Soft Set Theoretic Approach for Dimensionality Reduction 1

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1 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 Soft Set Theoretc pproch for Dmensonlty Reducton Tutut Herwn Rozd Ghzl Mustf Mt Ders Deprtment of Mthemtcs Educton nversts hmd Dhln Yogykrt Indones Fculty of Informton Technology nd Multmed nverst Tun Hussen Onn Mlys Johor Mlys bstrct reduct s subset of ttrbutes tht re ontly suffcent nd ndvdully necessry for preservng prtculr property of gven nformton system The exstng reduct pproches under soft set theory re stll bsed on Boolen-vlued nformton system However n the rel pplctons the dt usully contn non-boolen vlues In ths pper n lterntve pproch for ttrbute reducton n mult-vlued nformton system under soft set theory s presented Bsed on the noton of mult-soft sets nd ND operton ttrbute reducton cn be defned It s shown tht the reducts obtned re equvlent wth Pwlk s rough reducton Keywords: Informton system; Reduct; Soft set theory Introducton In lots of dt nlyss pplctons nformton nd knowledge re stored nd represented n n nformton tble where set of obects s descrbed by set of ttrbutes To ths one prctcl problem s fced: for prtculr property whether ll the ttrbutes n the ttrbute set re lwys necessry to preserve ths property [] sng the entre ttrbute set for descrbng the property s tme-consumng nd the constructed rules my be dffcult to understnd pply or verfy In order to del wth ths problem ttrbute reducton s requred The obectve of reducton s to reduce the number of ttrbutes nd t the sme tme preserve the property of nformton The theory of soft set [] proposed by Molodtsov 999 s new method for hndlng uncertn dt Soft sets re clled (bnry bsc elementry neghborhood systems [] The soft set s mppng from prmeter to the crsp subset of unverse From such cse we my see the structure of soft set cn clssfy the obects nto two clsses (yes/ or no/0 Ths mens tht the stndrd soft set dels wth Boolen-vlued nformton system The theory of soft set hs been ppled to dt nlyss nd decson support systems fundmentl noton supportng such pplctons s the concept of reducts The de of dmensonlty reductons under soft set theory hve been proposed nd compred ncludng the works of [4 8] The restrcton of those technques s tht they re pplcble only for Boolen-vlued nformton systems n erly verson of ths pper ppered n the Proceedng of Interntonl Conference DT 009 held s Prt of the Future Generton Informton Technology Conference FGIT 009 Jeu Islnd Kore December CCIS 64 Sprnger-Verlg pp

2 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 However n the theoretcl nd prctcl reserches of soft sets the stutons re usully very complex nd hence t my not suffce to represent dt n the form of Boolen-vlued nformton systems In the rel pplcton dependng on the set of prmeters gven prmeter my hve dfferent vlues (contn multple grdes For exmple the mthemtcs degree of student cn be clssfed nto three vlues; hgh medum nd low In ths stuton every prmeter determnes prtton of the unverse whch s contns more thn two dsont subsets nlke n Boolen-vlued nformton systems n multvlued nformton systems one cnnot drectly defne the stndrd soft set To ths we proposed the de of mult-soft sets to del mult-vlued nformton systems In ths pper we propose the de of dmensonlty reducton for mult-vlued nformton systems under soft set theory Three mn contrbutons of ths work re s follows: Frstly we present the de of mult-soft sets constructon from mult-vlued nformton system nd ND nd OR opertons on mult-soft sets Secondly we present the pplcblty of the soft set theory for dt reducton under mult-vlued nformton system usng multsets nd ND operton Lstly we show tht reducts obtned usng soft set theory re equvlent to tht rough set theory lthough some results re presented mor prt of ths pper s devoted to revelng nterconnecton between reducton n mult-vlued nformton systems under rough nd soft set theores The rest of ths pper s orgnzed s follows Secton descrbes relted works of dmensonlty reducton under soft set theory Secton descrbes Informton systems nd set pproxmtons Secton 4 descrbes fundmentl soft set theory Secton descrbes reduct n nformton systems usng soft set theory Fnlly we conclude our works n secton 6 Relted Works The de of reduct nd decson mkng usng soft set theory ws frstly proposed by M et l [4] In [4] the pplcton of soft set theory to decson mkng problem wth the help of Pwlk s rough mthemtcs ws presented The reducton pproch presented s usng Pwlk s rough reducton nd decson cn be selected bsed on the mxml weghted vlue mong obects relted to the prmeters Chen et l [-6] presented the prmeterzton reducton of soft sets nd ts pplctons They ponted out tht the results of reducton proposed by M s ncorrect nd observed tht the lgorthms used to compute the soft set reducton nd then to compute the choce vlue to select the optml obects for the decson problem proposed by M re unresonble They lso ponted out tht the de of reduct under rough set theory generlly cnnot be ppled drectly n reduct under soft set theory The de of Chen et l for soft set reducton s only bsed on the optml choce relted to ech obect However the de proposed by Chen s not error free snce the problems of the sub-optml choce s not ddressed To ths Kong et l [7] nlyzed the problem of suboptml choce nd dded prmeter set of soft set Then they ntroduced the defnton of norml prmeter reducton n soft set theory to overcome the problems n Chen s model nd descrbed two new defntons e prmeter mportnt degree nd soft decson prtton nd use them to nlyze the lgorthm of norml prmeter reducton Wth ths pproch the optml nd sub-optml choces re stll preserved Zou [8] proposed new technque for decson mkng of soft set theory under ncomplete nformton systems The de s bsed on the clculton of weghted-verge of ll possble choce vlues of obect nd the weght of ech possble choce vlue s decded by the dstrbuton of other obects For fuzzy soft sets ncomplete dt wll be predcted bsed on the method of verge probblty ll those technques re stll bsed on Boolen nformton systems s to ths dte no reserches hve been done on dmensonlty reducton n mult-vlued nformton systems under soft set theory Snce every rough set [9] cn be consdered s soft set s presented n [0] thus n lterntve 48

3 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 pproch wth potentl for fndng reduct n mult-vlued nformton systems s usng soft set theory Stll t provdes the sme results for rough reducton [ ] Informton Systems nd Set pproxmtons n nformton system s 4-tuple (qudruple S ( where { u u u L s non-empty fnte set of obects { L u s non-empty fnte set of ttrbutes V V V s the domn (vlue set of ttrbute f : V s n nformton functon such tht f( u V clled nformton (knowledge functon for every ( u n nformton system s lso clled knowledge representton systems or n ttrbutevlued system n nformton system cn be ntutvely expressed n terms of n nformton tble (see Tble Tble n nformton system k u f ( u f ( u f u k u f ( u f ( u f u k u f f f ( u u u f ( u f ( u k f ( u M M M O M O M u f( u f( u f ( u f ( u k The complexty for computng n nformton system S ( there re L s snce vlues of f ( u to be computed where L nd Note tht t nduces set of mps t f( u : V tuple t ( f( u f( u f( u L f( u Ech mp s where where L Note tht the tuple t s not necessrly ssocted wth entty unquely (see Exmple 6 In n nformton tble two dstnct enttes could hve the sme tuple representton (duplcted/redundnt tuple whch s not permssble n reltonl dtbses Thus the concept of nformton systems s generlzton of the concept of reltonl dtbses In mny pplctons there s n outcome of clssfcton tht s known Ths posteror knowledge s expressed by one (or more dstngushed ttrbute clled decson ttrbute; the process s known s supervsed lernng n nformton system of ths knd s clled decson system decson system [] s n nformton system of the form D ( { d where d s the decson ttrbute The elements of re clled condton ttrbutes The strtng pont of rough set pproxmtons s the ndscernblty relton whch s generted by nformton bout obects of nterest Two obects n n nformton system re clled ndscernble (ndstngushble or smlr f they hve the sme feture 49

4 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 Defnton (See [] Let S ( be n nformton system nd let B be ny subset of Two elements x y re sd to be B-ndscernble (ndscernble by the set of ttrbute B n S f nd only f f ( x f( y for every B Obvously every subset of nduces unque ndscernblty relton Notce tht n ndscernblty relton nduced by the set of ttrbute B denoted by IND ( B s n equvlence relton It s well known tht n equvlence relton nduces unque prtton B S V f denoted by / B nd the The prtton of nduced by IND n equvlence clss n the prtton / B contnng x denoted by [ x ] B lower nd upper pproxmtons of set cn be defned s follows The notons of Defnton (See [] Let S ( be n nformton system let B be ny subset of nd let X be ny subset of The B-lower pproxmton of X denoted by B ( X nd B-upper pproxmtons of X denoted by B ( X re defned by B( X x [ x] X nd B( X x [ x] I X φ respectvely B { B { The notons of rough pproxmtng of set cn be defned s follows: Defnton Let S ( be n nformton system nd let B be ny subset of rough pproxmton of subset X wth respect to B s defned s pr of lower nd upper pproxmtons of X e Defnton 4 Let S ( ( X B( X B be n nformton system nd let B be ny subsets of n nformton system S ttrbute b B s clled dspensble f / B b / Defnton Let S ( ( { B be n nformton system nd let B be ny subset of n nformton system S The subset B* B s clled reduct of B f B * stsfes the followng condtons: The core of B s defned s b / B* / B / where RED ( B s the set of ll reducts of B ( B* { b / B b B * ( B RED( B I CORE It s known tht the problem of fndng mnml reducts n nformton systems s NP-hrd Exmple 6 For smple exmple of rough reducton we consder smll dtset derved from [] Tble n nformton system from [] 4 low bd loss smll 0

5 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 low good loss lrge hgh good loss medum 4 hgh good loss medum low good proft lrge Let { 4 then we hve / {{{ { 4{ B { nd C { t cuses 4 The core s ( B C { {{{ { 4{ / B / C / CORE I The reducts of re The noton of soft sets nd ts fundmentl opertons re gven n the followng secton Much of the defnton nd exmples re quoted drectly from [4] 4 Soft Set Theory Throughout ths secton refers to n ntl unverse E s set of prmeters ( power set of nd E Defnton 7 (See [] pr gven by P s the F s clled soft set over where F s mppng F : P In other words soft set over s prmeterzed fmly of subsets of the unverse For ε F ( ε my be consdered s the set of ε -elements of the soft set ( F or s the set of ε -pproxmte elements of the soft set Clerly soft set s not (crsp set To llustrte ths de let we consder the followng exmple Exmple 8 Let we consder soft set ( E F whch descrbes the ttrctveness of houses tht Mr X s consderng to purchse Suppose tht there re sx houses n the unverse under consderton nd { h h h h h h E 4 { e e e e e 4 s set of decson prmeters where e stnds for the prmeters expensve e stnds for the prmeters beutful e stnds for the prmeters wooden e 4 stnds for the prmeters chep e stnds for the prmeters n the green surroundng Consder the mppng F : E P 6

6 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 gven by houses( where ( s to be flled n by one of prmeters Suppose tht e E F ( e { h F ( e { h F ( e { h h h F ( e { h h h { h 4 h 4 4 F e h Therefore F ( e mens houses (expensve whose functonl vlue s the set { h 4 Thus we cn vew the soft set ( F E s collecton of pproxmtons s below ( F E { h h4 { h h { h h4 h { h h h expensve houses beutful houses wooden houses chep houses { n the green surroundng houses h h Ech pproxmton hs two prts predcte p nd n pproxmte vlue set v For exmple for the pproxmton expensve houses { h h 4 we hve the predcte nme of expensve houses nd the pproxmte vlue set or vlue set f { h h 4 Thus soft set ( E F cn be vewed s collecton of pproxmtons below: ( F E { p v p v p v L p n v n Tble 4 Tbulr representton of soft set n the bove exmple e e e e 4 e h 0 0 h h 0 0 h h h Defnton 9 (See [4] The clss of ll vlue sets of soft set ( F E s clled vlue-clss of the soft set nd s denoted by C ( F E Clerly C P( F E Proposton 0 If ( F E s soft set over the unverse then ( E nformton system S ( Proof Let ( F E be soft set over the unverse we defne mppng F { f f Where L f : V nd x F( e f x 0 x F( e f n F s Boolen-vlued

7 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 Thus f E V e f f : V nd x F( e f x 0 x F( e : nd f ( x 0 V Ve M x F x F ( e ( e where V { 0 then soft set ( E e F cn be consdered s Boolen-vlued nformton system S V{ 0 f From Proposton 0 t s esly to understnd tht bnry-vlued nformton system cn be represented s soft set Thus we cn mke one-to-one correspondence between ( F E over nd S ( V{ 0 f Proposton Every rough set cn be consdered s soft set Let ( X B( X B B be the rough pproxmtng subset X We defne mppng B : P( P( s n Defnton Thus every rough set ( X B( X consdered pr of two soft sets ( F ( B P( B P( B cn be From the fct tht every rough set cn be consdered s soft set n the followng secton we propose n lterntve pproch for dmensonlty reducton n mult-vlued nformton systems under soft set theory Reduct n Informton Systems usng Soft Set Theory In ths secton we present the pplcblty of soft set theory for fndng reducts We show tht the reducts obtned re equvlent to the rough reducts s n [ ] s for frst step we need trnsformton from mult-vlued nformton system nto mult-soft sets In the mult-soft sets we present the noton of ND nd OR opertons For ttrbute reducton we employ n ND operton nd hve mnged to show tht the reducts obtned re equvlent to rough reducts Mult-soft sets constructon from mult-nformton systems In ths sub-secton we dscuss decomposton of mult-vlued nformton system S V f nto number of bnry-vlued nformton systems The decomposton of S ( nto the dsont- s bsed on decomposton of { L sngleton ttrbute { { { L t ths stge only complete nformton system s gven the consderton Let S ( V f( be n nformton system such tht for every s fnte non-empty set nd for every under th -ttrbute consderton v u f f ( u v such tht nd u f ( u For every v we defne the mp : { 0 V otherwse ( u 0 v v The next result we defne bnry- vlued nformton system s qudruple S V{ 0 f The nformton systems

8 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 S ( f L s referred to s decomposton of mult-vlued V{ 0 nformton system S ( depcted n Fgure Every nformton system S ( nto bnry-vlued nformton systems s determnstc nformton system snce for every L s nd for every u f ( u such tht the structure of mult-vlued nformton system nd number of bnryvlued nformton systems gve the sme vlue of ttrbute relted to obects k u f ( u f ( u f u k u f ( u f ( u f u k u f f f ( u u u f ( u f ( u k f ( u M M M O M O M u f( u f( u f ( u f ( k u Bnry-vlued nformton system- V V V k V n u u u M M M O M O M u M Bnry-vlued nformton system- V V V V k n u u u M M M O M O M u Fgure decomposton of mult-vlued nformton system Bsed on the noton of decomposton of mult-vlued nformton system n the prevous sub-secton n ths sub-secton we present the noton of mult-soft set S V f be mult-vlued representng mult-vlued nformton systems Let nformton system nd S ( L nformton systems From Proposton 0 we hve be the bnry-vlued 4

9 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 S ( S S ( V{ 0 f ( V { 0 S ( V{ f 0 ( F ( F L ( F We defne ( F E ( F ( F L ( F M M ( F ( F M ( F s mult-soft set over unverse representng mult-vlued nformton system S ( Exmple The mult Boolen nformton systems representng Tble 4 s gven below low hgh bd good loss proft smll lrge medum Fgure decomposton of mult-vlued nformton system From Fgure we hve the followng correspondng soft sets

10 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 ( F { low { hgh 4 ( F { bd { good 4 ( F { loss 4{ proft ( 4 { smll { lrge { medum 4 F Thus the mult-soft set representng Tble s ( F (( F ( F ( F ( F 4 { low { hgh 4 { bd { good 4 { loss 4{ proft { smll { lrge { medum 4 ND nd OR opertons n mult-soft sets The notons of ND nd OR opertons n mult-soft sets re gven below Defnton Let( F E ( F : L be mult-soft set over representng mult-vlued nformton system S ( The ND operton between ( F nd ( F s defned s where G ( F ND( F ( F ( V V F V I F V ( V V Exmple From Exmple let two soft-sets nd Then we hve for ( F { low { hgh 4 ( { bd { good 4 F ( F ND( F ( F (( low bd {( low good { ( hgh bd ( hgh good { 4 φ Defnton 4 Let( F E ( F : L mult-vlued nformton system S ( The OR operton between ( ( F s defned s where G be mult-soft set over representng ( F OR( F ( F ( V V F V F V ( V V Exmple From Exmple let two soft-sets for ( { bd { good 4 F F nd 6

11 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 nd ( { loss 4{ proft F Then we hve ( F OR( F ( F ( bd loss { 4( bd proft { ( good loss { 4( good proft { 4 ttrbute reducton In ths secton we propose the de of ttrbutes reducton under soft set theory The proposed pproch s bsed on ND operton n mult-soft sets s descrbed n the prevous secton Defnton 6 Let( F ( F : L mult-vlued nformton system S ( reduct for f C F( b L b C F( L be mult-soft set over representng B Exmple 7 From Exmple let two mult soft-sets set of ttrbutes B s clled ( F { nd ( F { For ( F { where ( F { low { hgh 4 ( F { bd { good 4 nd ( { loss 4{ proft Then we hve 4 F ( F ND( F ND( F ( F ( low bd loss {( low bd proft { ( low good loss { ( low good proft { ( hgh bd loss φ ( hgh bd proft φ ( hgh good loss { 4( hgh good proft { Notce tht F ( {{{ { 4{ C ( b For ( F { 4 where ( { loss 4 { proft ( F 4 { smll { lrge { medum 4 Then we hve F nd ( F ND( F 4 ( F 4 ( losssmll {( loss lrge { ( loss medum { 4 { proft smll φ proft lrge proft medum φ 7

12 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 Notce tht F ( {{{ { 4{ C ( 4 From ( nd ( we hve { nd { 4 re reducts of 6 Concluson The exstng reduct pproches under soft set theory re stll bsed on Boolen-vlued nformton system For the rel pplctons the dt usully contn non-boolen vlued In ths pper n lterntve pproch for ttrbute reductons n mult-vlued nformton systems under soft set theory hs been presented In the proposed pproch the noton of mult-soft set s used to represent mult-vlued nformton systems The ND operton s used n mult-soft sets to present the noton of ttrbute reducton It s founded tht the obtned reducts re equvlent to the rough reducts cknowledgement Ths work ws supported by the grnt of nverst Tun Hussen Onn Mlys References [] Zho Y Luo F Wong SKM nd Yo YY generl defnton of n ttrbute reduct Proceedng of Second Interntonl Conference on Rough Sets nd Knowledge Technology RSKT 007 LNI [] Molodtsov D Soft set theory-frst results Computers nd Mthemtcs wth pplctons [] Yo YY Reltonl nterprettons of neghbourhood opertors nd rough set pproxmton opertors Informton Scences [4] M PK Roy R nd Bsws R n pplcton of soft sets n decson mkng problem Compututer nd Mthemtcs wth pplcton [] Chen D Tsng ECC Yeung DS nd Wng X Some notes on the prmeterzton reducton of soft sets Proceedng of Interntonl Conference on Mchne Lernng nd Cybernetcs 00 IEEE Press [6] Chen D Tsng ECC Yeung DS nd Wng X The Prmeterzton Reducton of Soft Sets nd ts pplctons Computers nd Mthemtcs wth pplctons [7] Kong Z Go L Wng L nd L S The norml prmeter reducton of soft sets nd ts lgorthm Computers nd Mthemtcs wth pplctons [8] Zou Y nd Xo Z Dt nlyss pproches of soft sets under ncomplete nformton Knowledge Bsed Systems [9] Pwlk Z Rough sets Interntonl Journl of Computer nd Informton Scence

13 Interntonl Journl of Dtbse Theory nd pplcton Vol No June 00 [0] Herwn T nd Mustf MD drect proof of every rough set s soft set Proceedng of Interntonl Conference MS 009 IEEE Press 9-4 [] Pwlk Z Rough sets: theoretcl spect of resonng bout dt Kluwer cdemc Publsher 99 [] Pwlk Z nd Skowron Rudments of rough sets Informton Scences [] Pwlk Z Rough clssfcton Interntonl ournl of humn computer studes [4] M PK Bsws R nd Roy R Soft set theory Computers nd Mthemtcs wth pplctons Tutut Herwn He s PhD cnddte n Dt Mnng t nverst Tun Hussen Onn Mlys (THM Hs reserch re ncludes Dt Mnng KDD nd Rel nlyss Rozd Ghzl She receved her BSc (Hons degree n Computer Scence from nverst Sns Mlys nd MSc degree n Computer Scence from nverst Teknolog Mlys She obtned her PhD degree n Hgher Order Neurl Networks t Lverpool John Moores nversty K She s currently techng stff t Fculty of Informton technology nd Multmed nverst Tun Hussen Onn Mlys (THM Her reserch re ncludes neurl networks fuzzy logc fnncl tme seres predcton nd physcl tme seres forecstng Mustf Mt Ders He receved the BSc from nversty Putr Mlys MSc from nversty of Brdford Englnd nd PhD from nversty Putr Mlys He s professor of computer scence n the Fculty of Informton Technology nd Multmed THM Mlys Hs reserch nterests nclude dstrbuted dtbses dt grd dtbse performnce ssues nd dt mnng He hs publshed more thn 80 ppers n ournls nd conference proceedngs He ws pponted s one of edtorl bord members for Interntonl Journl of Informton Technology World Enformtk Socety revewer of specl ssue on Interntonl Journl of Prllel nd Dstrbuted Dtbses Elsever 004 specl ssue on Interntonl Journl of Cluster Computng Kluwer 004 IEEE conference on Cluster nd Grd Computng held n Chcgo prl 004 nd Mlysn Journl of Computer Scence He hs served s progrm commttee member for numerous nterntonl conferences/workshops ncludng Grd nd Peer-to-Peer Computng (GPP utonomc Dstrbuted Dt nd Storge Systems Mngement (DSM WSES Interntonl ssocton of Scence nd Technology ISTED on Dtbse etc 9

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Vol. 5, No. 5 May 2014 ISSN Journal of Emerging Trends in Computing and Information Sciences CIS Journal. All rights reserved. Vol. 5, No. 5 y 04 ISSN 079-8407 Jornl of Emergng Trends n Comptng nd Informton Scences 009-04 CIS Jornl. ll rghts reserved. http://www.csornl.org Notes on lt Soft trces D.Sngh, Onyeozl, I.., 3 lkl..j.,