute value doma, deote by IND(P), IND P u, vu U a P, au av, av s deoted as the value attrbute a object v; IND(P) determes the partto o U, deoted by U / IND u. The P ad the equvalece class of u, deoted by P lower approxmato set ad the upper approxmate set of X s defed as ad PX u U u X P U related to P C PX u U u X. D. Dubos ad others proposed the fuzzy rough set whch used fuzzy equvalet to approxmate the fuzzy sets. The decso table wth umercal attrbute doma DS U, C D, the relato R defed o U s called fuzzy smlarty relato f t satsfes the followg codtos [4]: ) Reflectvty: R x, x. 2) Symmetry: R x, y R y, x. 3) Max-m trastvty: R x, z m R x, y, R y, z x, y, z U. Let U be a o-empty fte set ad o U f for ay x, y U, we have: ) RQ x, y RQ x, y, P, for ay R P và 2) R RQ R x, y max x, y, RQ x, y, 3) R RQ R x, y m x, y, RQ x, y, 4) RQ x, y RQ x, y. R Q be a fuzzy equvalece relato 5

4 6 The relato matrx of where pj x, x j R P deoted by M ( ) [ pj ] p p2... p p2 p22... p 2 M( ), p p2... p s the fuzzy relato value of x ad Let DS U, C D s defed as x o P, 0, j p. be a decso table wth umercal attrbutes ad P, Q C. Accordg to [] we have Ra ad Q RQ, t meas that Q x, y m x, y, RQ x, y M R P ap for ay x, y U. Suppose that p ad M ( RQ) q are relatoal matrces of R P, R j j correspodg, the the relatoal matrx o the attrbute sets S P Q s defed as M ( R ) M R s S P Q j For P C, U x,..., x wth sj m pj, qj., the fuzzy partto / P geerated from the fuzzy equvalece relato R P : U / x x,..., x where 2 2 R R R P U R o U ca be, P P P x p / x p / x... p / x s a fuzzy set, s called a fuzzy equvalece of object x. The membershp fucto of objects s determed by x x, x x, x p for ay xj U. The, the cardalty of x R P j j j j x R P fuzzy equvalece s calculated [] as x p Let s called a set of all of fuzzy parttos o U whch determed by fuzzy equvalece o attrbute sets. The s called a fuzzy partto space o U. Thus, the fuzzy partto space s determed by fuzzy equvalece relato whch chose from the attrbute value doma. Let x,..., x partto where j j. be a fuzzy x p / x... p / x. Specally, f p 0,, j, the x 0 ad the fuzzy partto s called the fest oe, wrte as The, x,..., x x j / x j,, j, j 0. where j j j Q. If

5 p,, j, j the x U, the coarsest oe, wrte as j x / x j j,, j, j. relato, ad the fuzzy partto. The, x,..., x s a fuzzy partto space o U, for R P, RQ s called where, the partal order : R R x x, p q,, j, deoted by R Q. Furthermore, P Q RQ j j R P RQ x x, p q,, j, R R j j deoted by RQ; P R P R Q R P R Q ad RQ deoted by R Q. Example. Let,, x, x U x x Q, 2 R 2, P R RQ x P, x RQ 2 RS x, x2, x 0./ x 0.2 / x2 where x 0.2 / x 0.3 / x 2 2 R S R, x 0.2 / x 0.3 / x RQ 2, x 0.3 / x 0.4 / x, The we have 2 RS S R, Q, x 0.4 / x 0.6 / x. 2 2 x, x, RS x, x, RQ RS x x, x x, x x 0.3 RQ RQ x 0.3 / x 0.4 / x, 2 RQ 2 x , RQ x , RS 0.5, x x, x x, x x RQ RS 0.3 RS RQ RS RS 3. Fuzzy dstace betwee two fuzzy parttos ad ts propertes 3.. Fuzzy dstace betwee two fuzzy sets Frstly, we have proposed a dstace measure betwee two fuzzy sets, called a fuzzy dstace. Lemma. Let a, b, m be three real umbers wth a b. The, we have a b m a, m m b, m. 7

6 P r o o f: It s easy to see that a b m a, m m b, m m a, b m a, m b. Ths completes the proof. satsfes: Lemma 2. Let A, B, C be three fuzzy sets o the same uverse U. The, we have: ) If A B the B B C A A C. 2) If A B the C C A C C B. 3) A A B C C A C C B. P r o o f: ) Because of A B, for ay x U we have x B A x. Accordg to Lemma, we have x x m x, x m x, x B A B C A C U U U x x m x, x B A B C U m, A x x B A B C A C B B C A A C. 2) Because of A B, for ay x U B C, we have x x A m x, x m x, B C A C x x m x, x x m x, x U C A C C B C U U x m x, x U C A C C B C C C A C C B x m x, x 3) From A C A, accordg to property ) we have (*) A A B A C A C B.. Furthermore, from A B A, accordg to property 2) we have (**) C C A B C C A. From (*) ad (**) we have A A B C C A A C A C B C C A C A B C C C A. 8

7 Proposto. Let AB, be two fuzzy sets o the same uverse U. The, d A, B A B 2 A B s a dstace measure betwee A ad B. d A, B 0. P r o o f: Obvously, from A A B ad B A B to Furthermore, d A, B d B, A loss of geeralty, oe eeds to prove d A, B d A, C d B, C Lemma 2 (Part 3), we have:. We eed to prove the tragle equalty; wthout (***) A A B C C A C C B, (****) A A C B B A B B C. It s ferred from (***) ad (****), we have. Accordg to A B 2 A B A C 2 A C B C 2 B C, smlarly d A, B d A, C d B, C. Therefore, d A, B s a dstace measure betwee fuzzy set A ad fuzzy set B, called fuzzy dstace. We have proposed a dstace betwee two fuzzy parttos based o fuzzy dstace Fuzzy dstace betwee two fuzzy parttos ad ts propertes Theorem. Let DS U, C D be a decso table, where U x x x, RQ,,..., 2 be two fuzzy parttos duced by two fuzzy equvalece R P, o P, Q C. The () D, RQ s a fuzzy dstace betwee P r o o f: Obvously, D RQ x x 2 x x R R R R P Q P Q, ad RQ., 0 ad P, Q Q, P D R R D R R. We eed to prove the tragle equalty, wthout loss of geeralty, for ay R,, P R Q R S D, RQ D, RS D RQ, RS, ad we prove. ad R Q 9

8 It s ferred from Proposto, for ay x The U we have d x, x d x, x d x, x. RQ RS RQ RS P, Q P, S D R R D R R = x x 2 x x RQ RQ x x 2 x x RS RS It s easy to see that,, d x x RQ d x x RS, d x x R Q R S Q S D R, R. D, RQ 0 RQ ; D, RQ ad RQ, (or, ad RQ ). Therefore, 0 D, RQ. Proposto 2. Let be a fuzzy partto o. The, we have D, D,. P r o o f: Suppose that x, x2,..., x D, 2 x, D R R, K P 2 x we have D, D,.. The R R R P P P P Example 2. Cotug from Example. Accordg to Theorem, we have P, Q 0., D RQ RS D, RS D R R, 0.25,. The, 20

9 Therefore: P Q Q S P S P Q P S Q S Q S P S P Q D R, R D R, R D R, R, D R, R D R, R D R, R, D R, R D R, R D R, R. 4. A fuzzy dstace based attrbute reducto method decso tables wth umercal attrbutes I ths secto, we troduce a fuzzy dstace based attrbute reducto method whch performs drectly o the decso tables wth umercal attrbutes. The ew fuzzy dstace s defed betwee two fuzzy parttos (see Secto 3). DS U, C D be a decso table wth umercal attrbutes, Let U x, x2,..., x. We use a fuzzy equvalece relato defed o codtoal attrbutes. For ay p C, the followg fuzzy equvalece relato R p s ofte used to costruct relatoal matrx M R p p [6] j j j p x p x p x p x 4, 0.25, (2) pj pmax pm pmax pm 0 otherwse, where p x s the value of the attrbute p object x ; pmax, pm are maxmum value, mmum value of the attrbute p, correspodg. IND D ad a equvalece matrx We use a equvalece relato M IND D d j o the decso attrbute set, d j f x x f xj x. I other words, a equvalece class x D D equvalece class, deoted by x j D D ad d 0 j D ca be see as a fuzzy f x, the membershp fucto f x x x ad 0 x x j D The, the fuzzy partto deoted by. j D D x x,..., x. D D D x x j Based o the fuzzy equvalece relato, we propose a fuzzy dstace betwee the codtoal attrbute set ad the decso attrbute set. I Secto 3, the attrbute set P C determed a fuzzy partto. Thus, for smplcty we replace the cocept fuzzy dstace betwee two parttos wth the cocept fuzzy dstace betwee two attrbute sets. D j 2

10 Defto. Let DS U, C D attrbutes,, RQ relatos R P, R Q o P, Q C be a decso table wth umercal be two fuzzy parttos duced by two fuzzy equvalece. The, fuzzy dstace betwee P ad Q, deote by d P, Q, s defed a fuzzy dstace betwee two fuzzy parttos RQ, t meas that d P Q D RQ,,. Proposto 3. Let DS U, C D attrbutes, where U x x x ad be a decso table wth umercal, 2,..., ad R be a fuzzy equvalece relato determed o codtoal attrbutes. The, fuzzy dstace betwee two attrbute sets C ad C D whch s determed as x x C x R RC D (2) d C, C D. P r o o f: From Defto ad Theorem, we have: C CD d C, C D D R, R x x 2 x x RC RCD RC RCD x x x RC RC RD x x x 2 x x RC RC RD RC RD x x x RC RC D. It s easy to see that 0 d C, C D ; d C C D R C D ad d C C D., RC, 0 ad x x Proposto 4. Let DS U, C D attrbutes, where U x, x2,..., x, B C determed o codtoal attrbutes. The, d B B D d C C D D for be a decso table wth umercal ad R be a fuzzy equvalece relato,,. 22

11 P r o o f: From B C, accordg to [4] we have RC RB meas that t ca be ferred from x x that x x For ay x U, we have: RC RB R, whch C for. x x x R R D x x j x x j x x j C C RC RC D j j R B m,, x x x R R D x x j x x j x x j m,. B B RB RB D j j () For ay x x we have j D x x j D, therefore x x C x C x x B x B (2) For ay x x we have 0, therefore j D 0. R R D R R D x x j D x x x x x x x x. RC RC D RC RB From (), (2) we have: RB RB D x x x x x x RB RB D RC RC D x x B x R RB x D x R x R D C C d B, B D d C, C D. It s easy to see that d B, B D d C, C D x x RB RC ay x U. I ext part, we preset a attrbute reducto method of the decso table usg the fuzzy dstace measure Proposto 3. Our method cludes: defg the reduct based o fuzzy dstace, defg the mportace of the attrbute ad desgg a heurstc algorthm to fd the best reduct based o the mportace of the attrbute. DS U, C D be a decso table wth umercal attrbutes, B C If Defto 2. Let ad R be a fuzzy equvalece relato determed o codtoal attrbutes. ) d B, B D d C, C D, 2) bb, d ( B b, B b D) d ( C, C D), the B s a reduct of C based o fuzzy dstace. DS U, C D be a decso table wth umercal Defto 3. Let attrbutes, B C ad b C B. The mportace of attrbute b wth respect to B s defed as SIG b d B, B D B d B b, B b D. for 23

12 24 b. The mportace of b From Proposto 4, we have SIG B 0 SIG B characterzes the classfcato accuracy of attrbute b wth respect to decso attrbute D. It s used as the attrbute selecto crteral for heurstc algorthms to fd the best reduct. Algorthm F_DBAR (Fuzzy dstace based o attrbute reducto): A heurstc algorthm to fd the best reduct by usg fuzzy dstace. Iput: Decso table wth umercal attrbutes DS U, C D equvalece relato R. Output: The best reduct B Step. B ; M( RB ) ;, fuzzy Step 2. Calculate relato matrx M( R C ), calculate equvalece matrx MIND D, calculate fuzzy dstace d C, C D ; //Addg gradually to B a attrbute havg the greatest mportace d B, B D d C, C D do Step 3. Whle Step 4. Beg Step 5. For each a C B calculate SIG B a d B, B D d B a, B a D ; Step 6. Select am C B Step 7. B B a m ; Step 8. Ed; so that SIG B am Max SIG B a //Remove redudat attrbute B Step 9. For each a B Step 0. Beg d B a B a D ; ; ac B Step. Calculate, Step 2. If d B a, B a D d C, C D the B B a ; Step 3. Ed; Step 4. Retur B; DS U, C D be a decso table wth umercal attrbutes Example 3. Let (Table ) where U u, u, u, u, u, u, C c, c, c, c, c, c equvalece relato R s defed the Formula (2). Table. The decso table wth umercal attrbutes u c C C 2 C 3 C 4 C 5 C 6 D u u u u u u , the fuzzy

13 By usg steps of F_DBAR algorthm to fd the best reduct, we have B ; M( RB ) ; d, D 0.375, ad we calculate some relato matrces M ( Rc ), M ( Rc ), M ( Rc ), M ( Rc ), M ( Rc ), M ( Rc ), M ( R C ), ad the equvalece matrx IND M D : M( Rc ), M ( Rc2 ) 0 0 0, M( Rc3 ), M ( Rc4 ) 0 0 0, M ( Rc5 ), M ( Rc6 ), M( RC ), M(INDD) Calculate d C, C D 0, d c, c D , d c2, c2 D , d c c D d c4, c4 D 0., d c c D, ; SIG B c , d c6 c6 D , , 5, , SIG B c , 25

14 B c SIG B c , SIG SIG 3333, Attrbute c 4 s selected. B c SIG B c , Smlarty, d { c4, c},{ c4, c} D 0, checked d { c4, c},{ c4, c} D df C, C D 0, the algorthm fshed ad B c, c. Cosequetly, B c, c of DS. 5. Expermets 4 s the best reduct We select eght data sets wth umercal attrbutes from the UCI repostory [7] to test proposed algorthm Table 2. Evrometal testg s PC wth Petum Core 3, 2.4 GHz CPU, 2 GB RAM, usg Wdows 0 operatg system. 4 Table 2. Data sets the exprmetal aalyss Id Data sets Number of codtoal attrbutes Number of objects Ecol Ioosphere Wdbc (Breast Cacer Wscos) Wpbc (Breast Cacer Wscos) We Glass Soar (Coectost Bech) Heart We select FA_FPR algorthm (Fdg Reduct based o Fuzzy Postve Rego) ad FA_FSCE algorthm (fdg reduct based o fuzzy etropy) [4] to compare wth F_DBAR proposed algorthm o the classfcato accuracy of reduct. The FA_FPR algorthm s a mpovemet of FAR-VPFRS algorthm [8] o executed tme, the FA_FSCE s a mpovemet of FSCE algorthm [7] o executed tme. Accordg to fuzzy rough set approach, the classfcato accuracy of FAR-VPFRS algorthm [8], FSCE algorthm [7] are almost hgher tha the oes rough set approach after dscretzed data. However, authors [] have ot evaluated the classfcato accuracy for algorthms FA_FSCE, FA_FPR. For testg, we perform the followg tasks: ) Code FA_FPR, FA_FSCE ad F_DBAR algorthms by program C#. Algorthms used the fuzzy equvalece relato defed by the formula (2). 2) Execute three algorthms o eght data sets by evromet testg. 3) Use C4.5 algorthm WEKA [8] to evaluate the classfcato accuracy of three algorthms by selectg 2/3 frst objects as trag set ad the remader objects as testg set. 26

15 Table 3 shows the testg results of eght data sets where U s the umber of objects, C s the umber of the codtoal attrbute, R s the umber of attrbute of the reduct for each algorthm. Table 3. The exprmetal result of three algorthms FA_FSCE, FA_FPR, F_DBAR Id Data set U C FA_FSCE Algorthm FA_FPR Algorthm F_DBAR Algorthm R Classfcato accuracy of C4.5 (%) R Classfcato accuracy of C4.5 (%) R Classfcato accuracy of C4.5 (%) Ecol Ioosphere Wdbc Wpbc We Glass Soar Heart The average classfcato accuracy of C FA_FSCE FA_FPR F_DBAR Fg.. The classfcato accuracy C4.5 of FA_FSCE, FA_FPR ad F_DBAR The exprmetal results Table 3 ad Fg. show that the average classfcato accuracy of F_DBAR (used the fuzzy dstace) s hghest, ext to FA_FPR (used fuzzy postve rego) ad FA_FSCE s lowest (used fuzzy etropy). For each data set, the classfcato accuracy of three algorthms are dfferet. Cosequetly, the classfcato accuracy of algorthm F_DBAR s the best oe of three algorthms. 6. Cocluso The am of attrbute reducto the decso table s to mprove the accuracy of classfcato model. Amog attrbute reductos the decso table wth umercal value doma, related researches show that attrbute reducto methods based o fuzzy rough set approach have the classfcato accuracy more hgher tha oe based o tradtoal rough set. I ths paper, we propose a attrbute reducto method o the decso table wth umercal attrbute value whch uses fuzzy dstace based o fuzzy rough set. Our research cludes: proposg a ew 27

16 fuzzy dstace betwee two fuzzy parttos, defg reduct ad mportace of attrbutes based o fuzzy dstace, proposg a heurstc algorthm to fd the best reduct. The expermetal results from data sets show that the classfcato accuracy of fuzzy dstace method s hgher tha that of the oes usg fuzzy postve rego ad fuzzy etropy. Our further research approach ssue s fdg the relato betwee reduct obtaed by dfferet methods to subgroup ad overall evaluato of methods based o fuzzy rough set approach. Ackowledgemets: Ths research has bee fuded by the Research Project, VAST 0.08/6-7. Vetam Academy of Scece ad Techology. R e f e r e c e s. D u b o s, D., H. P r a d e. Rough Fuzzy Sets ad Fuzzy Rough Sets. Iteratoal Joural of Geeral Systems, Vol. 7, 990, pp D e m e t r o v c s, J., V. D. Th, N. L. G a g. A Effcet Algorthm for Determg the Set of All Reductve Attrbutes Icomplete Decso Tables. Cyberetcs ad Iformato Techologes, Vol. 3, 203, No 4, pp D e m e t r o v c s, J., V. D. Th, N. L. G a g. O Fdg All Reducts of Cosstet Decso Tables. Cyberetcs ad Iformato Techologes, Vol. 4, 204, No 4, pp D e m e t r o v c s, J., N. Th, L. H u o g, V. D. Th, N. L. G a g. Metrc Based Attrbute Reducto Method Dyamc Decso Tables. Cyberetcs ad Iformato Techologes, Vol. 6, 206, No 2, pp T s a g, E. C. C., D. G. C h e, D. S. Y e u g, X. Z. W a g, J. W. T. Lee. Attrbutes Reducto Usg Fuzzy Rough Sets. IEEE Tras. Fuzzy Syst., Vol. 6, 2008, pp Da, J., Q. X u. Attrbute Selecto Based o Iformato Ga Rato Fuzzy Rough Set Theory wth Applcato to Tumor Classfcato. Appled Soft Computg, Vol. 3, 203, pp H u, Q., D. R. Y u, Z. X. Xe. Iformato-Preservg Hybrd Data Reducto Based o Fuzzy- Rough Techques. Patter Recogt. Lett., Vol. 27, 2006, No 5, pp H u, Q., Z. X. Xe, D. R. Y u. Hybrd Attrbute Reducto Based o a Novel Fuzzy-Rough Model ad Iformato Graulato. Patter Recogt., Vol. 40, 2007, pp J e s e, R., Q. S h e. Sematcs-Preservg Dmesoalty Reducto: Rough ad Fuzzy- Rough-Based Approaches. IEEE Tras. Kowl. Data Eg., Vol. 6, 2004, No 2, pp J e s e, R., Q. S h e. Fuzzy-Rough Attrbute Reducto wth Applcato to Web Categorzato. Fuzzy Sets Syst., Vol. 4, 2004, pp J e s e, R., Q. S h e. Fuzzy-Rough Sets Asssted Attrbute Reducto. IEEE Tras. Fuzzy Syst., Vol. 5, 2007, No, pp J e s e, R., Q. S h e. New Approaches to Fuzzy-Rough Feature Selecto. IEEE Tras. Fuzzy Syst., Vol. 7, 2009, No 4, pp B h a t t, R. B., M. G o p a l. O Fuzzy-Rough Sets Approach to Feature Selecto. Patter Recogt. Lett., Vol. 26, 2005, pp Q a, Y. H., Q. W a g, H. H. C h e g, J. Y. L a g, C. Y. D a g. Fuzzy-Rough Feature Selecto Accelerator. Fuzzy Sets ad Systems, Vol. 258, 205, pp P a w l a k, Z. Rough Sets: Theoretcal Aspects of Reasog about Data. Lodo, Kluwer Academc Publsher, P a w l a k, Z., J. W. G r z y m a l a-b u s s e, R. S l o w s k, W. Z a k o. Rough Sets. Commu. ACM, Vol. 38, 995, No, pp The UCI Mache Learg Repostory