Fuzzy Time Series Forecasting based on Grey Model and Markov Chain

Transcription

1 IAENG Iteatoal Joual of Appled athematcs 46:4 IJA_46_4_08 Fuzzy Tme Sees Foecastg based o Gey odel ad akov Cha Xagya Zeg a Shu Jg Jag Abstact Ths atcle deals wth the foecastg of teval ad tagula fuzzy umbe sees based o accumulato method G ( ) (AG ( )) ad akov cha Because AG ( ) oly sut pecse umbe the teval ad tagula fuzzy umbe sees ae tasfomed to pecse umbe sees The tasfomato pocess matas the tegty ad the elatve posto of the bouday pots of the fuzzy umbe Because G ( ) s ot sutable fo the log-tem ad stogly fluctuatg pedcto akov cha theoy s appled to modfy AG ( ) The teval pedctos of the Cosume Pce Idex ad the powe load show the effectveess of the models poposed the pape Idex Tems fuzzy umbe gey model accumulato method akov cha T I INTRODUCTION HE tadtoal pedcto models such as the autoegessve movg-aveage model the blea model ad the olea autoegessve model ae oly sutable fo the pecse umbe sees I may applcatos such as the electc powe load the pces of ol stock o gold the daly a tempeatue ad the exchage ate the obseved data fluctuate at ay tme The usg teval fuzzy umbes to epeset these data s moe easoable tha usg pecse data I [] Zadeh toduced fuzzy umbe fstly ad fuzzy mathematcs has bee appled to may felds [] The commo types of teval fuzzy umbes ae bay teval fuzzy umbe tagula fuzzy umbe (o tay teval fuzzy umbe) ad tapezodal fuzzy umbe I ths atcle we cosde the foecastg of teval ad tagula fuzzy umbe sees Aothe poblem wth the tadtoal pedcto appoaches s that they eque a lage copus of data because these pedctos ae costucted o the assumpto about the dstbuto of the populato [3-5] It s dffcult to cofm auscpt eceved Decembe 05; evsed Jue 0 06 Ths wok was fuded by the Scece ad Techology Reseach Poects of Guagx Colleges (Gat No KY05YB3) the Natoal Scece Foudato of Cha (Gat No ) Guagx Dstct Natual Scece Fud of Cha (Gat No 04GXNSFAA8003 ad 04GXNSFAA800) the Natual Scece Foudato Poect of CQ CSTC (Gat No cstc04cya00054) ad the Fudametal Reseach Fuds fo Chogqg Educato Commsso (Gat No KJ503) X Y Zeg s wth the School of athematcs ad Computatoal Scece Gul Uvesty of Electoc Techology Gul Cha (e-mal: zegxyhbyc@ 63com) Shu s wth the School of athematcal Scece Uvesty of Electoc Scece ad Techology of Cha Chegdu Cha (e-mal: shul@uestceduc) J Jag s wth Chogqg Uvesty of Ats ad Sceces Chogqg Cha the populato dstbuto wth small sample Fo ths poblem Deg poposed gey models focusg o ucetaty chaactezed by poo fomato ad a small sample [6 7] G ( ) s oe of the basc gey models It ca be bult o the bass of at least fou data pots ad has good pecso [8-] Howeve ust as may othe foecast methods gey models ae oly applcable to pecse umbes I the modelg pocess of G ( ) some calculatos about the fuzzy umbes such as accumulated geeatg opeato vese accumulated geeatg opeato ad matx multplcato ad veso caot be acheved Theefoe the classcal modelg pocess s ot sutable fo the fuzzy umbe sees I the pape the teval fuzzy sequece s tasfomed to eal sees Fst G ( ) models ae bult o these pecse sees ad the though the estoato pocess the pedcted values of the teval fuzzy umbe sees ae obtaed The pedcto cuve of G ( ) ca oly eflect the tegal developmet tedecy of the sees Thus t oly suts the sees wth weak fluctuato [3-5] Fo the fluctuatg sees G ( ) should be modfed I [6] the akov cha foecastg model s combed wth G ( ) ad the combed model has good effect o the fluctuatg eal sees I ths pape the pedcto esults of G ( ) model wll be futhe modfed va the akov-cha foecastg model whe the fuzzy sees has stogly fluctuato The est of the atcle s ogazed as follows I Secto we povde the paamete estmatos based o accumulato method (A) ad pove some of ts popetes I Secto 3 the tasfomato pocesses of the teval ad tagula fuzzy umbe sees ae povded The modelg pocess of the accumulated method G ( ) (AG ( )) s povded Secto 4 I secto 5 the modfed pocess va akov cha pedcto model s poposed I Secto 6 two applcable examples ae gve to llustate the effectveess of the model poposed the pape Fally coclusos ae dscussed Secto 7 II BASIC THEORY OF ACCUUATION ETHOD The accumulato method (A) was fst toduced by Itala mathematca P achs 778 It s a ew method of the paamete estmato ad has bee appled to may felds I [7] Zeg ad Xao toduced the A method to G ( ) ad aalyzed ts mobdty poblem They cocluded that the codto umbe of the A method s less tha the least squae method et X { x() x() x( )} be a ogal sees The each-ode accumulato sum s defed as (Advace ole publcato: 6 Novembe 06)

2 IAENG Iteatoal Joual of Appled athematcs 46:4 IJA_46_4_08 () x( ) x() x() x( ) x( ) () x( ) x() ( x() x()) ( x() x() x () ( )) x( ) ad so o Thus each-ode accumulato sum s defed as follows whee ( ) ( ) x( ) x( ) () ( ) s called the -ode basc accumulato sum Geeally the tedous calculato s ot coveet fo the pactcal applcato I [8] Cao ad Zhag have povded the followg coveet fomula fo the accumulato sums ( ) x( ) C x( ) ( )( ) ( ) x( ) ( )! () ad ( ) C ( ) ( ) (3)! Next gve the pocess of paamete estmato usg the A method et { x ( t) x ( t) x ( t) y( t)} t m be goups of samples Based o these samples the paametes m of the followg lea model wll be estmated y( t) x ( t) x ( t) ( t) 0 (4) whee () t s a stochastc dstubace wth zeo mea value Fst the accumulato sum s opeated o both sdes of (4) Wth +m paametes the maxmum ode of the accumulato sum should be +m The we get whee m Y X (5) m () () ( m) Y y( t) y( t) y( t) t t t X x ( t) xm ( t) x ( t) xm ( t) x ( t) xm( t) () () () t t t () () () t t t ( m) ( m) ( m) t t t 0 m ad t t t Thus the paametes ae estmated as () t () t () t () () ( m) ˆ X Y (6) It ca be easly show that ˆ s the lea ubased mmum vaace estmato of Fom (6) we ca see that A s based dectly o the samples whch avods the assumptos about the eo Next we gve the geometc meag of the A method based o the theoy of the cete of gavty Assume that each data of X { x() x() x( )} epesets a pot of the umbe axs The cete of gavty of oe pot s the pot tself: x x() The cete of gavty of two pots ( x() ad x () ) s The cete of gavty of thee pots ( x () () dvdes the segmet betwee amely x() x() x() x() x x ad x (3) ) T ad x(3) to : x() x() x ( x(3)) ( x() x() x(3)) 3 3 ad so o the cete of gavty of X { x() x() x ( )} s () x x ( x () x () x ( )) (Advace ole publcato: 6 Novembe 06)

3 IAENG Iteatoal Joual of Appled athematcs 46:4 IJA_46_4_08 () ( ) x( ) Fom () ad (3) we have x ad () () () x ( ) s called the fst-ode cete opeato Nomally the cete of gavty of ( m ) pots dvdes the segmet betwee the cete of gavty of the top m pots ad the cete of gavty of the latte pots to : m Thus the cete of gavty of the followg data: x () s x() x() x() x() x(3) 3 x () () () ( () ()) ( ) ( ) x x x x x x () whch s called the -ode cete opeato Ad so o ( ) ( ) x ( ) s called the -ode cete opeato Fom the above aalyss we kow that the geometc meag of A s fo a gavty-cete le based o the kow sample pots oeove the method of A acheves the data smoothg at most Thus the cosevatsm s ehaced ad the ll effects of outles ae weakeed III TRANSFORATION OF THE FUZZY NUBER SERIES The foecastg models ca be bult o the boudaes of the fuzzy umbe dectly Ths dea howeve wll esult the boke tegty of the fuzzy umbes ad the elatve postos of the dsodeed boudaes of the teval fuzzy umbe the foecastg esults Theefoe we cosde tasfomg the teval fuzzy umbe fst ad the usg the tasfomed sees to buld the foecastg model A Tasfomato of Iteval Fuzzy Numbe Sees ~ et X { ~ x() ~ x () ~ x ( )} be the teval fuzzy sequece whee x ( ) [ x ( ) x ( )] Hee x () ad x () U ae the lowe ad uppe boudaes espectvely The md-pot ad the legth of the teval umbe espectvely ae U et the md-pot sequece ( ) ad the legth sequece ( ) espectvely be { m() m() m( )} { l() l() l( )} Ad ow tasfom the teval fuzzy sequece to two pecse sequeces as ~ X { ~ x() ~ x () ~ x ( )} { m() m() m( )} { l() l() l( )} The bouds of the teval fuzzy umbe ca be estoed espectvely as l () x ( ) m( ) ad Obvously x ( ) x ( ) U l () xu ( ) m( ) (9) whch esues the elatve postos of the bouds of the teval umbe the foecastg esults B Tasfomato of Tagula Fuzzy Numbe Sees et the tagula fuzzy sequece be whee ~ X { ~ x() ~ x () ~ x ( )} x ( ) [ x ( ) x ( ) x ( )] U The gavty cete of the fuzzy umbe o the mea value seves as the dex of the compaso ad sot ode of the fuzzy umbe Fo the tagula fuzzy umbe the gavty cete s calculated as f() x( ) x ( ) xr( ) 3 (0) The legths betwee the thee bouds ae calculated as p( ) x ( ) x ( ) ad q( ) x ( ) x ( ) () et the gavty cete sequece ( F ) ad the tow legth sequeces ( P ad Q ) espectvely be U ad m () x( ) xu( ) (7) l( ) x ( ) x ( ) (8) U F { f () f () f ( )} P { p() p() p( )} Q { q() q() q( )} ad the tasfom the tagula fuzzy sequece to thee pecse sequeces as (Advace ole publcato: 6 Novembe 06)

4 IAENG Iteatoal Joual of Appled athematcs 46:4 IJA_46_4_08 ~ X { ~ x() ~ x () ~ x ( )} F { f () f () f ( )} P { p() p() p( )} Q { q() q() q( )} The educto pocess s gve by p( ) q( ) x ( ) f ( ) 3 3 p( ) q( ) x ( ) f ( ) () 3 3 p( ) q( ) xu ( ) f ( ) 3 3 Obvously x ( ) x ( ) x ( ) whch esues the U elatve postos of them the foecastg esults Fom (7) (8) (0) ad () the ew tasfomed sees ae smultaeously affected by the bouds of the fuzzy umbe ad actually they ae the weghted mea values of the bouds These chaactestcs mata the tegty of the fuzzy umbe ad weake the umpg degee of the boudaes Thus the smoothess of the tasfomed sees s bette tha the aw bouday sees I addto the educto pocesses (9) ad () esue the elatve postos of the boudaes of teval umbes ae ot dsodeed the pedcto IV PREDICTION PROCESS BASED ON AG ( ) The accumulato method G ( ) (AG ( )) s bult usg the tasfomed sees ad the though the educto pocess the pedcted values of the fuzzy umbes ae obtaed Fst the modelg pocess fo the md-pot sees { m() m() m( )} s gve as follows The gey dffeetal equatos of G ( ) based o ae whee ad m a z (3) m () () () ( ) ( ) () () () z m m ( ) 05( ( ) ( )) 3 (4) m() m() m( ) m( ) (5) Fst the accumulato sum s opeated o both sdes of (3) Assume that the hghest ode of the accumulato sum s deoted by Because (3) has tow paametes t s ceta that () () () () m a z ( ) ( ) () () () () m a z ( ) ( ) (6) Usg the esults [8] descbed eale we have the calculato fomulas of the followg accumulato sums () () () z ( ) z ( ) () m ( ) m( ) () () () () z ( ) C z ( ) ( ) z ( ) et X ad () m( ) C m( ) ( ) m( ) () () C ( ) ( ) C ( ) ( ) () () () z () () () z a A Y The the matx fom of (6) ca be expessed as ad the paamete estmato s obtaed as ˆ () () m () m () X A Y (7) aˆ A X Y ˆ (8) Fom (3) the pedctve fomula s deduced as follows Due to tasfomed as () () z m ( ) 05( ( ) m () ( )) (3) s a () () m( ) ( m ( ) m ( )) Fom (5) we have a () m( ) ( m( ) m ( )) (Advace ole publcato: 6 Novembe 06)

5 IAENG Iteatoal Joual of Appled athematcs 46:4 IJA_46_4_08 The Fom (5) () a m ( ) m () 05a () () m m m ( ) ( ) ( ) so () a m ( ) m () 05a a m m 05a () ( ( ) ( )) () a m ( ) a m( ) 05a 05a a m( ) m ( ) 05a a m ( ) a a m() a a a m() a 05a ( a ) ( a m()) ( a ) (9) Afte the estmato s obtaed fom (8) ad m () s adopted as the statg value (9) ca be used as the pedcto fomula Namely the pedcted values of m () s ˆ ˆ ˆ ( aˆ ) ( a ) ( a m()) m ˆ () 3 (0) Fo the othe tasfomed sequeces ( F P ad Q ) AG ( ) ca be establshed smlaly Fally fom (9) ad () the pedcted values of the bouday values of the fuzzy umbe ca be acheved V ODIFIED PROCESS BASED ON ARKOV CHAIN The pedcto cuve of G ( ) s a smooth cuve whch ca eflect the geeal developmet ted of the aw sees The fluctuato ule of the sees s ot eflected Theefoe fo the stogly fluctuatg sees the foecast pecso of G ( ) s ot satsfactoy The akov-cha foecastg ca eflect the fluctuato ule of the sees ad ca be used to modfy the above poposed models Fo AG ( ) poposed Secto 4 the pedcto sees of the md-pot sees { m() m() m( )} s ˆ { mˆ () mˆ () mˆ ( ) } The modfed pocess of ˆ s gve as follows Fo the othe tasfomed sequeces ( F P ad Q ) the modfed pocess of the pedcto sees ( ˆ ˆF ˆP ad ˆQ ) s smla as ˆ Step Status patto Based o the atos m( ) mˆ ( ) the system s dvded to s statuses E : [ A B ] s Step Establsh the tasto pobablty matx et N be the umbe of aw data tasfeed fom Status E to umbe of E by k steps s E by k steps et N be the occuece E The the tasto pobablty fom N P s N The tasto pobablty matx s P P P s P P P s Pk ( ) Ps Ps Pss Step 3 -step pedcted values detfcato If m( ) mˆ ( ) s the status of E h ad max P () P () h the modfed values of m ˆ ( ) s calculated as If hl E to () m( ) mˆ ( ) ( Al Bl) () m( ) mˆ ( ) s the status of E h ad max P () does ot exst the modfed values of m ˆ ( ) s take as the expectato s m( ) mˆ ( ) [ Ph () ( A B)] (3) Step 4 -step pedcted values detfcato If m( ) mˆ ( ) s the status of E h ad max P () P () h hl h (Advace ole publcato: 6 Novembe 06)

6 Boudaes of CPI IAENG Iteatoal Joual of Appled athematcs 46:4 IJA_46_4_08 the modfed values of m ˆ ( ) s calculated as m( ) mˆ ( ) ( Al Bl) (4) fuzzy umbes ae used the pecso s satsfactoy Howeve we ca see that the fluctuato of the gve sees s weak Whe the sample s lage ad has stog fluctuato ths example caot llustate the effectveess of the poposed model If m( ) mˆ ( ) s the status of E h ad max P () s ot exst the modfed values of m ˆ ( ) s the expectato h 0 5 Raw Value AG() s m( ) mˆ ( ) [ Ph () ( A B)] (5) 0 05 Smlaly the mult-step pedcted values ca be obtaed 00 VI IUSTRATIVE EXAPES Example Small sample wth weak fluctuato The State Statstcs Bueau of Cha povdes Cosume Pce Idex (CPI) of each moth (See Cha Statstcal Yeabook) We take the mea value of twelve moths of oe yea as the md bouday of the tagula fuzzy umbe The mmum ad maxmum of twelve moths ae take as the lowe ad uppe boudaes of the tagula fuzzy umbe espectvely We take the ecods fom 00 to 005 as the aw sees to establsh AG ( ) ad the value 006 wll be pedcted The aw data fom 00 to 005 ae gve TABE I TABE I THE FORECAST RESUTS OF CPI Yea Raw Data AG ( ) 00 [ ] [ ] 003 [ ] [ ] 004 [ ] [ ] 005 [ ] [ ] 006 [ ] [ ] ARE 098% Fst due to the tasfomatos (0) ad () the thee tasfomed sees ( F P ad Q ) ae obtaed AG ( ) s bult usg F P ad Q espectvely The paamete estmates of AG ( ) by method of A ae F : aˆ F 0003 ˆ F Q : P : aˆ Q 0090 ˆ Q 734 aˆ P ˆ P 300 Fom (0) the pedcted values of F P ad Q ca be obtaed Fom estoato fomula () the pedcted sees fom 00 to 006 s calculated ad ae show Table The esults show that the aveage elatve eo (ARE) of AG ( ) s oly 098% I Fg the pedcto cuve s gve ad has good fttg effect Although oly fou tagula Yea Fg Raw values ad esults of AG() fo CPI Example age sample wth stog fluctuato Powe load keeps chagg ad ts foecastg s ot sutable to be expessed as a exact umbe Based o the eal umbe sees [9] ad [0] obtaed the teval pedcto by the degee of cofdece o coveage pobablty We have got the powe load data of oe dstct of Gul Cty Cha fom Septembe to Septembe 5 04 Fst oe day s dvded to fou tme buckets: 00:00-06:00 06:00-:00 :00-8:00 ad 8:00-4:00 whch ca epeset fou stages of the lfe of oe day The mmum value of powe load oe tme bucket s as the left bouday of the teval umbe ad the maxmum value s as the ght bouday The aw teval sees s show TABE II The pedcto cuves of AG ( ) ae show Fg We ca see that the tegal developmet ted of the teval sees s eflected but the fluctuato ule s ot show The aveage elatve eo (ARE) of the fttg values fom Septembe to Septembe 4 s 79% ad the aveage elatve eo (ARE) of the pedcto values o Septembe 5 s 4% Theefoe the accuacy of AG ( ) s ot satsfactoy fo the fluctuato sees Next the esults of AG ( ) wll be modfed va akov cha foecastg method Fst fom (7) ad (8) the tasfomed sees ( ad ) of the aw sees ad the tasfomed sees ( ˆ ad ˆ ) of the foecastg sees of AG ( ) ae calculated espectvely The calculated values ad the atos ( / ˆ ad / ) ˆ ae all show TABE II Step Status patto Based o the atos m( ) mˆ ( ) ad l( ) lˆ ( ) show TABE II the system s dvded to fou statuses show TABE III Step Establsh the tasto pobablty matx Fom TABE II the -step tasto pobablty matx of / ˆ ad / ae ˆ espectvely as follows (Advace ole publcato: 6 Novembe 06)

7 IAENG Iteatoal Joual of Appled athematcs 46:4 IJA_46_4_08 No Tme TABE II THE TRANSFORED SERIES AND THEIR RATIOS Raw Sees AG() Rato (%) Raw values ˆ ˆ / ˆ / ˆ 9-: 00:00-06:00 [9 43] : 06:00-:00 [3 59] : :00-8:00 [43 73] : 8:00-4:00 [6 79] : 00:00-06:00 [37 48] : 06:00-:00 [37 60] : :00-8:00 [54 83] : 8:00-4:00 [7 88] : 00:00-06:00 [46 6] : 06:00-:00 [47 53] : :00-8:00 [49 66] : 8:00-4:00 [48 74] : 00:00-06:00 [30 4] : 06:00-:00 [35 56] :0 6:00-:00 [43 63] : 8:00-4:00 [5 63] : 00:00-06:00 [3 4] : 06:00-:00 [33 50] : :00-8:00 [4 55] : 8:00-4:00 [48 73] TABE III THE STATUS PARTITION / ˆ / ˆ Status Rage Status Rage E 66%-85% E 3%-6% E 85%-05% E 6%-9% E3 05%-5% E3 9%-% E4 5%-45% E4 %-54% P P / 3 / 3 0 () / 0 / 4 / / / / / 4 0 / 4 () / 5 0 / 5 Step 3 -step pedcted values detfcato I TABE II mˆ (7) 5 ad l ˆ(7) 404 Next the values ae modfed espectvely Due to m(6) / mˆ (6) 0863% t s E3 (05%-5%) Fom the 3d ow of P () the ato should be tasfeed to E (66%-85%) Thus the modfed value of m ˆ (7) s calculated as m(7) mˆ (7) (66% 85%) Due to l(6) / lˆ (6) 859% t s E (6% - 9%) Fom the d ow of P () the ato should be tasfeed to E (3% - 6%) Thus the modfed value of l ˆ(7) s the expectato ˆ l(7) l(7) (3% 6%) The due to (9) the educto fomula the foecastg values of x(7) ad xr(7) based o AG ( ) ad akov cha (AG ( )) ae acheved as follows (0) x (7) (0) x R (7) Step 4 -step pedcted values detfcato Fom TABE II the mult-step tasto pobablty matx of / ˆ ad / ae ˆ espectvely as follows P 0 / 3 / 3 0 / / 5 / 5 () / 4 3 / (Advace ole publcato: 6 Novembe 06)

8 Powe oad IAENG Iteatoal Joual of Appled athematcs 46:4 IJA_46_4_ Raw Sees AG() AG ( ) No Fg Iteval fuzzy pedcto cuves of AG ( ) ad AG ( ) fo powe loads P P P P P / 4 0 / / 4 () / 4 / 0 / 4 / 3 0 / 3 / 3 / 4 / / 4 0 (3) 0 3 / 4 / / 4 / 4 / (3) 0 / 3 / 3 / 3 / 3 0 / 3 / / 4 / / 4 0 (4) 0 / 4 / / / 0 / 0 / 3 0 / 3 (4) / 3 0 / 3 / 3 0 / 3 / 3 / 3 Smla to Step 3 the modfed values of xˆ () ad xˆ R() 890 ae obtaed ad show TABE IV The aveage elatve eos (ARE) ae deceased fom 4% to 987% I Fg the pedcto cuve of AG ( ) eflects the fluctuato ule of the powe load Thus the modfed pocess based o akov cha s effectve TABE IV RESUTS OF AG ( ) FOR POWER OADS (Ut: W) No AG() RE (%) 7 [ ] [ ] [ ] [ ] ARE 4% No AG() RE (%) 7 [ ] [ ] [ ] [ ] ARE 987% VII CONCUSION I the pape the teval fuzzy sequece s tasfomed to two pecse sequeces ad the tagula fuzzy umbe sees to thee pecse umbe sees These tasfomed sees actually ae the weghted meas of the bouday pots of the fuzzy umbe It ot oly matas the tegty of the fuzzy (Advace ole publcato: 6 Novembe 06)

9 IAENG Iteatoal Joual of Appled athematcs 46:4 IJA_46_4_08 umbe but also weakes the umpg degee of the fuzzy sees ad the the foecastg accuacy ca be ceased oe mpotatly the tasfomato esues the elatve posto of the boudaes of the fuzzy umbe the foecastg esults of AG ( ) I Example the tagula fuzzy umbe sees has oly fou data but the poposed model based o AG ( ) has good pecso Howeve AG ( ) s ot sut the pedcto of the sees wth stog fluctuato I Example the fuzzy sees about powe load has stog fluctuato The pedcto cuve of AG ( ) ca oly eflect the tegal developmet ted of the sees but ot eflect the fluctuato ules But the modfed pocess va akov cha has good effectveess The pedcto pecso s geatly mpoved though the modfed pocess Thus the combed model based o AG ( ) ad akov cha suts the pedcto of fluctuatg fuzzy sees [0] X Zhao ad W Cha The key techology fo gd tegato of wd powe: dect pobablstc teval foecasts of wd powe Southe Powe System Techology vol 7 pp 8 03 REFERENCES [] A Zadeh Fuzzy sets Ifomato ad Cotol vol 8 pp [] R owe athematcs ad Fuzzess Fuzzy Sets ad Systems vol 7 pp [3] K Wedewad ad C Adks Ivetoy of load models electc powe systems va paamete estmato Egeeg ettes vol 3 pp [4] S P eeaksh ad S V Raghava Foecastg ad evet detecto teet esouce dyamcs usg tme sees models Egeeg ettes vol 3 pp [5] S P Sdoov A Revutsky ad A Fazlev Stock volatlty modellg wth augmeted GARCH model wth Jumps IAENG Iteatoal Joual of Appled athematcs vol 44 pp 0 05 [6] J Deg Cotol poblems of gey system System ad Cotol ettes vol pp [7] J Deg Itoducto to gey system theoy Joual of Gey System vol pp [8] S F u ad J Deg The age sutable fo G ( ) Joual of Gey System vol pp [9] SF u J Foest Y Yag Advaces gey systems eseach Joual of Gey System vol 5 pp 8 03 [0] Wu S F u ad Yao The effect of sample sze o the gey system model Appled athematcal odellg vol 37 pp [] C Chag D ad Y Huag A ovel gay foecastg model based o the box plot fo small maufactug datasets Appled athematcal Computato vol 65 pp [] Wu S F u ad Z G Fag Popetes of the G() wth factoal ode accumulato Appled athematcal Computato vol pp [3] Z ao J Su Applcato of Gey-akov model foecastg fe accdets Poceda Egeeg vol pp [4] Y H C C Chu ad P C ee Applyg fuzzy gey modfcato model o flow foecastg Egeeg Applcatos of Atfcal Itellgece vol 5 pp [5] H X u ad D Zhag Aalyss ad pedcto of hazad sks caused by topcal cycloes Southe Cha wth fuzzy mathematcal ad gey models Appled athematcal odellg vol 36 pp [6] Z ao ad J Su Applcato of Gey-akov model foecastg fe accdets Poceda Egeeg vol pp [7] X Y Zeg ad X P Xao A eseach o mobdty poblem accumulatg method G ( ) odel Poceedgs of the Fouth Iteatoal Cofeece o ache eag ad Cybeetcs pp [8] D Cao ad S Zhag Itoducto to accumulato method Beg: Scece Pess 999 [9] Z J Dg ad D Wu A esemble model of the exteme leag mache fo load teval pedcto Joual of Noth Cha Electc Powe Uvesty vol 4 pp (Advace ole publcato: 6 Novembe 06)