Trend Relational Analysis and Grey-Fuzzy Clustering Method *

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1 d Rlatoal alyss ad Gy-Fuzzy Clustg tod * Zj C Wz C 3 Ql C 4 ad a-yu C stat *34 I ts pap t latst sas o td latoal aalyss ad gy-fuzzy lustg mtod a pstd Gy systms ad fuzzy systms a foud vyw sad sults a op w pospts fo t dvlopmt ad applato of systms mtodology to data mg ywods d latoal aalyss gy-fuzzy lustg mtod gy systms fuzzy systms Itoduto It s wll ow tat t oto of a systm s at oad ad a tad to atquty So to spa ay a ojt vstgatd su as t moto of a maosop patl o som soooom pomo may qualfd as a systm Systms su as soal oom agultual dustal ologal ad ologal systms a usually tos of gat omplxty s omplatd ojts appatly av t followg aatst fatus: * s pojt was suppotd y t Natoal Natual S Foudato of Ca GF-R-58 Sool of Eooms ad aagmt Wua Uvsty of S ad Egg Wua Ca Emal:_zj@tomom Dpatmt of Cotol S ad Egg uazog Uvsty of S ad ology Wua Ca Emal:my@malustdu Emal:wz_005@6om 3 Dpatmt of Ifomato ad Eltal Egg Wua olyt Uvsty Wua Ca Emal:wz@wpudu 4 Dpatmt of Comput S Wua Uvsty of S ad Egg Wua Ca Emal:ql@wusdu 34 - s o pysal pototyp; - opato masm s ot la; - latosps tw t puts ad t outputs a ot ovous; - dtmatss s vy stog; - Oft tms oly a fw of dst data osvd a otad us t studs of su ojts ow to uld a systm modl ow to foast ow to ma dso ad ow to otol ly to a gat xtt o t us of fomato fom ojtv alty Fom a patal pot of vw owv t s vy dffult o mpossl to gt t adquat o omplt fomato fom vstgatd ojt may stuatos pomo wt omplt fomato tfo s usually outd Iomplt fomato s t fudamtal mag of g gy am of gy systm s os asd o t amout of ow fomato Cosd a la ox stads fo a ojt su tat ts tal stutu s totally uow to t vstgato t wod la psts uow fomato wt fo ompltly ow fomato ad gy fo tos fomato w a patally ow ad patally uow odgly systms wt ompltly ow fomato a alld as wt systms Systms wt ompltly uow fomato as la systms ad t systms wt patally ow ad patally uow fomato as gy systms sptvly [] S 98 t as a qu dvlopmt gy systms toy[34] Ca ad t s also vy sussful t applato of t toy to may al pojts su as agultu soty ooms gg I data mg maagmt ologal

2 potto ology vomtal studs t ft ov 0 yas of apd dvlopmt t toy of gy systms ossts of t followg ma los of opts ad sults: - Foudato osstg of gy ums gy lmts ad gy latos; - Gy systms aalyss ludg gy d aalyss gy statsts gy lustg t; - Gy systms modlg toug t us of gato of gy ums o futos so tat dd patts a foud; - Gy pdto; - Gy dso mag; - Gy otol; - Gy poss; - Itgal gatg tasfom; - d latoal aalyss; ad - Systms louds ad wo tas a tst ts sujt sould f to yts Vol33 No 004 Gy Systms oy ad pplatos Gust Edtos: a-yu C Sfg u ad Y I ts pap t latst sas o td latoal aalyss ad gy-fuzzy lustg mtod a pstd sad sults a op w pospts fo t dvlopmt ad applato of systms mtodology to data mg O Gy oss xssvly omplx o omplatd ojt w gally sows a la of ompltd modl fomato may lood upo as a gy systm at s wt t ad of t gy systms appoa w a al to solv t polms of t aalyss ad dsg of omplatd systms o xssvly omplx systms ludg t data systms Su a systm mgt lood upo as a data ogazg famwo aodg to w som data a osdd to lvat ots ot Fom a gy systm s pot of vw all t dtmat o adom opts a gadd to gy I od to ds a vstgatd ojt wt omplt fomato ad to gt a asoaly stal ptu w a ommuatd som gy opts a dfd as follows: - most as gdt w xsts a gy systm wt omplt fomato s alld t gy lmt dotd y ; - dtmat vaal wos ampltud vas ov a sutal ag s alld a gy vaal dotd y ; - futo dfd o t Catsa podut spa dotd y t s alld a gy poss w s a gy lmt ad t psts tm; - osval output of a vstgatd ojt w s a futo of t dotd y t s alld a wtg futo of gy poss t t o oft ta ot oly a fw of t dst data osvd fom t vstgatd ojt a otad su as { 0 w s alld t ogal tm ss ospodg to t W osd tat fo a vstgatd ojt all avoal fomato s otad t ad all lvat fomato toug osvato s otad t o us t o as povdd t ass fo ostutg systms modl[5] 3 d Rlatoal alyss fo Gy Systms t us osd dyam latosps tw fatos tat a pst a vstgatd ojt Natually w a gt 35

3 Dfto 3 W all t f fato : [ 0] ξ oos fly t ompad fato { Cospodgly { s alld t f tm ss { ompad tm ss t I od to xpss t appoxmatss ad smlaty tw { ad { w pod to possg of data as follows: Dfto 3 ll of t followg tasfomatos a alld to mappg of quatty: w : : : { { { 3 { 3 { Dfto 33 t a mappg If t w all ξ t td latoal futo of ot { ad { om 3 Basd o{ ad{ f ξ ξ { { [ β γ 0 γ [ 0] 3 β ] t ξ s a d of td latoal futos oof mmal amout of fomato aout { ad { s volvd ξ us ξ a xpss t dyam latosp tw { ad { sufftly ad mt t odtos Dfto 33 smlaty; appoxmatss o ta smlaty; { { [6-] ξ Coollay 3 If ξ ost 3 t { s ompltly smla to { Dfto 34 ξ s a td latoal futo tw { ad { t 36

4 ξ m / m s dfd as t td latoal gad w m m[ ] as a masu Coollay 3 ag m t td latoal gad s / ξ Cospodgly oof Omttd om ad os dyam modlg os s t optmal smla a t fft tools of gy Cosd tat ad stad fo owldg sts w ad uow aout modl W av s ow aout modl Cosd tat j w om 34 mmum lmt s t a ota t td latoal gad matx as follows: j j j optmal dotd y oof wt t owldg dsovy to m m d Omttd { { j om 3 maxmum lmt s t d s a fft tool fo dsovg w optmal dotd y wt t systm modlg to max max om oof Omttd om 33 t { { j { ξ ξ m m { ξ ξ max max If ξ s satsfd wt t to low J m{ ξ ξ opt max m t t s alld t optmal smla ξ dotd owldg 4 pplato of to Gy Dyam odlg t us fst osd a gy systm wt a sgl fato ad w a ota a alzato toug osvato say a sgl tm ss Ou tas s to ostut t systm foastg modl w s at satsfatoy asd o { t fst w may tasfom { { to toug t Itgal Gatg asfom y ξ w 3 os IG 37

5 IG { { w a R W may w m m ogz tat f s mpls t ss of td lato satsfatoly f s s 05 3 dtmay of { { s stog ta ssum tat t s a st of ow futo dotd y S f { f S f j j m oug t td latoal aalyss tw { ad f f f sptvly w a gt t td latoal gads as follows: { m xf xf xfm I od to fd a satsfatoly ompag futo w av to aalyz t td latoal futos ad t td latoal gads us asw s xplt Suppos tat t f s { f s xfs max { xf j j m w s sougt out may stad fo a s latt law appoxmatly s a dtmat futo dfd o tm doma fo xampl f s a a laty of { w may us data of to ft l a glov ad fd t f s systm foastg modl as follows: w 0 a a a l [ ]/[ ] 3 3 a a a a s s a systm foastg modl y am SCG SCG [-40] Now lt us osd a gy systm wt fatos ad w a gt t ogal tm ss { Ou tas s to ostut t systm modl asd o { followg tom s statd aov w summaz t om 4 t { t ogal tm ss Cospodtly t a t ma-valu tm ss { ad ts gatg ma-valu tm ss 38

6 { w If f j { t { f j w s ow psts t oomogous xpotal futo wt spt to dst tm j m ad t stads fo a satsfatoly td lato t t gy modl of systms louds a ostutd as & t ad ts soluto s o t U SCG y am t 0 otuously 4- t t U U 4- U U 4-3 w l[ ] ; B 4-5 C B 4-6 U C ; B C ; [ ] [ t t ] t ; t oof Sltg t multplyg ot sds of Eq 4- y t d dt [ & t [ t ] t t U t t t 0 to t tgal fato ad t] t t t dtu 0 U t w av ft aagg Eq 4- oms Eq 4- o Eq4-3 w Eq4-3 s a dst soluto of Eq4- t C U ad B C Cosdg t gv odtos w df B C 0 I B 39

7 W 3 w a ostut t matxs: ad If s osgula ad a t Eq4-4 olds Baus { s satsfatoly latd to t Cosdg Eq4-3 Eq4- Eq4- a otad y dfftatg ot sds of Eq4- wt spt t om 4 o w Â ad SCG t t B B 0 foastg modl s B a gv y Eq4-4 ad Eq4-5 fttg uvs dtmd y Eq4- o Eq4-3 w a gt B C 0 sptvly oof Omttd Coollay 4 W SCG modl Eq4- oms w Y Y I Y Y Y w I s t dtty matx Eqs4-5 ad 4-6 old us w av U ad Eq4-3 Rmas aamts dtfato of o l SCG s w ow t ogal tm ss w a usd to ostut oly flt t past ad pst avo o vstgatd ojt Wt t laps of tm sd ad outsd may agg ospodgly a um of w data a otad I od to ma al to flt ad ta t ag of avo w sould mpov t ogal SCG SCG SCG osvd so tat w a p modl usg t w data all t tm to vald ad ps Cosdg t paamts Â B ad Ĉ of SCG modl oly w B ad Ĉ SCG s sstal pat w a dv fom t dtal s t followg Suppos tat t s a ogal tm ss 40

8 { w s satsfd fo modlg odg to om 4 w av SCG t ad a t a W a w data vto osvd s otad w av ] [ [ ] [ ] [ ] [ ] 4-7 t t a a [ ] a a 4-8 w [ ] [ ] fot w a gt B B C Eqs4-7 ad 4-8 w a two u fomulas a sougt out { may atay tm ss t ps of utaty I may stuatos a vstgatd ojt may gadd as a galzd gy systm ad t s sutal to pst su systm avo y xpotal futo us t ppl of gy dyam modlg ad modl a al to povd t fftv tools fo modlg t tas of t al wold SCG 5 Gy-Fuzzy Clustg tod s s wll ow tat t opt of a fuzzy st fst aos fom t study of polms latd to patt lassfato s t ogto of patts s a mpotat aspt of uma ppto w s a fuzzy poss atu Fuzzy lassfato mtods may lassfd to t followg fou atgos[5]: 4

9 - tods asd o fuzzy lato; - tods asd o fuzzy patt matg podus; - tods asd o fuzzy lustg podus; ad - Ot mtods O lustg mtods t pmay ojtv of lustg tqus s to patto a gv data st to so-alld omogous lusts tm omogous mas tat all pots t sam goup a los to a ot ad a ot los to pots ot goups Clustg algotms may usd to uld patt lasss o to du t sz of a st of data wl tag lvat fomato[6] Fom a patal pot of vw gy systms ad fuzzy systms a foud vyw ad t pstato of lusts y gy-fuzzy lustg mtod low may sm mo appopat ta stuatos Suppos { s a sampl st w a lmt s alld a sampl I tms of patal osvato vy sampl as dxs { us asd o om 3 ad Coollay 3 w a al to ostut t td latoal gad matx dotd y { w dss t appoxmatss ad smlaty amog sampls o sampl st Ovously t appoxmatss ad smlaty a fuzzy opts ad t td latoal gad s a fuzzy quatty 0 ad t td latoal gad matx s a fuzzy lato o st Baus satsfs tat s t s flxv symmt ad tastv tfo s a fuzzy quvalt lato o I pat w usually us t tat s [ γ ] ξ s a smlaty lato o I ts as t xsts ad p U p Cosd tat t α -ut s a fuzzy quvalt lato o s a oday lato α [ 0] us w a asly pov tat quvalt lato f α α of s a lato Basd α [ 0 ] w a α s a quvalt al to patto a gv data st to so-alld omogous lusts[7-9] 6 Dmostatg Exampls 6 Foastg t floodwat ua Rv ua Rv s loatd t sout of Fuyag 4

10 Dstt of u ov Ca s v was a pla famous fo ts mo flood stoy I 990 usg t SCG modl JSu t al [0] sussfully foastd t atastop flood appd ua Rv 99 ad ot yas s a xampl f t dsag of v s ov 7000 s t avy flood wll app ua Rv a aomal valu ξ as 7000 yas as follows: 3 / 3 / s ad wt dow t patula { w t dsag of t ua Rv s mo ta 7000 Basd 3 / o s W may tasfom to { 8 { w a ostut t SCG foastg modl w ag a 0 a a w a ota t foastg sult of t tm w t floodwat wll ta pla xt tm tat wll app 9964 O t ass of foastg sult aov t flood otol statgy was ta adva so t avy losss w dud gatly Fuyag Dstt 6 Dyam gy-fuzzy lustg Suppos { s t st of fv popl ad a lato o s famla odg to gy-fuzzy lustg mtod mtod aov w av t td latoal gad matx It s la tat w ad Su a matx s a fuzzy smlaty lato o us U p p w s a fuzzy quvalt lato Basd o a dvdd to som lusts Coos α ad 04 w av ad ospodg lust: α 43

11 { 3 { 4 { 5 ; { { { { { { { ; { { { { ; { { ; { 3 4 W a tat t lustg sults otad aov a t sam as usual fuzzy lustg aalyss[5] s s y tos mtod aov t gy-fuzzy lustg mtod dvlopd ts pap s a vy usful tool of data mg 5 5 Rfs []YC Gy systm ad ts dyam modlg odgs of t Itatoal SE Cof o Sgal Data Systms todologs & pplatos CaluttaIda D7-999SE ss Vol []Y YC ad SFu oy of gy systms: aptug utats of gy fomato yts Vol33004 No pp96-8 [3]JDg Cotol polms of gy systms Systms & Cotol tts Vol98 No5 pp88-94 [4]YC Gy dyams of t systm of a og ma Joual of uazog Uvsty of S ad ology Cs Vol098 No6 pp7- [5]YC Systm loud ad ts gy modl odgs of t Fout Japas-So Sappoo Itatoal Cof o Comput pplatos oado Uvsty Sappoo 990 pp38-4 [6]YC d latoal aalyss ad gy dyam modlg odgs of t wlft Euopa tg o Cyts ad Systms Rsa Uvsty of Va Wold Stf 994 pp9-4 [7]YC WZC ZJC Gy stat modl ad foastg otol of omplatd systms odgs of t 5 t Itatoal Cof o Systms S Spt Wolawolad pp4-50 [8]YC Utaty aalyss ad gy modlg Utaty odlg ad alyss ISU 90 IEEE Comput Soty ss 990 pp [9]WZC ZJC Z Zou ad YC ultfato poss aalyss ad SCG mv foastg modl Joual of Wua Uvsty of ology Cs Vol8004 No pp

12 [0]YC Jog ad ZJC appoa to modal otol of gy systms SS 998 Vol33 pp47-56 []YC Fuzzy otol systm wt td latoal pdto Joual of Fuzzy atmats Vol994 No4 pp []YC Gal systms studs ad gy dyam modlg Gal Systms Studs ad pplatos uazog Uvsty of S ad ology ss 997 pp-9 [3]ZJC WZC QC ad YC oo sgal possg ad gy modlg tqu odgs of Egtt Itatoal Cof o Systms Egg ugust as Vgas NvadaUS pp5-56 [4]WZC ZJC QC ad YC ppoas to gy pdto ad otol of vomtal systms ossgs of Egtt Itatoal Cof o Systms Egg ugust as Vgas NvadaUS pp5-57 [5]Gas ad FDspot ompaso of som mtods of fuzzy lassfato o al data odgs of IIZU 999pp [6]DDus ad ad Fuzzy Sts ad Systms-oy ad pplato Nw Yo 980 [7]ZJC QC WZC ad YNWag Gy la pogammg yts Vol33004 Nopp38-46 [8]Jog YC ad Qa pplato of td latoal gad to fuzzy lustg dvas Systms S ad pplatos Spal Issu 996pp38-43 [9]YYC Fuzzy atmats Cs uazog Uvsty of S ad ology ss 986 [0]JSu ad JZao applato of t modl of tm gwa aa odgs of t Itatoal Cof o Systms Cotol Ifomato todologs & pplatos SCI 94 uazog Uvsty of S ad ology ss 995 pp98-30 SCG aout foastg t floodwat t ua as ad t stog floodwat 45

CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane.

CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane. CBSE CBSE SET- SECTION. Gv tht d W d to fd 7 7 Hc, 7 7 7. Lt,. W ow tht.. Thus,. Cosd th vcto quto of th pl.. z. - + z = - + z = Thus th Cts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl.