BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND IORMATION TECHNOLOGIES Volume 6, No 4 Sofa 206 Prt ISSN: 3-9702; Ole ISSN: 34-408 DOI: 0.55/cat-206-0064 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
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
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
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
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, 0. 0.2 0.3 RS 2 0.2 0.3 0.5 x, x, 2 0.3 0.4 0.7 RQ 0.3 0.4 0.7 RS x x, x x, x x 0.3 RQ 2 2 0.5 RQ x 0.3 / x 0.4 / x, 2 RQ 2 x 0.2 0.3 0.5, RQ x2 0.4 0.6, RS 0.5, x x, x x, x x 2 2 0.7 RQ RS 0.3 RS RQ RS 2 2 0.5. 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
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
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. 3.2. 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
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 0.225. D R R, 0.25,. The, 20
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
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
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
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 2 3 4 5 6 equvalece relato R s defed the Formula (2). Table. The decso table wth umercal attrbutes u c 2 3 4 5 6 C C 2 C 3 C 4 C 5 C 6 D u 0.8 0.2 0.6 0.4 0 0 u 2 0.8 0.2 0 0.6 0.2 0.8 u 3 0.6 0.4 0.8 0.2 0.6 0.4 0 u 4 0 0.4 0.6 0.4 0 u 5 0 0.6 0.6 0.4 0 u 6 0 0.6 0 0 0, the fuzzy
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 ), 2 3 4 5 6 ad the equvalece matrx IND M D : 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M( Rc ), 0 0 0 0 M ( Rc2 ) 0 0 0, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M( Rc3 ), 0 0 0 0 0 M ( Rc4 ) 0 0 0, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2 0.2 0.2 0 0 0.2 0.2 0.2 0 0 0 0 0 M ( Rc5 ), 0 0 0 0 0 M ( Rc6 ), 0 0.2 0 0 0.2 0 0 0.2 0 0 0.2 0 0 0.2 0 0 0.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M( RC ), M(INDD). 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Calculate d C, C D 0, d c, c D 0.66667, d c2, c2 D 0.66667, d c c D d c4, c4 D 0., d c c D, ; SIG B c 0.208333333, d c6 c6 D 0.22222 3, 3 0.66667, 5, 5 0.22222, SIG B c 2 0.208333333, 25
B c 3 0.20833 SIG B c 4 0.263888889, SIG 7778. SIG 3333, Attrbute c 4 s selected. B c 6 0.25277 SIG B c 5 0.252777778, 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 7 336 2 Ioosphere 34 35 3 Wdbc (Breast Cacer Wscos) 30 569 4 Wpbc (Breast Cacer Wscos) 32 98 5 We 3 78 6 Glass 9 24 7 Soar (Coectost Bech) 60 208 8 Heart 3 270 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
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 336 7 6 8.50 7 82.45 7 82.45 2 Ioosphere 35 34 88.72 3 9.52 5 94.25 3 Wdbc 569 30 6 95.2 7 90.46 9 92.84 4 Wpbc 98 32 6 65.32 7 73.60 8 74.60 5 We 78 3 5 88.72 9 9.57 0 89.25 6 Glass 24 9 6 80.5 7 8.56 7 8.56 7 Soar 208 60 8 75.40 2 70.60 3 76.25 8 Heart 270 3 8 74.62 9 76.95 0 78.65 The average classfcato accuracy of C4.5 8.2 82.33 83.73 00 80 60 40 20 0 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
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. 9-209. 2. 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. 8-26. 3. 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. 3-0. 4. 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. 3-5. 5. 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. 30-4. 6. 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. 2-22. 7. 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. 44-423. 8. 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. 3509-352. 9. 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. 457-47. 0. 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. 469-485.. J e s e, R., Q. S h e. Fuzzy-Rough Sets Asssted Attrbute Reducto. IEEE Tras. Fuzzy Syst., Vol. 5, 2007, No, pp. 73-89. 2. 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. 824-838. 3. 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. 965-975. 4. 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. 6-78. 5. P a w l a k, Z. Rough Sets: Theoretcal Aspects of Reasog about Data. Lodo, Kluwer Academc Publsher, 99. 6. 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. 89-95. 7. The UCI Mache Learg Repostory. http://archve.cs.uc.edu/ml/datasets.html 8. https://sourceforge.et/projects/weka/ 28