Quach Xuan Trudng vd Dtg T?p chi KHOA HOC & CONG NGHE 135(05): 155-160
MQT S 6 D p ©O SV* PHV THUOC TRONG RUT GON THUOC TINH TREN Ti^P THO M d
Qudch Xuan Truong'*, \A Vdn P h u n g ' 'Truomg Dgi hge Cong ngh^ thong tin & TruySn thdng ~ DH Thdi Nguyen.
^Vi4n Cong ngh? thdng tin
TOM TAT
Rut gpn thudc u'nh Id mdt trong nhiing nfli dung quan trpng nhdt ciia ly thuySt tdp tho uTig dyng trong ITnh virc khai phd dtt lidu. Bdi bdo trinh bdy tong quan vS ly thuydt tap thd vd mpt hudng md rong ciia nd Id tap thd md, trong do tap trung nghien ciiu cac ttnh chat va k5' thuat nit gon thu^c tinh trong tap tho md. Phan dau bai bao chiing toi gidi thieu tdng quan vd tdp tho va tho md, phdn thd hai bdi bao theo trinh bdy mpt thudt toan nit gpn thudc tinh da dupc dd xudt dyra trgn dp do su phu thu^c cua cdc tdp thudc tinh trong tap tho md. Tir thuat toan n6i tren, phdn thii ba cOa bdi bdo chiing tdi de xudt mdt cdch tinh dd quan trpng cua thupc tinh dua tren su tinh todn dp phy thupc cua cdc tap thupc tinh len moi mpt thupc tinh, ky thuat ndy nay hieu qua hon trong mOt so trudng hpp ciia bp di^ li^u md nhSn tdp quyet dinh c6 the khd xdc dinh hoac khong day dii.
TIT kh6a: tgp tho; tgp md; igp tho md; rut ggn thugc tinh, dp phu thupc GlOlTHieU
Ly thuyet tap thd do Z.PawIak [1] dd xudt dugc phat tridn tren ndn tang ca sd toan hge vimg chde cung cdp cdng cy hiru ich dd giai quyet cac bai toan phan ldp, phat hien luat,..
chda dii li?u ma ho, khdng chde chdn [10][11][12]. Mgt trong nhiing dng dung quan trgng ciia ly thuyet tap thd la nit ggn thugc tinh, muc tieu cua nit ggn thugc tinh la loai bd cac thugc tinh du thura de tim ra cac thugc tinh cdt ydu va can thidt trong ca sd dir li?u ma vdn bao toan dugc thdng tin nhu ban dau.
Tuy nhien, trong thirc td gia tri cua cac thudc tinh cd the la gia trj lien tuc ho§c ngii nghia va iy thuyet tap thd tmyen thong gap khd khdn trong viec bidu dien vd xii ly cdc d^ng dii lieu nay [9], Mgt trong nhiing each giai quyet la rdi rac hda cac gia trj ciia thugc tinh de tao ra bg gia tri mdi vdi nh&ng gia trj tugng trung. Tuy nhidn, each giai quyet nay gdp phai han che Id mdt thdng tin trong qua trinh rdi rac hda. Mgt phuang phap khac la su dyng t^p tho md de bieu dien tinh md va tinh khdng phan biet nhu mgt diuig dii lieu khdng chde chdn trong ly thuyet tap tho. Trong tap thd md mgt quan hd tuong tu md dugc sir dung dd bidu didn muc dg tugng tu nhau giira
' Tel- 0989 090832: Email: qxtruong'aictu.edu.v,
hai ddi tugng thay cho quan he tuong duang trong tap thd truyen thdng. Do tuang tu cua hai doi tugng cd gia tri trong khoang [0,1].
Ndu do tuong tu la 1 thi chung khdng the phan biet, va chung cd the phan biet dugc neu cd dg tuong tu bdng 0. Ndu dg tucmg tu cd gia tri trong khoang [0,1] thi hai ddi tugng cd su tuang tu d miic do nhat dinh. Tuy nhien, vdi quan he tucmg tu khac nhau cd the tao ra do tuong tu khac nhau vi thd vide xdc djnh mgt quan he tuang tu md hgp ly la can thiet de dam bao tinh todn ven thdng tin (Dya tren do tuong tu chap nhan dugc).
Hien nay, cdc nghien ciru vd tap thd md tap trung chu yeu vdo xap xi cua tdp md [2][3][4][5][6][13], chua cd nhidu cdc nghien cuu dd xuat cac ky thudt nit ggn thugc tinh cho tap thd md, mgt trong nhung vdn de quan trgng trong khai pha dii lidu. Phan tiep theo cua bai bdo se trinh bay cac khai niem ca ban cua tap thd, tap thd md, cac vdn dd ca ban trong nit gpn thugc tinh dua tren do phu thugc cua tap cac thugc tinh va gidi thieu mgt thuat toan nit ggn thugc tinh su dung dp do s\r phu thugc da dugc dh xudt trudc day vd dugc xem nhu la mdt trong ky thuat dau tien trong nit ggn thudc tinh tap thd md [9]. Dua tren cac tinh chdt co ban ciia tap thd md va thuat toan da trinh bay, bai bao de xuat them mgt md 155
Qudch Xudn Trudng vd Dtg T^p chi KHOA HQC & CONG NGHE 135(05): 155-160 rgng ciia mdt thuat toan trong tap thd truyen
thdng dua tren dp phy thugc cua cac tap con cac thugc tfnh vdo mdi mgt thugc tinh de ap dung cho bg dii lipu ma trong dd cd the khdng xac djnh dugc nhan ldp quyet djnh hodc ldp quyet dinh khdng ddy du.
KHAI NIEM CO B A N TAP THO Cho h? thdng tin I = (U, A), trong dd U la tap hiiu ban khac rdng cac ddi tugng; A la tap hihi han khac rdng cac thupc tinh va a:
U—•Va Va e ^ , Va la tip cac gia tri ciia thugc tinh a E A. Mgt ldp ddc bidt cua he thdng tin cd vai trd quan trgng trong nhidu ung dyng la bang quyet dinh. Bang quyet dinh la mgt h$
thdng tin trong dd A dugc chia thanh hai t^p khac rdng rdi nhau C vd D, lan lugt ggi la tap thupc tinh dieu ki?n vd tap thupc tinh quyet dmh. Vdi bdt kyP^ A xdc dinh mdt quan he tucmg duang IND(P):
INPiP) = [(r.y) E U'\ Va £ P . o ^ ) = a( y)} ( ] ) Mpt phan hoach cua U dugc sinh bdi IND(P) gpi la U/P va dugc tinh nhu sau:
U/P =®{a eP:U flNDiia}) (2) Trong dd
A (s> B = [Xfw-.vx E A.VY e B.xnv ^ e)
rND(P) la quan h? P-khdng phan biet dugc, ndu (x. y) e IND(P} khdng phan biet dugc bdi cac thugc tinh trong P. Ldp tucmg duong trong phan hoach P/U dugc ky hieu Id [x]p, khi dd:[x\p = {yEU\ix,y)elNDiP)] (3) Cho X EU, P-xdp xi dudi va P-xSp xi tren cua
tdp X dugc xac djnh nhu sau:
PX = {x\[x]^£xlPX=ixmpnx = i>} ^^j
Cho P.Q £.4, midn ducmg, midn phii dinh, mien bien cua Q Id t^p xac dinh sau:POSp(.Q) = \J PX
NEGp(Q) = U-Uxsu/pPX ^^^
BNDpiQ) = Uxeu/pPX - Ujieu/pPX ^^^
Mpt npi dung quan trpng trong viec phan tich dii lieu la phat hien ra sir phy thupc giiia cac 156
thudc tinh. Tap cac thudc tinh Q phy thupc hoan toan vao tap thupc tinh P (ki hieu la P =^ Q) neu tat ca cac gia tri cua thupc tinh Q dugc xac dinh duy nhdt bdi cac gia trj ciia thugc tinh P. Sy phy thudc cd thd dugc dinh nghTa nhu sau:
Cho P.Q^A,Q phu thugc vao P vdi dd phy thudc k (0 < Jc < 1), ky higu \aP=>-^Q , neu:
\POSpi.Q)\
1^1 (8) k = y^iQ) --
Neu k=l Q phy thugc hoan toan vao P, ndu 0<k<l Q phu thudc mgt ph^n vao vao P vd neu k=0 Q khdng phy thugc vao P.
Bang vi^c tinh toan su thay ddi ciia su phy thupc khi mgt thugc tinh bj logi bd khdi t^p cac thupc tinh dieu kien, mpt muc do cua tdm quan trgng cua thugc tinh dugc xem xet, ndu mirc dp thay ddi ciia sy phu thugc cang ldn thi tam quan trgng ciia thugc tinh cang ldn. ndu miic do tam quan trgng bdng 0 thi thudc tinh do la khdng can thiet.
Cho P, Q vd thugc tinh x G f .Ta cd thd djnh nghTa mirc do tdm quan trpng ciia thugc tinh x vao Q nhu sau:
opiQ.x) = rp(Q^ - yp-t*-/?) (9) Riit gpn thugc tinh dugc thyc hi?n bdng each so sanh quan he tuang duang dugc sinh ra bdi tap cac thugc tinh. Cac thugc tinh bi loai bo sao cho tap riit ggn van cung cdp ket qua phan ldp nhu tap thugc tinh ban ddu. Nhu v|.y, tap nit ggn cd the dugc djnh nghTa Id mot tap con R cac thupc tinh trong t^p thugc tinh didu kidn C sao cho Ys Co) = YcO^), va mdt bp dii lieu co the cd nhieu tap thupc tinh nit ggn. Tap tat ca cac nit ggn dugc dinh nghTa nhu sau:
Rsd(d = &lsq7Bii)i = >cto>vB = R.yaii)s:ic(D;)(lO) Va tip cac nit ggn tdi uu la:
/?pri(r)m... =-f»FPffrf)Vfl' Fffprf,|ff| < Ifl'l} / ] ] \ TAP THO MCJ
Trong ly thuydt tap thd truyen thong [7] quan hd tuang duong la khai nigm nguyen thiiy va quan trgng phan hoach nen cac ldp tucmg duong. Trong tap thd md, quan hg tuong ty md
Qudch Xudn Tnrdng vd Dig Tgp chi KHOA HQC & CONG NGHE 135(05): 155-160 dugc sir dung de thay the quan he tuang duong
de phan hogch cdc ldp tuang duong md.
Cho U la tap vu tru khac rdng, mdt quan he nhi phan S tren U dugc gpi la quan he tuang tu md neu S thda man tinh chat:
Phanxa: S(x,x)=l;
Ddi ximg: S(x,y)=S(y,x);
Bac cau sup-min:
Six.y) > sirpagt,mm{5(r,z),5(r,y)}.
Theo mpt sd cdng trinh nghien ciru [2][3][4][5][6][13][14] da cd, P-xdp xi dudi yd P-xdp xi tren tap thd md dugc djnh nghTa nhu sau:
!>fjU) = «ff jK^/OinCif-W.'"/, ef'"3x(l -fJ^ GO-.Uj (>•))) (12)
f^/j) = sapfj.^p!nin(f(j.()r),5iipj^j,miii[p^>j),Uj{y)|) Trong do F la tap cdc ldp tuang duang md thugc U/P.
Trong qua trinh duyet cac ddi tugng, khdng phdi tat ca cdc ddi tugng )• e [/ deu can dugc xem xet den, chi nhiing ddi tugng ma ."f 0') "* ^i nghTa la ddi tugng y la mpt thanh vien md ciia ldp tuang duong md F. Mdt bd
< PX.PX > dugc ggi la tgp thd md. Nhu vay, CO the th^y rang tap thd md cd thd suy bien thanh tap thd truyen thdng neu tdt ca cac ldp tuang duang la tap ro. Khi dd xap xi dudi dugc ddc tnmg bdi ham thanh vien sau:
^ p / W - [ o fchic (13) Nhu vay, ta khdng djnh rdng mdt doi tugng x thudc vao P-xap xi dudi cua tap X ndu nd thudc vdo mgt Idp tuang duang vd id tgp con ciia tap X. Vd hien nhien la cac tinh toan trdn xap xi thap md phdi chinh xdc tren cac tinh toan cua tdp rd. Do dd, ta cd thd vidt lai xap xi dudi md nhu sau:
jifjix) = stip,g[;.^min(^.Ci).ifif[u.GO ^ %0'))) ^j^j Trong dd "—*" Id todn tii keo theo md. Trong trudng hgp tap ro, u^ix) va ^j.(r)se nhgn gid tri tCr {0,1}. Nhu vay,_u„Cr) se nhgn gia trj khdng chi khi cd it nhdt mpt doi tugng trong idp tuong duong F thugc hoan toan vao tgp F
nhung khdng thugc tap X, ket qua nay hoan toan gidng nhu trong dinh nghTa xdp xi dudi trong tap ro.
Riit ggn thugc tinh trong tap thd md xay dyng dua tren xdp xi dudi md cho phep nit ggn tren cac bd dii lieu chua thudc tinh cd gid trj thyc hodc lien tyc, qua trinh nay toan toan gidng vdi each tidp can tren tap rd. Trong ly thuydt tap thd truydn thdng, midn duong dugc dinh nghTa la hgp ctia cac xdp xi dudi, md rdng tu djnh nghTa trong tap ro [8], muc thanh vien ciia mpt ddi tugng x s U thugc vdo mien duong md dugc dinh nghTa bang:
.^?CS^..,Cx) = SUpj.^y,,^MpA-W (15) Doi tugng x se khdng thugc vao mien duang neu ldp tuong duong ma nd thudc vdo khdng la mgt phan ciia midn duong. Vdi midn ducmg md, ham thugc md dugc djnh nghTa nhu sau:
^P^^^- \U\ ~ |t/| ( , 6 j Be qua trinh nit ggn thudc tinh ciia tap thd md cd hieu qua, can phdi thuc hidn vdi nhieu thugc tinh de tim do phu thugc giira cdc tap con khac nhau trong tap thudc tinh ban dau.
Vi dy cho P=^{a,b}, ta can phai xdc dinh miic dg phu thudc ciia cdc thudc tinh quyet djnh ddi vdi P={a,b}. Trong trudng hgp tap rd, U/P chiia tap cac ddi tugng khdng phan biet theo ca ddi tugng a va ddi tugng b. Nhung trong trudng hgp md, cac doi tugng cd the thudc vao mgt sd Idp tuong duong, vi vay cdn sii dung tich Descartes ciia U/lND({a}) va U/IND({b}) trong viec xdc djnh U/P. Ta cd tdng qudt:
U/P =® {a E P'.U/lNDCia))) (17) trong dd
A®B=[Xr\YzVX EA,\fYeB.XnY:!i: Q}.
Mdi tap trong U/P la mdt ldp tuong duong.
Vi du vdi P={a,b}, U/IND({a})={Na, Z^} vd U/IND({b})={Nb,Zb}thi
U/P = IN a n A'j,. w^ fl z^, Za n jv^, z^ n Zj,}
Ham thudc r ' dugc su dyng de tinh todn tap nit ggn R trong tap day dii C, ndu y'Cfl) = /(Hu{o}) vdi vaeC-R thi R la tgp nit ggn ciia C. Vdi tu tudng nay cd mgt sd 157
Quach Xuan Trudng vd Dig Tap chi KHOA HQC & CONG NGHE 135(05): 155-160
thuat toan d u g c phat tridn dd tim tap thudc tinh nit ggn.
Thugt todn FQuickReduct
Trong cac nghien c u u ve tap thd m d hidn nay, thuat todn F Q u i c k R e d u c t [9] la m g t t r o n g nhirng dd xudt tidn p h o n g giai q u y e t bai todn tim t a p t h u d c tinh n i t g p n t r o n g tap t h d md. T h u a t toan F Q u i c k R e d u c t sir d u n g h a m thugc m d
^''^ m ~ \u\ (18)
dd lya chgn thudc tinh them vao tap thudc ti'nh ling vien, thuat toan d u n g Igi khi bd^ sung thudc tfnh cdn lai ma khdng Iam tai^g gia tri
ham thugc. ' Thugt toan:
FQuickReduct (C,D)
//C Id tap thugc tinh diiu kien; \ //D la tgp thugc tinh quyet dinh;
il)R *- Q; r'-o,,r = Oi r'orsi,- = 0:
(2) do ( 3 ) r - K (4)/pr«.- ^y'-cf^t (5)VQ G(C-ff) ( 6 ) i / y ' s y [ s ; ( D ) > y V ( ^ ) i7)l ^RK) [a]
( 8 ) y W *-rVC^) (9)R - T ilQ)iBitHY'ee!T = r ' p m '
(11) return R
Oe xac dinh d o d o phy thugc ciia tap thupc tinh quydt djnh D vao tap thupc tinh dieu kien, thuat toan sir d y n g ham thupc m d y' (trong c d n g t h u c cua y'pCQ) vdi mau sd |U| thi Y' tidn tdi 1) de t h u c hien vide lya chgn thugc tinh them vdo tap iing vien. T u t u d n g cua thuat todn FQuickReduct hoan toan t u o n g t u nhu thuat toan Q u i c k R e d u c t trong tap thd truydn thong [9]. Thuat toan ket thiic khi khdng cdn bat ky thugc tinh ndo ma lam tdng miic d o phy thudc t h d n g qua gid tri ciia ham thugc md.
Phdf triin thugt todn EFQuickReduct Trong t h u c te cac iing d u n g khai pha dii lieu, bp dii lipu cd the khdng xac dinh d u g c thugc
158
tinh quyet dinh hoac gid tri c u a tgp thudc tinh nay khdng ddy du. V d i nhiing t r u d n g hgp nay, dp d y n g thuat todn d u g c trinh bay d phan trdn se k h d n g hieu q u a . T r o n g phan nay, c h u n g tdi phat trien mgt thugt toan m d rgng tu thuat todn n i t ggn t h u g c tinh da cd trong tgp thd truyen t h d n g d u a tren y t u d n g danh gid d$
do phu t h u d c ciia c a c tgp con thugc tinh vdo mdi thugc tinh t r o n g bd dir lieu.
T r o n g thugt todn F Q u i c k R e d u c t d u g c trinh bay d phan tren cd hai tham sd dau vao: tgp thudc tinh dieu kien va tgp thugc tinh quyet djnh. Vide d a n h gid d o do ciia gid tri ph|^
thupc d u a vdo tap t h u g c tinh quydt dinh. Doi vdi thuat toan E F Q u i c k R e d u c t chi cd duy nhat mgt tham so dau vdo la t|ip thudc tinh dieu kien. Tai day, chiing ta ti'nh todn gia trj ciia dp phy p h u g c cho cdc t a p con thudc tinh vdo mdi mdt thudc tinh didu kien va tinh gia trj trung binh ciia d g phu thupc cua tat ca thupc tinh didu kien. Thugt todn E F Q u i c k R e d u c t su d y n g c d n g t h u c sau:
dd tinh gia tri d o d o sy phu thugc cua tap thupc tinh vao mdi t h u g c ti'nh dieu kien, va sil d y n g c d n g t h u c tinh gia trj trung binh ciia cdc gia tri d o d o s y phu t h u d c :
^-— ^ c . ^ y ' p ( Q J
' ' ^ ^ ' ' ^ = U i (19)
d e lira c h o n thuoc tinh img vien tot nhat.
Thuat toan:
E F R Q u i c k R e d u c t ( C ) / / C la tap thuoc tinh dieu ki?n;
CDR ^ 0 C2)di) ( 3 ) 7 ^ K ( 4 ) v a = ( C - n ) C5)vli E C
WJ V SulBJ^'J [J]
(.g)? .-R u la}
(fli! - -
( l O ) i m t i l l ' j C a J . V a e t = y ^(xiWa e i.
( l l ) r e t u r n R
Quach Xudn Tnrdng vd Dig Tgp chi KHOA HQC & CONG NGHE 135(05); 155-160 Tuang tu nhu tu tudng ciia thugt todn
FQuickReduct, xult phat tu tgp rdng. Tgi moi thdi didm tap con cdc thugc tinh dugc xem xet trong dd tap con thugc tinh ma cd gia tri trung binh do do phy thugc tdt nhat se dugc bd sung vdo tap ung vien cho den khi do do phu thudc cua tgp ung vidn dat bdng gid trj trung binh do do phy thugc cua tap todn bg thupc tinh didu kien C.
Vid^ minh hga
De minh hga cho tu tudng ciia thugt todn dugc trinh bay d trdn, chung ta xem xdt mgt bg dii lieu vi dy nhd nhu sau:
BSng I. Bo da- lieu minh hpa object
1 2 3 4 5
a 0 0.2 0.6 0.3 0.2
b 0.1 0.1 0.7 0.4 0.7
c 0.1 0.6 0.3 0.8 0.9
d 0.5 0.9 0.9 0.6 0.2
Dau tien cac xap xi dudi cua mgt thudc tinh nhat djnh vdi mdi mdt thudc tinh khde nhau trong bd dii lieu sd dugc tinh toan. Cdc gia tri nay se dugc su dyng de ttnh toan dg phy thugc vd gid trj trung binh cua cac bg gia trj phy thudc ciia timg tap con thudc tinh theo cdng thirc (19) vd (20). Vi du tai thugc tinh b, ta xet su phy thudc ciia thudc tfnh b len thugc tmh a. Qud trinh nay lap Igi cho mgi ddi tugng de ti'nh toan xap xi dudi. Sau dd chung dugc sir dung dd tinh cac gid trj mien duong ciia b vao a:
fposia^wi^) = 0.5
/^TOv,oj{e)(2) = 0.5
^'•ov,„,(b)(3) = 0.67
"po^t^iwC*) = 0-67 ''/'o.Wi>](5) = 0-67
/ { G 5 W = \U\ (20)
Tii cac kdt qua tren theo (19) ta cd dp phy thugc ciia thugc tinh b vao thugc tinh a ia:
Tuong ty nhu tren, chiing ta tiep tyc tinh cdc do phy thudc ciia b len cac thudc tinh cdn lai dd tinh gia trj phu thugc trung binh Y'/W '^"^
b theo cdng thiic (20).
Qua trinh tren duac tiep tyc cho cac thudc tinh tidp theo cua bd dii lieu, sau mdi budc tap thugc tinh ciia gia tri phu thugc trung binh Idn nhat se dugc chgn bd sung vao tap iing vien. Thuan toan dung khi gid tri trung binh cua do phy thugc tap thudc tinh con theo (20) khdng Idn hon gia trj trung binh dg phy thudc cua cac thugc tinh vao tap toan bg thugc tinh dieu kien. Ket qua cudi cung Id tap con {b,d}
la tap nit ggn ciia bd dii lieu theo vi dy tren.
KET LUAN
Trong bdi bdo nay, chiing tdi chii yeu gidi thieu tdng quan ve mdt sd khai nidm, tich chat CO ban cua tap thd va tap thd md, su md rdng cua tap thd truyen thdng. Trong dd tap trung chii yeu phan tich va trinh bay chi tidt vdo van de nit gpn thugc ti'nh tren tap t\\6 md, mdt ung dung rat quan trpng ciia tap thd trong ITnh vuc khai phd dii lieu.
Trong ndi dung gidi thidu, chiing tdi cd trinh bay thugt toan FquickReduct da dugc R.Jensen and Q.Shen [9] de xuat trudc day va dugc xem nhu Id mdt trong ky thuat ddu tidn trong nit gpn thugc tinh tap thd md. Ky thugt nay Id su md rdng cua ky thuat nit ggn thudc tinh trdn tap thd truydn thdng sang tgp thd md dua trdn cdc khdi niem ham thudc, mien duong. Thdng qua dd, chiing tdi xem xet, nghien ciiu de xuat mdt md rgng thdm mdt thuat todn dya tren do do su phu thudc ciia cdc tap con thudc tinh vao mdi mgt thudc ti'nh trong tgp tat ca cac thudc tinh dd dp dung cho bd dii lieu ma trong dd khdng the xdc djnh nhan iap thugc tinh quydt djnh hogc gid trj ldp thudc tinh quyet djnh khdng ddy dii.
Trong vi du minh hpa, ket qua tinh toan cho thay thugt toan de xuat md rdng ciia chiing tdi la kha thi vd cd thd logi bd cdc thudc tinh khdng can thiet. Tgp con thugc tinh cho dugc bdi thuat toan EFQuickReduct cd kinh thudc 159
Quach Xuan Tnrdng vd Dtg Tap chi KHOA HQC & CONG NGHE 135(05): 155-160
t u o n g t u n h u kdt qua cho bdi thuat toan FquickReduct.
Trong cdc nghien c u u tiep theo, chiing tdi se tgp t r a n g nghien c u u vd phat trien mdt sd thuat todn nhanh, hieu qua hon trong bai toan nit ggn thudc tinh cua tgp thd md.
T A I LIEU THAM K H A O 1. Z. Pawlak, "Rough sets," Int. J. Comput.
Inform. Sci, vol. 11, no. 5, pp. 341-356, 1982.
2. D. S. Yueng, C. Degang, E. C. C. Tsang, J. W.
T. Lee, and W. Xizhao, "On the generalization of fuzzy rough sets," IEEE Trans. Fuzzy Syst, vol.
3. no. 3, pp. 343-361, Jun. 2005.
3. T. Y. Lin, Sets with partial memberships: A rough sets view of fuzzy sets, Proc. 1998 World Congr. Computational Intelligence (WCCI98), Anchorage, AK, May 4-9, 1998, pp. 785-790.
4. W. Z. Wu and W.-X. Zhang, Constructive and axiomatic approaches of fuzzy approximation operators. Information Science, vol. 159, no. 3-4, (2004) pp. 233-254.
http://dx.doi.Org/10.1016/j.ins.2003.08.005.
5. W. Z.Wu, J.-S. Mi, and W.-X. Zhang, Generalized fuzzy rough sets, Information Science, vol. 151 (2003), pp. 263-282.
http://dx.doi.org/10.l016/S0020-0255(02)00379-l.
6. N. N. Morsi and M. M. Yakout, Axiomatics for fiizzy r o u ^ sets. Fuzzy Sets Syst, vol. 100 (1998).
pp. 327-342. http://dx.doi.org/10.1016/SOI65- 0114(97)00104-8.
7. Z. Pawlak, "Rough sets," Int. J. Comput.
Inform. Set., vol. 11, no. 5, pp. 341-356, 1982.
8. James F. Peters, Christopher Henry, Approximation spaces in off-policy Monte Carlo learning, Engineering Applications of Artificial Intelligence 20 (2007) 667-675.
9. R. Jensen and Q. Shen, "Fuzzy-rough attributes reduction with application to web categorization,"
Fuzzy Sets Syst., vol. 141, pp. 469-485.
10. Skowron and L. Poikowski, Rough Sets in Knowledge Discovery. Berlin, Germany: Springer- Verlag, 1998, vol. 1,2.
11. R. Slowinski, Ed., Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory. Boston, MA: Kluwer, 1992.
12. W. P. Ziarko, Ed., Rough Sets, Fuzzy Sets and Knowledge Discovery. London, U.K.: Springer- Verlag, 1994, Workshop in Computing.
13. D. S. Yueng, C. Degang, E. C. C. Tsang, J. W.
T. Lee, and W. Xizhao, "On the generalization of fuzzy rough sets," IEEE Trans. Fuzzy Syst., voi.
13, no. 3, pp. 343-361, Jun. 2005.
14. D. Dubois and H. Prade. Putting rough sets and fuzzy sets together. In [22] 203-232. 1992
S U M M A R Y
S O M E M E A S U R E S O F T H E D E P E N D E N C E O F T H E A T T R I B U T E R E D U C T I O N O N T H E F U Z Z Y R O U G H S E T
Quach Xuan Truong'', Le Van Phung*
'College oflnformaion and Communication Tchnology - TNU,
^Institute oflnformalio Technology
Attribute reduction, which is widely used in data mining, is one ofthe most important issues in the rough set theory. In this paper, we present an overview of extended rough set theory which is based on the combination of rough set and the fiizzy set theory to illustrate the opacity and uncertainty of data. In particular, in the first section, we present an overview of rough set and fiizzy set theory, in the next section, we remind an attribute reduction algorithm which have been proposed on the measures ofthe dependence of atffibute sets in the fiizzy rough set. Accordingly, we propose an extent algorithm which is based on the caculation of dependence of attribute sets on each attribute. Our algorithm is more efficiency in some cases where the decision class labels are difFicuU to determine or insufficient.
Keyword: Rough set; fuzzy set; fuzzy rough set; attribute reduction; measure ofthe dependence
Ngdy nhdn bdi:24/2/20I5; Ngdy phdn bi?n:I2/3/20I5; Ngdy duyil ddng: 31/5/2015
Phdn bien khoa boc: TS. VU Due Thdi - Trudng Dgi hpc Cong nghe Thong tin & Truyin thong - DHTN Tel- 0989 090832; Email: [email protected] w
160
