Constructing Rough Set Based Unbalanced Binary Tree for Feature Selection

Transcription

1 hnese Journal of Electroncs Vol.23, No.3, July 2014 onstructng Rough Set Based Unbalanced Bnary Tree for Feature Selecton LU Zhengca 1,QINZheng 1,2,JINQao 1 and LI Shengnan 1 (1.Department of omputer Scence and Technology, Tsnghua Unversty, Bejng , hna) (2.School of Software, Tsnghua Unversty, Bejng , hna) Abstract Feature selecton s one of the challengng problems facng data analyss n areas such as pattern recognton, data mnng, and decson support. Many rough set algorthms for feature selecton have been developed, most of whch are essentally dependent on the defnte nformaton contaned wthn the lower approxmaton. Ths paper proposes a novel approach, called Unbalanced bnary tree based feature selecton (UBT-FS), whch utlzes the ndefnte nformaton contaned wthn rough set boundary regon for reducton. UBT-FS desgns the underlyng mechansm for obtanng the boundary regon from the unbalanced bnary tree and adopts the boundary regon based sgnfcance for determnng the optmal search path as well as the boundary regon based evaluaton crteron for dentfyng feature subsets. These allow UBT-FS to have consderable ablty n fndng an optmal or suboptmal reduct whlst smultaneously achevng obvously better computatonal effcency than other avalable algorthms, whch s also supported by the expermental results. Key words Feature selecton, Rough set theory, Boundary regon, Unbalanced bnary tree. I. Introducton Feature selecton, also called attrbute reducton, s vewed as a common problem n the felds of pattern recognton, data mnng, and decson support [1,2]. Its task s just to elmnate reducble or dspensable attrbutes such that the reduced set retans the dentcal dscernble ablty as that of the unreduced set, consderably decreasng the tme complexty of analyss algorthms and remarkably ncreasng the generalzaton capablty of nduced patterns. Rough set theory (RST) [3] s a powerful tool to deal wth mprecson, uncertanty, and vagueness. Much use has been made of RST to perform feature selecton and many consequent technques have been developed. Generally, they can be dvded nto two man categores: dscernblty matrx based soluton and search based soluton. The former obtans feature subsets by constructng a knd of dscernblty matrx from the consdered dataset and smplfyng the correspondng dscernblty functon [4,5]. It s ntutve, concse and suffcent to fnd the mnmal reduct but suffers from a poor computatonal performance. The latter dscovers a reduct by the use of search strateges, such as exhaustve search, ncomplete search [6], random search [7], and heurstc search [8 10]. Snce the heurstc search strategy has the competence to obtan optmal or suboptmal results along wth an acceptable computatonal complexty, t plays an mportant role n the feature selecton communty. omputatonal effcency prompts research nto heurstc search algorthms [11 17]. houchoulas and Shen [11] proposed the Quck reduct algorthm, whch employs the postve regon as heurstcs for reducton. Meng and Sh [12] put forth a faster verson by usng decomposton and sortng technques to calculate the postve regon. Wth consderaton of the boundary regon as auxlary nformaton, Parthaln et al. [13] presented the dstance metrc asssted feature selecton algorthm, whch combnes the lower approxmaton and the boundary regon to create a new evaluaton functon. Qan et al. [14] took hybrd attrbute measures for feature selecton, whch reflect the sgnfcance of an attrbute n postve regons and boundary regons. Dfferent from the algorthms mentoned above, Qan and Lang [15] ntroduced the concept of combnaton entropy for descrbng the uncertanty of nformaton systems and used ts condton entropy to select a feature subset. Sun et al. [16] utlzed rough entropy based uncertanty measures to evaluate the roughness and accuracy of knowledge, and then constructed a heurstc search algorthm wth low computatonal complexty for feature selecton from ncomplete decson systems. One can observe that most of the exstng heurstc feature selecton algorthms are essentally relant upon the defnte nformaton contaned wthn the lower approxmaton and very lttle work has htherto consdered the ndefnte nformaton contaned wthn the boundary regon as the leadng role for mnmzaton. In ths paper, we propose a novel soluton, called Unbalanced bnary tree based feature selecton (UBT-FS), whch captures the ndefnte nformaton contaned wthn the boundary regon to uncover the potental feature subsets. UBT-FS desgns an effcent mechansm to construct a rough set based unbalanced bnary tree, of whch each Manuscrpt Receved Apr. 2013; Accepted June Ths work s supported by the Specalzed Research Fund for the Doctoral Program of Hgher Educaton of hna (No ).

2 onstructng Rough Set Based Unbalanced Bnary Tree for Feature Selecton 475 nconsstent node corresponds to a certan boundary regon. Ths mechansm, workng n conjuncton wth the boundary regon based sgnfcance and the boundary regon based evaluaton crteron, makes UBT-FS capable of dentfyng the sgnfcant features effcently. The remander of ths paper s organzed as follows. In Secton II, we revew the theoretcal background to RST. Secton III explores the rough set based unbalanced bnary tree. Based on t, UBT-FS s desgned n Secton IV. In Secton V, some experments are practced to valdate the effectveness of UBT-FS. Fnally, we gve a concse concluson n Secton VI. II. Background Let S = (U, D) be an ncomplete decson system, where U s a non-empty fnte object set (the unverse), s a condton attrbute set, and D s a decson attrbute set. For each a D, there s a mappng f : U V a. V a s the value doman of a. V = {V a a } { } (where stands for mssng values) and V D = {V d d D} are the value domans of and D, respectvely. For any subset P, there exsts an assocated tolerance relaton SIM(P ): SIM(P )={(u, v) U U : a P, f(u, a) =f(v, a) f(u, a) = f(v, a) = } (1) The famly of tolerance blocks generated by SIM(P )s denoted as π P and can be calculated by π P = {X X U/SIM(P )}, wherex s a tolerance block, depctng the collecton of objects whch are possbly ndscernble from each other wth respect to P. If there does not exst another tolerance block Y such that X Y, X s called a maxmal tolerance block [5]. Generally, a tolerance class of an object u, denoted as S P (u), descrbes the maxmal set of objects whch are probably ndstngushable to u wth respect to P,namely,S P (u) = {v U (u, v) SIM(P )}. It can be also expressed wth the maxmal tolerance blocks such that S P (u) = S X π mp (u) X [5], where π mp (u) s the famly of maxmal tolerance blocks contanng u. Furthermore, t s easy to prove that S P (u) = S X π P (u) X, where πp (u) s the famly of tolerance blocks contanng u. onsder a partton π D = {D =1, 2,,j} of U determned by D. D s called the decson class and can be approxmated by a par of precse concepts whch are known as the lower and upper approxmatons: P (D )={u U S P (u) D } (2) P (D )={u U S P (u) D } (3) P (D )sregardedasthemaxmalp -defnable set contaned n D,whereasP (D ) s consdered as the mnmal P -defnable set contanng D. If P (D ) = P (D ), D s an exact set, otherwse t s a rough set. By the dual approxmatons, the unverse of the decson system s parttoned nto two mutually exclusve crsp regons: postve regon and boundary regon, defned respectvely as POS P (π D)= BND P (π D)= j[ P (D ) (4) =1 j[ (P (D ) P (D )) (5) =1 It s apparent that POS P (π D) BND P (π D) = U and POS P (π D) BND P (π D)=. If BND P (π D)= or POS P (π D)=U, wesaythats s consstent, otherwse t s nconsstent. III. Rough Set Based UB-tree The Unbalanced bnary tree (UB-tree) dscussed n ths work s of great sgnfcance to construct an effcent feature selecton algorthm. Ths secton ams to structure a UB-tree usng RST and explore some mportant propertes of t. To ths end, some related concepts are ntroduced. 1. Related concepts Defnton 1 Let S =(U, D), P, andx π P. Then X s sad an Inconsstent tolerance block (IT-block) f λ(x) > 1 or a onsstent tolerance block (T-block), where λ(x) ={f(u, d) u X} and λ(x) s the cardnalty of λ(x). The famly of IT-blocks from π P, denoted by πp nc, s called the nconsstent famly, whle the famly of T-blocks from π P, denoted by πp con, s called the consstent famly. Obvously, πp nc = {X π P λ(x) > 1}, πp con = {X π P λ(x) =1}, πp nc πp con = π P,andπP nc πp con = { }. Defnton 2 Let S =(U, D), P, a P, and X π P. Then the famly of sub-blocks determned by a on X s defned as θ(x, a) ={{u X f(u, a) =b f(u, a) = } b V a}. The nconsstent and consstent sub-famles of θ(x, a) are defned as θ nc (X, a) ={Y θ(x, a) λ(y ) > 1} and θ con (X, a) ={Y θ(x, a) λ(y ) =1}, respectvely. Theorem 1 Let S =(U, D), P, a P,and X πp con.thenanyy θ(x, a) s a T-block. Proof Y θ(x, a) gves Y X. Snce X πp con, λ(x) = 1, whch yelds λ(y ) = 1. Ths means Y s a T-block. The theorem holds. Defnton 3 Let S =(U, D), P, a P,and W a famly of tolerance blocks. Then the famly of sub-blocks determned by a on W s defned as ω(w, a) ={θ(x, a) X W }. The nconsstent and consstent sub-famles of ω(w, a) are defned as ω nc (W, a) ={Y ω(w, a) λ(y ) > 1} and ω con (W, a) ={Y ω(w, a) λ(y ) =1}, respectvely. Theorem 2 Let S =(U, D), P, a P, and W a famly of T-blocks. Then any Y ω(w, a) sa T-block. Proof For any X W, X s a T-block. By Theorem 1, any Z θ(x, a) s also a T-block. ω(w, a) ={θ(x, a) X W }, whch mples that any Y ω(w, a) s a T-block. The theorem holds. Theorem 3 Let S =(U, D), P, a P. Then P {a} = ω nc (πp nc,a). Proof Assume P = {a 1,a 2,,a t}. By Defnton 2 and 3, we have π P = ω( (ω(θ(u, a 1),a 2), ),a t) and

3 476 hnese Journal of Electroncs 2014 π P {a} = ω(ω( (ω(θ(u, a 1),a 2), ),a t),a). Then π P {a} = ω(π P,a)=ω(πP con πp nc,a)=ω(πp con,a) ω(πp nc,a). P {a} ={X ω(πp con,a) ω(πp nc,a) λ(x) > 1} ={X ω(πp con,a) λ(x) > 1} {X ω(πp nc,a) λ(x) > 1} =ω nc (πp con,a) ω nc (πp nc,a) =ω nc (πp nc,a) Thus, P {a} = ωnc (πp nc,a). The theorem holds. 2. UB-tree A UB-tree s a bnary tree n whch the rght (or left) node keeps growng whle the left (or rght) node has exactly no chldren. In ths study, Our objectve s to use RST to buld a UB-tree wth flourshng nconsstent nodes played by nconsstent famles for an ncomplete decson system. Let S =(U, P ), = {c 1,c 2,,c n}, = a 1,a 2,, a n, and = a 1,a 2,,a, where s an arbtrary permutaton of. WesaythatUBT( )saub-treeofs wth respect to, whch s constructed accordng to the followng process: (1) The root of UBT( ) s the famly that contans only the unverse. Snce λ(u) > 1, U s an IT-block. Ths means that the root s an nconsstent node. (2) Let π con and be the consstent and nconsstent chld nodes of the root, respectvely. Then π con 1 = ω con ({U},a 1)and = ω nc ({U},a 1). From Theorem 2, we know that any sub-block derved from π con s also a Tblock. In other words, nconsstent tolerance sub-blocks are dervable only from. So πnc s consdered as the father node for dervaton whle π con s regarded as the leaf node, called a consstent leaf node. (3) Let π con 2 and 2 be the two chldren of the node 1. Accordng to Theorem 3, πcon 2 = ω con ( 1,a2) and 2 = ωnc (,a2). Smlarly, only πnc 2 s selected as the father node for contnuously growng. (4) Lkewse, we can recursvely arrve at π con n and πnc n, representng the two sub-nodes of n 1 such that πcon n = ω con ( n 1,an) andπnc n = ωnc ( n 1,an). Snce πnc n has no chldren, t s called an nconsstent leaf node. Fg.1 vsualzes constructon of UBT( ). Fg. 1. UBT( ) 3. Propertes of UB-tree Theorem 4 Let S =(U, D), = {c 1,c 2,,c n}, and = a 1,a 2,,a n, where s an arbtrary permutaton of. Then each nconsstent node of UBT( ) determnes the boundary regon wth respect to such that BND (π D)= {X },where =< a 1,a 2,,a >. Proof For any u BND (π D), there doesn t exst any decson class D 0(D 0 π D) such that S P (u) D 0. Snce S P (u) = S X π P (u) X, there must exst at least one IT-block X contanng u. Ths means X. In other words, BND (π D) {X }. On the other hand, for any u X (X ), SP (u) = S X π P (u) X. Snce λ(x) > 1, λ(s P (u)) > 1, whch means there doesn t exst any decson class D 0(D 0 π D) such that S P (u) D 0. So u/ POS (π D), mplyng u BND (π D). In other words, {X } BND (πd). Accordng to dscusson stated above, BND (π D)= {X }. The theorem holds. Theorem 4 unravels the relatonshp between the nconsstent node and the boundary regon, whch ndcates that the boundary regon s n fact a unon of IT-blocks from the nconsstent node. Theorem 5 Let S =(U, D), = {c 1,c 2,,c n}, and = a 1,a 2,,a n, where s an arbtrary permutaton of. If j, BND (π D) BND j (π D), where = a 1,a 2,,a, j = a 1,a 2,,a j. Proof Assume k = j, then = j + c j+1, c j+2,,c j+k. Then = ω nc ( (ω nc (ω nc ( j,cj+1), c j+2, ),c j+k ). For any Y ω nc ( j,c j+1), there must exst Z j such that Y Z. Then {Y ω nc ( j,cj+1)} {Z j }. Accordngly, we have {X πnc } {X j }, whch yelds BND (πd) BND j (πd). The theorem holds. The theorem mples that the deeper an nconsstent node s, the narrower the correspondng boundary regon s. Snce each nconsstent node corresponds to one boundary regon, the UB-tree, n fact, offers the sequence of gradually reduced boundary regons, whch portrays n detal the evoluton of boundary regon becomng narrower and narrower untl reachng that of the system. IV. Feature Selecton Based on UB-tree The UB-tree based feature selecton algorthm (UBT- FS) s structured on the UB-tree, marryng boundary regon based sgnfcance and boundary regon based evaluaton crteron. The boundary regon based sgnfcance s defned as Sg(a, P, D) =( BND P (π D) BND P {a} (π D) )/ U, whch expresses how the boundary regon wll be affected f the attrbute a s added nto the set P. It acts as a router to determne the optmal search path. The boundary regon based evaluaton crteron determnes whether an attrbute subset P s a reduct or not by two condtons: (1) BND P (π D)= BND (π D), and (2) BND P (π D) BND (π D) for any P P. It works as the rule of dentfyng feature subsets. Table 1 gves the detaled descrpton of UBT-FS algorthm. UBT-FS starts wth the root and keeps selectng optmal attrbutes to splt the nconsstent node repeatedly. An optmal attrbute s acheved by ths way. Each of unselected

4 onstructng Rough Set Based Unbalanced Bnary Tree for Feature Selecton 477 attrbutes s used to work on the current nconsstent node and then the new nconsstent node s generated, from whch the correspondng boundary regon s derved. By evaluatng the boundary regon based sgnfcance, the attrbute wth the bggest sgnfcance value s selected as the expected one. Termnaton of the loop occurs when the newly generated boundary regon s dentcal wth that of the system. At ths tme, the set of selected attrbutes s a wanted reduct. Table 1. UBT-FS algorthm Input: S =(U, D), where = {c 1,c 2,,c n} and D = {d} Output: a reduct R begn: Let =, = {U}, BND (π D )=U, andp = ompute BND (π D ) Repeat untl BND (π D )=BND (π D )orp = Select the optmal attrbute c opt by the functon SelectOptAttr(π nc,bnd (π D ),P) Use c opt to derve the nconsstent node π nc from +c opt such that = ω nc ( +c opt,c opt) Induce the boundary regon from π nc +c opt such that BND +c opt (π D )= {X Let = + c opt, +c opt } = +c opt, BND (π D )=BND +c opt (π D ), and P = P {c opt} Let R = end Functon SelectOptAttr(,BND (π D ),P) begn: For each attrbute c n P ompute π nc +c = ωnc (π nc,c) ompute BND +c(π D )= {X π nc +c } ompute Sg(c,,D)=( BND (π D ) BND +c(π D ) )/ U Return c opt wth Sg(c opt,,d)=max{sg(c,,d) c P } end There are several hghlghts decoratng UBT-FS. Frst, each nconsstent node, although determned by a certan attrbute set, s dervable only from ts predecessor wth a sngle attrbute, whch greatly mproves the computatonal effcency. Second, every nconsstent node corresponds to a certan boundary regon, makng t easy to create the boundary regon. Thrd, obtanng a seres of boundary regons needs to traverse the entre condton attrbute set just only once. Fnally, the boundary regon s useful to set up the measure of attrbute sgnfcance and the crteron of dentfyng reducts. These advantages allow UBT-FS to acheve tme complexty much less than O( 2 U ). ompared wth exstng feature selecton algorthms for ncomplete decson systems wth tme complextes no less than O( 2 U log U ) [12],UBT-FS acheves an obvously lower tme complexty. V. Expermental Evaluaton In ths secton, we carry out several experments on a personal computer wth Wndows XP, 2.53GHZ PU, and 2.0G memory so as to evaluate the effectveness of UBT-FS. UBT-FS s compared wth three knds of typcal algorthms whch conduct feature selecton by employng dscernblty matrx [4], postve regon [12], and combnaton entropy [15].For convenence, they can be called Dscernblty matrx based feature selecton algorthm (DM-FS), Postve regon based feature selecton algorthm (PR-FS), and ombnaton entropy based feature selecton algorthm (E-FS), respectvely. DM- FS consders dscernblty of tolerance relaton and uses dscernable attrbutes obtaned by parwse comparsons of objects to buld the dscernblty matrx. It has tme complexty no less than O( 2 U 2 ). PR-FS focuses on ndscernblty of tolerance relaton and captures ndscernble objects as tolerance classes to construct postve regon for a feature subset. Its tme complexty s O( 2 U log U ). E-FS ntroduces combnaton entropy as the measure of attrbute sgnfcance. It uses the core attrbutes as the ntal reduct and keeps addng one attrbute of hgh sgnfcance to the set each tme untl the condtonal combnaton entropy of the current set s equal to that of the whole attrbute set. The tme complexty of E-FS s O( 2 U 2 ). The four algorthms (UBT- FS, DM-FS, PR-FS, and E-FS) are appled to eght publcly accessble datasets from UI Repostory of machne learnng databases [18]. The expermental results, ncludng the sze of the selected subset and runnng tme expressed n seconds, are presented n Table 2. Table 2 shows that the sze of selected features by DM-FS s the smallest of four algorthms. Ths ndcates that DM- FS can undoubtedly fnd the mnmal reduct, whle the other three algorthms can t guarantee selecton of the optmal subset. One can also observe that the results of UBT-FS, PR-FS, and E-FS are almost the same and very close to that of DM- FS, whch mples that UBT-FS, PR-FS, and E-FS have the smlar ablty to seek out the optmal or suboptmal feature subsets, though they can t guarantee the best performance every tme. On the other hand, from Table 2, we can observe that when dataset scale s relatvely small, UBT-FS takes tme as much as PR-FS, whle DM-FS and E-FS are more tme consumng. As the ncrease of the dataset scale, DM-FS and E-FS consume too long tme to be acceptable, PR-FS also costs much Table 2. Expermental results by four algorthms DataSet Objects Features DM-FS E-FS PR-FS UBT-FS Sze Run tme Sze Run tme Sze Run tme Sze Run tme Lung cancer Standardzed audology ongressonal votng Balance scale weght Tc-Tac-Toe endgame ar evaluaton hess end-game Nursery

5 478 hnese Journal of Electroncs 2014 tme, and only UBT-FS keeps a relatvely low tme consumpton. Evdently, the runnng tmes of DM-FS, E-FS, and PR-FS ncrease much more rapdly than that of UBT-FS. The dfferences can be llustrated by plottng the ratos of DM- FS, E-FS, and PR-FS to UBT-FS, respectvely, as shown n Fg.2. work on ten new datasets derved from the expanded Standardzed Audology dataset. As the Standardzed Audology dataset contans only 200 objects, randomly generated objects are appended to t untl ts sze s up to Attrbutes of the new dataset are segmented nto ten parts of equal sze. We thnk of the frst part as the frst dataset, treat the unon of the frst dataset and the second part as the second dataset, take the unon of the second dataset and the thrd part as the thrd dataset, and so on. These ten datasets have the same number of objects but dfferent attrbute number. Fg.4 formulates the correspondng expermental results. Fg. 2. Ratos of DM-FS, E-FS, and PR-FS to UBT-FS Fg.2 llustrates that the larger the dataset scale s, the greater the rato s. For example, the rato of PR-FS to UBT- FS on Dataset 3 s almost equal to 1, whle that on Dataset 8 reaches up to 62. Ths means that PR-FS and UBT-FS runnng on the ongressonal Votng dataset almost share the same tme consumpton, whereas the tme consumpton of PR- FS s 62-fold as much as that of UBT-FS when they work on the Nursery dataset. Not to menton DM-FS and E-FS. One can also observe that although the slope of each curve tends to ncrease wth dataset sze, the curves fluctuate sgnfcantly. For example, the rato of DM-FS to UBT-FS on Dataset 2 s hgher than that on Dataset 3. The case arses from dfferent attrbute numbers of datasets (e.g., the Standardzed Audology dataset has 69 attrbutes but the ongressonal Votng dataset has 16). In fact, tme consumpton of a feature selecton algorthm s determned by both the object number and the attrbute number of a dataset. To further demonstrate the nfluence of each ndvdual factor on runnng tme, we desgn the next two experments. To llustrate the nfluence of the object number on runnng tme, we run the four algorthms on ten sub-datasets of the Nursery dataset, whch are generated accordng to the followng rules. Objects of the Nursery dataset are dvded nto ten groups of equal sze. The frst group s consdered as the frst dataset, the combnaton of the frst dataset and the second group s regarded as the second dataset, the combnaton of the second dataset and the thrd group s referred to as the thrd dataset, and so on. Therefore, these ten sub-datasets of the Nursery dataset have the same attrbutes but dfferent number of objects. The correspondng expermental results are represented n Fg.3. Fg.3 shows that the curves correspondng to DM-FS,E- FS, and PR-FS are approxmately quadratc, whle that correspondng to UBT-FS s almost lnear. Ths experment verfes that tme consumpton of DM-FS, E-FS, and PR-FS ncreases much more rapdly than that of UBT-FS when the attrbute number s constant. The other experment ams to demonstrate the nfluence of the attrbute number on runnng tme. The four algorthms Fg. 3. Runnng tme on ten sub-datasets of Nursery dataset Fg. 4. Runnng tme on ten new datasets Fg.4 shows that the curves correspondng to DM-FS and E-FS are approxmately quadratc, whle those correspondng to PR-FS and UBT-FS are almost lnear but the slope of the curve correspondng to PR-FS s steeper than that correspondng to UBT-FS. Ths means tme consumpton of DM- FS, E-FS, and PR-FS ncreases much more rapdly than that of UBT-FS, whch verfes that UBT-FS provdes the least runnng tme when the object number s constant. VI. onclusons Incomplete data are common n real world applcatons. Many feature selecton algorthms avalable for ncomplete decson systems have been proposed. Snce ther tme complextes are not less than O( 2 U log U ), they are ntolerable when dealng wth volumnous data. Ths paper has developed UBT-FS for ncomplete decson systems. Its tme complexty s not more than O( 2 U ). Unlke other algorthms based on the lower approxmaton, UBT-FS adopts the UB-tree technque, whch specalzes n acceleratng calculaton of the boundary regon. Ths mechansm, along wth the use of the boundary regon based sg-

6 onstructng Rough Set Based Unbalanced Bnary Tree for Feature Selecton 479 nfcance to determne the optmal search path as well as the boundary regon based evaluaton crteron to dentfy feature subsets, makes UBT-FS capable of fndng a reduct excellently. Both theoretcal studes and expermental results demonstrate that UBT-FS ndeed outperforms other avalable algorthms wth regard to tme effcency. References [1] J. Zhong, et al., A novel feature selecton method based on probablty latent semantc analyss for hnese text classfcaton, hnese Journal of Electroncs, Vol.20, No.2, pp , [2] B. Leng, et al., MATE: A vsual based 3D shape descrptor, hnese Journal of Electroncs, Vol.18, No , [3] Z. Pawlak, Rough sets, Internatonal Journal of omputer and Informaton Scences, Vol.11, No.5, pp , [4] M. Kryszkewcz, Rough set approach to ncomplete nformaton systems, Informaton Scences, Vol.112, No.1-4, pp.39 49, [5] Y. Leung, D.Y. L, Maxmal consstent block technque for rule acquston n ncomplete nformaton systems, Informaton Scences, Vol.153, No.1, pp , [6] Y.M. hen, et al., A rough set approach to feature selecton based on power set tree, Knowledge-Based Systems, Vol.24, No.2, pp , [7] M.E. ElAlam, A flter model for feature subset selecton based on genetc algorthm, Knowledge-Based Systems, Vol.22, No.5, pp , [8] M. Dash, H.A. Lu, onsstency-based search n feature selecton, Artfcal Intellgence, Vol.151, No.1-2, pp , [9] Y. Qan, et al., Postve approxmaton: An accelerator for attrbute reducton n rough set theory, Artfcal Intellgence, Vol.174, No.9-10, pp , [10] Y.H. Qan, et al., An effcent accelerator for attrbute reducton from ncomplete data n rough set framework, Pattern Recognton, Vol.44, No.8, pp , [11] A. houchoulas, Q. Shen, Rough set-aded keyword reducton for text categorzaton, Appled Artfcal Intellgence, Vol.15, No.9, pp , [12] Z.Q. Meng, Z.Z. Sh, A fast approach to attrbute reducton n ncomplete decson systems wth tolerance relaton-based rough sets, Informaton Scences, Vol.179, No.16, pp , [13] N.M. Parthalán, et al., A dstance measure approach to explorng the rough set boundary regon for attrbute reducton, IEEE Transactons on Knowledge and Data Engneerng, Vol.22, No.3, pp , [14] J. Qan, et al., Hybrd approaches to attrbute reducton based on ndscernblty and dscernblty relaton, Internatonal Journal of Approxmate Reasonng, Vol.52, No.2, pp , [15] Y.H. Qan, J.Y. Lang, ombnaton entropy and combnaton granulaton n rough set theory, Internatonal Journal of Uncertanty, Fuzzness and Knowledge-Based Systems, Vol.16, No.2, pp , [16] L. Sun, et al., Feature selecton usng rough entropy-based uncertanty measures n ncomplete decson systems, Knowledge- Based Systems, Vol.36, No.1, pp , [17] J. Zhou, et al., Analyss of alternatve objectve functons for attrbute reducton n complete decson tables, Soft omputng, Vol.15, No.8, pp , [18] A. Asuncon, D.J. Newman, UI Machne Learnng Repostory, mlearn/ MLRepostory.html, LU Zhengca receved B.S. and M.S. degrees from the Academy of Equpment ommand and Technology, hna, n 1997 and 2000, respectvely. He s currently workng toward the Ph.D. degree n Department of omputer Scence and Technology, Tsnghua Unversty, hna. Hs research nterests nclude machne learnng and data mnng. (Emal: luzc09@mals.tsnghua.edu.cn) QIN Zheng was born n Hunan Provnce, hna, n He s a professor n the School of Software, Tsnghua Unversty, hna. Hs major research nterest ncludes software archtecture, data fuson and artfcal ntellgence. (Emal: qngzh@mal.tsnghua.edu.cn) JIN Qao receved B.S. degree from School of Software, Tsnghua Unversty, hna, n He s currently a M.S. canddate at Department of omputer Scence and Technology, Tsnghua Unversty, Bejng, hna. Hs research nterests nclude data mnng and other artfcal ntellgence based technques. (Emal: jnq07@mals.tsnghua.edu.cn) LI Shengnan receved B.S. degree n Software engneerng from School of Software, NanKa Unversty, hna, n She s currently a Ph.D. canddate at Department of omputer Scence and Technology, Tsnghua Unversty, Bejng, hna. Her research nterests nclude machne learnng and data fuson. (Emal: xaolng4329@gmal.com)