Attribute Reduction Algorithm Based on Discernibility Matrix with Algebraic Method GAO Jing1,a, Ma Hui1, Han Zhidong2,b

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1 Inernaonal Indusral Informacs and Compuer Engneerng Conference (IIICEC 05) Arbue educon Algorhm Based on Dscernbly Marx wh Algebrac Mehod GAO Jng,a, Ma Hu, Han Zhdong,b Informaon School, Capal Unversy of Economcs and Busness, Beng Beng Informaon Cener of Chna UnonPay Co., Ld, Beng 0093 a emal:gaonggao@6.com, bemal:hanzhdong@sna.com Keywords: ough Se; Posve egon; Dscernbly Marx; Algebra Mehod; Complexy Absrac. There are many algorhms of arbue reducon based on dscernbly marx, where he elemen of he dscernbly marx s used as heursc nformaon. Bu here are few algebra mehods abou arbue reducon based on dscernbly marx. In hs paper, we proposed an algorhm of arbue reducon wh algebra mehod. Frs of all, we should desgn an algorhm o compue he heursc nformaon. Then we used hs heursc nformaon o desgn an new algorhm of arbue reducon based on dscernbly marx, whose me complexy s O( C U ), and whose space complexy s O( U ). A las, we used an example o llusrae he new algorhm s valdy and hgh effcency. Inroducon Arbue educon s one of he researches n rough se heory [], and whch based on dscernbly marx has araced a los of aenon as s smplfy and nuve []. On one hand, he research on Arbue educon Algorhm Based on Dscernbly Marx s more nclned o be desgned by he heursc nformaon, whch s composed by elemens of he dscernbly marx. On he oher hand, Professor Pawlak has pu forward many desgn mehods of Arbue educon Algorhm of Arbue educon model. Namely, hey are respecvely algebra mehod [3], dscernbly marx mehod [4-6] and marx [7].ec. So s arbue reducon algorhm of arbue reducon model based on nformaon enropy proposed by Professor Wang, however, Bu he desgn of arbue reducon algorhm of dscernbly marx n algebrac mehod s less used, hs s manly because he model has no drec algebrac defnon eference [0-] llusraes a new arbue reducon model based on nformaon. eference [] has proved ha he model s equal o he Arbue educon Algorhm Based on Dscernbly Marx. Of whch eference [3] has pu forward a marx mehod. So, he paper desgns an algebra mehod of Arbue educon based on dscernbly. The Specfc seps are as follows: Frsly, we desgn a algorhm o calculae a heursc nformaon wh rapd response, hen based on he heursc nformaon we desgn Arbue educon algorhm based on dscernbly marx. Whose me complexy s O( C U ), and space complexy s O( U ). A las, an example s provded o llusrae he new algorhm. Basc Theory Defnon : Decson Table s defned as follows: S = (U, C, D, V, f, d ). In hs formula, U = {x, x,, xn } s he doman, C = {c, c,, cr } s he condons arbue ses. D s he decson arbue ses. f : U C V and d : U D V s nformaon funcons, n he, V = Va, a F, Va means range of a. funcons, F = C D, C D = Defnon : Defne dscernbly marx as M = ( m ) n S = (U, C, D, V, f, d ). The elemen s as follows: 05. The auhors - Publshed by Alans Press 349

2 { ck ck C, f ( x, ck) f ( x, c ), (, ) (, )} k d x D d x D m = () else In hs formula, k=,,..., r ;, =,,, n. Defnon 3: In decson able, S= ( U, C, DV,, f, d), we hypohesze ha M = ( m ) s HU dscernbly marx, B C,f B sasfes followng condons: () m M, B m ;() b B, B {} b doesn sasfy (), B s consdered as arbue reducon based on dscernbly marx. all arbue reducons s shor for HEDC ( D ). Defnon 4: n decson able, S= ( U, C, DV,, f, d), any arbue n U B C D ( knowledge, Equvalence relaon famly, U / B= { B, B,, B }) s a random varable conssed n subse of U n algebra., he Probably dsrbuon s confrmed by he followng mehods: B B B [ B: p] = pb ( ) pb ( ) pb ( ) () In he formula, pb ( ) = B / U, =,,, Defnon 5: In decson able, S= ( U, C, DV,, f, d), he knowledge nformaon of decson arbues ses U / D = { D, D,, Dk} respecng o condon arbues ses U / C = { C, C,, Cm} s defned as follows: m k X D D X D I( D C) = (3) = = U U U k X = X D X / D= { X, X, X } (4) Defnon 6: In decson able, S= ( U, C, DV,, f, d), for b B C, f I( D B ) = I( D ( B { b})), b s Omed for B wh respec o D. or, b s canno be omed for B wh respec o D. for B C, f any elemen s necessary for D, hen B s ndependen for D. Defnon 7: In decson able, S= ( U, C, DV,, f, d), f B C, I( D B ) = I( D C ) and B s ndependen for D, hen B s arbue reducon based on knowledge for C wh respec o D. mark all he arbue reducon as NewEDC ( D ). Theorem, n he decson able, S= ( U, C, DV,, f, d), Arbue reducon based on dscernbly marx s equvalen o arbue reducon based on amoun of knowledge. We wll desgn arbue reducon based on dscernbly marx wh algebrac mehod under he condon of Theorem ; Theorem, n he decson able, S= ( U, C, DV,, f, d), C A B, I( D A) I( D B) s deduced. Defnon 8: n he decson able, a C B, defne he heursc nformaon as sg ( ) ( ) ( { }) B a = I D B I D B a. Obvously, sgb ( a) 0, and sgb ( a ) ge bgger, he ncremenal amoun of nformaon of he a propery s greaer Defnon 9: In he decson able, S= ( U, C, DV,, f, d), X U, X /{ a} = { X, X, X r }, r k X D X D X D g( X, a) = s he funcon we defne. = = U U U Theorem 3, n he decson able, S= ( U, C, DV,, f, d), U / B= { B, B,, B}, a C B, 350

3 sg ( a) = g( B, a) can be deduced from defnon 8 and defnon 9. B = Hgh-effcen arbue reducon algorhm In order o ge he hgh-effcen arbue reducon algorhm, frsly we ge he algorhm of quck response calculaon heursc nformaon. Algorhm: calculae g( X, a ) Inpu: decson able S= ( U, C, DV,, f, d), X Ua, C Oupu: g( X, a ) ). Calculang X /{ a} wh adx sor n he reference [3] ).calculang X/ DX, / D wh radx sor n he reference [3] 3).calculang g( X, a ) wh he formula n defnon 9. Complexy analyss of algorhm : The me complexy of he frs sep of algorhm s O( X ), The me complexy of second sep s O( X ), so he me complexy of algorhm s O( X ),The wors-case complexy of space s O( X ). Algorhm : calculae sgb ( a ). Inpu: n decson able, S= ( U, C, DV,, f, d), U / B= { B, B,, B }, a C B Oupu: sgb ( a ). ) calculae gb (, a ) wh algorhm. ) calculae sgb ( a ) wh algorhm 3. Complexy prncpal analyss of algorhm : The me Complexy of he frs sep of algorhm s O( B ), so he wors-case complexy of me s O( B ) = O( U ), he wors-case complexy of space s O( U ). = Algorhm 3: arbue reducon algorhm Inpu: decson able S= ( U, C, DV,, f, d). Oupu: arbue reducon ; ) = ; )for any a C, calculae sg ( a ) wh algorhm. 3)make sg ( b) = max sg ( a), f sg ( b ) > 0, hen = {} b, ge o he sep, or ge o a C he sep 4. 4) oupu he arbue reducon. Complexy prncpal analyss of algorhm 3: The wors-case me complexy of second sep of algorhm 3 s O( C U ) wors-case me complexy of algorhm 3 s space complexy s O( U ). C = 0 = O C U = O C U ( ) ( ), so he, he wors-case Example analyss Decson able n he leraure [3] s used o llusrae he new algorhm. 35

4 Table decson able a b c d D X 0 X X3 0 X4 X5 3 0 X Calculae sg ( a) = ( + ) + ( + ) + ( + + ) + ( + + ) = wh algorhm By he same oken, sg ( b) =, sg () c = = sg ( d). We choose arbue b accordng he hrd sep, so we ge he equaon = {} b, and hen ge o he sep. Calculae 8 4 sg{ b} () a =, sg{ b} () c = sg{ b} ( d) =, we choose arbue accordng o he hrd sep. So we ge he expresson- = {, ba}, hen ump o sep, calculae sg{ ba, }() c = sg{ ba, }( d) = 0 by sep 4, arbue reducon = {, ba} s oupued. Concluson In hs paper, algebrac expresson arbue reducon defnons based on dscernbly marx s pu forward, and hen sarng from he arbue reducon defnons of algebrac mehods, he heursc nformaon s defned, and he algorhm of heursc nformaon calculang wh rapd reacon s provded, fnally desgn an arbue reducon algorhm based on dscernbly marx, wch me complexy s O( C U ) and space complexy s O( U ). Fnally, an example s used o llusrae he effecveness and effcency of he new algorhm. Acknowledgemen In hs paper, he research was suppored by he Beng alened persons ranng scenfc research proec n 0 (Proec name: Algorhm of daa mnng of ncomplee nformaon sysems ; Proec No.0D ); he research was suppored by he mporaon and developmen of hgh-calber alens proec of Beng Muncpal Insuons n 03(Proec No.CIT&TCD030435;Proec name: Decson ree generaon algorhm and s opmzaon of ncomplee nformaon sysems); he research was suppored by Beng Educaon Commee funds on how o mprove scence research level; he research was suppored by Naonal Naural Scence Foundaon of Chna (Gran No ). eferences [] Pawlak Z. ough Ses [J]. Inernaonal Journal of Compuer and nformaon Scence, 98, (5): [] Hu Xao Hua, Cercone N. Learnng n relaonal daabases: a rough se approach [J]. Compuaonal Inellgence, 995, (): [3] Zhang Yan Xu, Zuo Peng Lu, Bng u Yang, We Song. A Quck Arbue educon Algorhm wh Complexy of max(o( C U ),O( C U/C )) [J]. Chnese Journal of Compuers, 006, 9 (3) : [4] Ja Yang Wang, Can Gao. Improved Algorhm for Arbue educon Based on Dscernhly Marx [J]. Compuer Engneerng, 009, 35(3):

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