A Fuzzy Statistics based Method for Mining Fuzzy Correlation Rules

Transcription

1 WSES TRNSTIONS o MTHEMTIS Nacy P L Hug-Je he Hao-E hueh We-Hua Hao hug-i hag Fuzzy Statstcs based Method o Mg Fuzzy oelato Rules NNY P LIN HUNG-JEN HEN HO-EN HUEH WEI-HU HO HUNG-I HNG Depatmet o ompute Scece ad Iomato Egeeg Tamag Uvesty Yg-chua Road Tamsu Tape TIWN RO acyl@maltuedutw cheh@malsuedutw @s90tuedutw 88990@s89tuedutw tatdc@maltuedutw bstact: - Mg uzzy assocato ules s the tas o dg the uzzy temsets whch equetly occu togethe lage uzzy dataset but most poposed methods may dety a uzzy ule wth two uzzy temsets as teestg whe act the pesece o oe uzzy temsets a ecod does ot mply the pesece o the othe oe the same ecod To pevet geeatg ths d o msleadg uzzy ule ths pape we costuct a ew method o dg elatoshps betwee uzzy temsets based o uzzy statstcs ad the geeated ules ae called uzzy coelato ules I ou method a uzzy coelato aalyss whch ca show us the stegth ad the type o the lea elatoshp betwee two uzzy temsets s used y usg thus uzzy statstcs aalyss the uzzy coelato ules wth the omato about that two uzzy ot oly equetly occu togethe same ecods but also ae elated to each othe ca be geeated Key-Wods: - Fuzzy assocato ules Fuzzy temsets Fuzzy statstcs Fuzzy coelato aalyss Lea elatoshp Fuzzy coelato ules Itoducto Mg assocato ule s a mpotat data mg tas whch s ote deed as dg the temsets whch equetly occu togethe lage databases [ 4 7 ] popula applcato o assocato ule mg s the maet baset aalyss whch detes the buyg behavous o customes It s wdely used to d the poducts whch ae equetly puchased togethe by same customes tasacto databases Ths d o omato s clealy useul o may maetg decsos ut pactcal databases may data may be descbed by uzzy temsets but useul watg to be exploed; hece methods to dscove assocato ules om uzzy temsets ae eeded too To ths ed may eseaches tu to popose methods o mg uzzy assocato ules om vaous uzzy dataset ecetly [ ] Most uzzy assocato ules mg methods poposed up to ow employ a suppot-codece amewo whch uses two measuesuzzy suppot ad uzzy codece to d the uzzy temsets whch equetly occu togethe ad to dety the uzzy assocato ules as teestg Howeve a stuato eeds to be otced a uzzy temset almost occus all ecods the t may equetly occu wth othe uzzy temsets Theeoe the suppot-codece amewo ca be msleadg that t may dety a uzzy ule wth two uzzy temsets as teestg whe act the pesece o oe uzzy temsets a ecod does ot mply the pesece o the othe oe the same ecod To pevet geeatg ths d o msleadg ule hee we adopt a ew method o dg useul elatoshps betwee uzzy temsets based o uzzy statstcs ad the geeated ules ae called uzzy coelato ules I ou method the aalyss o uzzy coelato [8 0 8 ] s used The uzzy coelato aalyss whch s deved om the covetoal statstcs ad uzzy set theoy ca show us the stegth ad the type o the lea elatoshp betwee two uzzy temsets y usg the uzzy coelato aalyss the uzzy coelato ules wth the omato about that two uzzy ot oly equetly occu togethe same ecods but also ae elated to each othe ca be geeated The est o ths pape s ogazed as ollows: The cocepts o mg uzzy assocato ules ae metoed secto The cocepts o the uzzy coelato aalyss ae peseted secto The uzzy coelato ules mg method s toduced secto 4 expemet s dsplayed secto The coclusos ae gve secto 6 Fuzzy ssocato Rules ISSN: Issue Volume 6 Novembe 007

2 WSES TRNSTIONS o MTHEMTIS Nacy P L Hug-Je he Hao-E hueh We-Hua Hao hug-i hag Fuzzy assocato ules mg s a tas o dg the uzzy temsets whch equetly occu togethe lage databases [ 4 7 ] Most methods o mg uzzy assocato ules employ a suppot-codece amewo whch uses uzzy suppot ad uzzy codece to dety the uzzy assocato ules as teestg Let F L } be a set o uzzy tems m T t t L t ) be a set o uzzy ecods ad each uzzy ecod t s epeseted as a vecto wth m values t ) t ) L t )) whee t ) m s the degee that uzzy tem appeas ecod t ) [0] The a uzzy assocato ule s t deed as a mplcato om such as F whee F F FY F ae two uzzy temsets ad F F all x x all y y Y The uzzy assocato ule F holds T wth the uzzy suppot supp F FY }) ad the uzzy codece co F )) The uzzy suppot ad the uzzy codece ae deed as ollows: supp F FY }) m t) F FY }) ) supp F FY }) co F F Y ) ) supp F }) I the supp F FY }) s geate tha o equal to a pedeed theshold mmal uzzy suppot s ) ad the co F ) s also geate tha F o equal to a pedeed theshold mmal uzzy codece c ) the F s cosdeed as a teestg uzzy assocato ule ad t meas that the pesece o uzzy temsets F a ecod ca mply the pesece o uzzy temsets F Y the same ecod Now let us cosde a specal stuato a uzzy temset s commo ad t almost occus all uzzy ecods the accodg the above amewo we may dety may uzzy assocato ules as teestg but act the pesece o ths obseved uzzy temsets does ot mply the pesece o othe uzzy temsets whch ae also cluded these Y uzzy assocato ules Theeoe these dscoveed ules ae msleadg actually Some eseaches have also otced ths poblem ad thus tued to adopt alteatve measue whch ca show exta omato about the elatoshps betwee temsets mg pocesses [7 4 7 ] The amous measue s deed as ollows [7 4 7]: Suppot thee ae two temsets ad a gve ecod set; the pobablty that occus s expessed as P ) ; the pobablty that occus s expessed as P ) ; the pobablty that ad occu both s expessed as P ) The the coelato o the assocato ule ca be expessed as coel ) coel ) P ) P ) P ) ) The value computed om ) les betwee [0 ] I P ) P ) P ) the coel ) s equal to meag ad ae o elated ad the pesece o oe s depedet o the pesece o the othe oe ut coel ) s close to 0 o tha t meas that ad ae hghly elated ad the pesece o oe ca mply the pesece o the othe oe lthough the above pobablty-based omula ca be used to aalyze the elatoshp betwee csp temsets t s ot sutable o aalyzg the elatoshp betwee uzzy temsets I ode to aalyze the elatoshps betwee uzzy temsets a useul uzzy statstcs aalyss uzzy coelato s adopted The uzzy coelato aalyss s deved om the covetoal statstcs ad uzzy set theoy ad t ca show us the stegth ad the type o the lea elatoshp betwee two uzzy temsets The cocepts o uzzy coelato aalyss ad how to use the uzzy coelato aalyss ou poposed method wll be explaed the ext secto Fuzzy oelato alyss The coelato aalyss o uzzy sets s called uzzy coelato aalyss May methods have bee poposed to calculate the uzzy coelato coecet [8 0 8 ] I ou method we adopt the omula deved by L [0] because t ca povde the exta omato we eed Suppose thee ae two uzzy temsets F whee F s a uzzy space ad ae deed o ISSN: Issue Volume 6 Novembe 007

3 WSES TRNSTIONS o MTHEMTIS Nacy P L Hug-Je he Hao-E hueh We-Hua Hao hug-i hag a csp uvesal set wth membeshp uctos ad ad the uzzy temsets ad ca be expessed as ollows: x x)) x ) 4) x x)) x ) ) whee [ 0 ] ssume that thee s a adom sample x x L ) aloe wth a sequece o paed data x ) )) L } whch coespod to the gades o the membeshp uctos o uzzy temsets ad deed o The the uzzy coelato coecet betwee the uzzy temsets ad s: whee s s 6) s s ) ) ) ) 7) ) 8) ) 9) S S S x ) ) x ) ) 0) ) s ) S s ) The value computed om 6) les betwee [- ] I 0 the the uzzy temsets ad > ae postvely elated I 0 the the uzzy < temsets ad ae egatvely elated ut 0 the the uzzy temsets ad have o elatoshp at all ccodg to these mpotat popetes we ca obta the stegth ad type o the lea elatoshp betwee two uzzy temsets hece the uzzy coelato aalyss s geat useul o mg the teestg uzzy coelato ules 4 Mg Fuzzy oelato Rules The uzzy coelato ules mg method wll be toduced ths secto ssume that F L } be a set o m uzzy tems; T t t L t ) be a adom sample wth uzzy data ecods ad each sample ecod t s epeseted as a vecto wth m values t ) t ) L t )) whee t ) s the degee that m uzzy tem occus ecodt t ) [ 0] d ext thee pedeed thesholds ae eeded to be deed Hee s s the mmal uzzy suppot; c s the mmal uzzy codece; s the mmal uzzy coelato coecet The pocedue o mg uzzy coelato ules s descbed as the ollows: Step : The uzzy suppot o each uzzy tem Fsupp ) s computed by usg omula ) Step : Let L F supp ) s } p p p be the set o equet uzzy temsets whose sze s equal to Step : Let F F )} be the set o all combatos o two elemets belog to L whee F F L F F That s s geeated by L ot wth L ecause F ad F ae the elemets o L the sze o each elemet o s Step 4: Fo each elemet o F F ) the uzzy suppotsupp F F}) s computed by usg omula ) ad the the uzzy coelato coecet betwee F ad F s computed by usg omula 6) too Sce om the adom sample T s computed s eeded to be tested to deteme t s eally geate tha the mmal uzzy coelato coecet The omula o testg s as ollows [4]: ISSN: Issue Volume 6 Novembe 007

4 WSES TRNSTIONS o MTHEMTIS Nacy P L Hug-Je he Hao-E hueh We-Hua Hao hug-i hag t 4) obseved uzzy tems Hee s s set to 00; c s set to 080; s set to 00; α s set to 0 ad thus t 09 0 s equal to 7 ompae the computed t value to t α ) th whee t α ) s the α) pecetle the t dstbuto wth degee o eedom I we obta the t value whch s geate tha t α ) the we ca coclude that s geate tha the pedeed mmal uzzy coelato coecet [4] Step : Fo each elemet whose uzzy suppot s geate tha o equal to s ad uzzy coelato coecet passes the test o the t s a elemet o L Hece L s the set o the equet combatos o two uzzy temsets ad stll the sze o each elemet o L s Step 6: Next each s geeated by L ot wth L ssume that F F ) ad W F Y F Z ) ae two elemets o L whee F FY I the sze o the combato F F F }) s W Z ad F W FZ ) s also a equet combato o two uzzy temsets the the combato F F F }) W Z s a elemet wth sze o Fo each elemet o ts uzzy suppot ad uzzy coelato coecet ae stll used to d the elemets o L Step 6: Whe each L s obtaedo each elemet o L F F ) two caddate uzzy G H FG FH ad H FG coelato ules F ca be geeated I the uzzy codece o a ule s geate tha o equal toc the t s cosdeed as a teestg uzzy coelato ule The algothm wo t stop utl o ext + ca be geeated Example expemet wll be dsplayed ths secto ssume that T t t t t4 t t6 t7 t8 t9 t0 t t} s a adom sample wth uzzy ecods show Table ad F } s the set o 4 Table : adom sample wth uzzy ecods F 4 T t t t t t t t t t t t t Fst the uzzy suppot o each uzzy tem o F s computed ad lsted Table ecause all supp ) L ae geate tha s the set o the equet uzzy temsets whose sze s equal to s L } 4 Table : The uzzy suppot o each uzzy tem o F F supp Next the set o all combatos o two elemets o L s geeated by L ot wth L ) ) 4) ) ) ) ) 4) ) 4) ) 4)} Fo each elemet o the uzzy suppot the uzzy coelato coecet ad t value o ISSN: Issue Volume 6 Novembe 007

5 WSES TRNSTIONS o MTHEMTIS Nacy P L Hug-Je he Hao-E hueh We-Hua Hao hug-i hag testg the uzzy coelato coecet ae computed ad lsted Table Table : The uzzy suppotuzzy coelato coecet ad t value o testg the uzzy coelato coecet o each elemet o supp t } }) } }) } 4 }) } }) } }) } 4 }) } }) } 4 }) } }) } }) I Table a elemet whose supp s geate tha o equal to s 00) ad t value s geate tha o equal to t ) s cosdeed a elemet o L Thus L } }) } }) } })} Whe L s obtaed s geeated by L ot wth L } }) } }) } }) } Smlalyo each elemet o the uzzy suppot the uzzy coelato coecet ad the t value o testg the uzzy coelato coecet ae also computed ad dsplayed Table 4 Table 4: The uzzy suppotuzzy coelato coecet ad t value o testg the uzzy coelato coecet o each elemet o supp t } }) } }) } }) I Table 4 because all elemets o satsy s ad t 09 0 all elemets o ae elemets o L Thus L No ext 4 ca be geeated by L ot wth L so the mg pocedue stops hee y usg the elemets o L ad L caddate uzzy coelato ules ca be geeated ad lsted Table Table : The uzzy codeces o the caddate uzzy coelato ules } co } 097 } } 076 } } 08 } } 08 } } 078 } } 09 } } 080 } } 086 } } 06 } } 096 } } 08 } } 08 ccodg to Table we deteme 9 teestg uzzy coelato ules as ollows because the uzzy codeces ae geate tha o equal to c 080) } } ) } } 6) } } 7) } } 8) } } 9) } } 0) } } ) } } ) } } ) } } 4) ISSN: Issue Volume 6 Novembe 007

6 WSES TRNSTIONS o MTHEMTIS Nacy P L Hug-Je he Hao-E hueh We-Hua Hao hug-i hag Fom above expemet we ca see that the umbe o elemets o each L s eectvely educed Fo example total umbe o the elemets o s 0 ad the umbe o elemets whose uzzy suppots satsy s s 8 but ate testg the uzzy coelato coecet the umbe o elemets whch belog to L s oly Theeoe we ca coclude that oly eally teestg elatoshps betwee uzzy temsets ca be dscoveed by usg ou poposed method 6 ocluso I ths pape a uzzy statstcs based method o mg uzzy coelato ules s poposed Most methods o mg uzzy assocato ules employ a suppot-codece amewo whch adopts uzzy suppot ad uzzy codece to dety the uzzy assocato ules as teestg Howeve the suppot-codece amewo may dety may uzzy assocato ules as teestg but act the uzzy temsets o these ules have o elatoshp at all I ou method a uzzy coelato aalyss whch ca show us the stegth ad the type o the lea elatoshp betwee two uzzy temsets s used Sce the uzzy coelato coecet s computed om a adom sample ou method s ecet ad ca be used lage uzzy dataset y usg the uzzy coelato aalyss the uzzy coelato ules wth the omato about that two uzzy ot oly equetly occu togethe same ecods but also ae eally elated to each othe ae geeated Reeeces: [] R gawal T Imels ad Swam Mg ssocato Rules betwee Sets o Items Lage Databases Poceedgs o the M SIGMOD Iteatoal oeece o Maagemet o Data Washgto D May 99 pp07-6 [] R gawal H Mala R Sat H Tovoe ad I Veamo Fast Dscovey o ssocato Rules dvaces Kowledge Dscovey ad Data Mg hapte I/MIT Pess 99 [] R gawal ad R Stat Fast algothms o mg assocato ules Poceedgs o the 0th Iteatoal oeece o Vey Lage Databases Satago hle Septembe 994 pp [4] S F old Mathematcal Statstcs Petce- Hall New Jesey 990 [] W-H u ad K ha eectve algothm o dscoveg uzzy ules elatoal databases Poceedgs o the IEEE Wold ogess o omputatoal Itellgece 998 pp4 9 [6] P osc D Dubos O Pvet ad H Pade O uzzy assocato ules based o uzzy cadaltes Poceedgs o the IEEE Iteatoal Fuzzy Systems oeece Melboue 00 [7] S R Motwa ad Slveste eyod maet basets: Geealzg assocato ules to coelatos Poceedgs o the M SIGMOD Iteatoal oeece o Maagemet o Data 997 pp 6-76 [8] H ustce ad P ullo oelato o teval-valued tutostc uzzy sets Fuzzy sets ad systems Vol pp7-44 [9] K ha ad K Wog Mg Fuzzy ssocato Rules Poceedgs o the sxth teatoal coeece o Iomato ad owledge maagemet 997 Las Vegas Nevada Uted States Novembe 997 pp09- [0] D hag ad N P L oelato o Fuzzy Sets Fuzzy Sets ad Systems Vol pp-6 [] M De oc oels ad EE Kee Elctato o uzzy assocato ules om postve ad egatve examples Fuzzy Sets ad Systems Vol pp7 8 [] M Delgado N Maí D Sáchez ad M Vla Fuzzy ssocato Rules: Geeal Model ad pplcatos IEEE Tasactos o Fuzzy Systems Vol No pl 00 pp4- [] D Dubos E Hullemee ad H Pade Note o Qualty Measues o Fuzzy ssocato Rules Poceedgs o the 0th Iteatoal Fuzzy Systems ssocato Wold ogess IFS-0) Lectue Notes tcal Itellgece 7 Spge-Velag 00 pp46- [4] M H Duham Data mg Itoductoy ad dvaced Topcs Peaso Educato Ic 00 [] Fu M Wog S Sze W Wog W Wog ad W Yu Fdg uzzy sets o the mg o uzzy assocato ules o umecal attbutes Poceedgs o the Fst Iteatoal Symposum o Itellget Data Egeeg ad Leag Hog Kog Octobe 998 pp6-68 ISSN: Issue Volume 6 Novembe 007

7 WSES TRNSTIONS o MTHEMTIS Nacy P L Hug-Je he Hao-E hueh We-Hua Hao hug-i hag [6] JM de Gaa W Kostes ad JJW Wttema Iteestg Fuzzy ssocato Rules Quattatve Databases Poceedgs o the PKDD 00 The th Euopea oeece o Pcples o Data Mg ad Kowledge Dscovey) Spge Lectue Notes ompute Scece 68 Febug Gemay Septembe 00 pp40- [7] J Ha ad M Kambe Data mg: ocepts ad Techques cademc Pess 00 [8] D H Hog ad S Y Hwag oelato o tutostc uzzy sets pobablty spaces Fuzzy Sets ad Systems Vol7 99 pp77-8 [9] H Ishbuch T Naashma ad T Yamamoto Fuzzy assocato ules o hadlg cotuous attbutes Poceedgs o the IEEE Iteatoal Symposum o Idustal Electocs 00 pp8- [0] M Kuo Fu ad M H Wog Fuzzy assocato ules lage databases wth quattatve attbutes M SIGMOD Recods Mach 998 [] R Sat ad R gawal Mg Geealzed ssocato Rules Futue Geeato ompute Systems Volume -) Decembe 997 pp6-80 [] Wu Zhag ad S Zhag Ecet mg o both postve ad egatve assocato ules M Tasactos o Iomato Systems Vol Issue July 004 pp8 40 [] Yu oelato o uzzy umbes Fuzzy Sets ad Systems Vol 99 pp0-07 [4] W Zhag Mg Fuzzy Quattatve ssocato Rules Poceedgs o the th IEEE Iteatoal oeece o Tools wth tcal Itellgece 999 pp99-0 ISSN: Issue Volume 6 Novembe 007