Intrnational Confrnc on Computr Systms and chnologis - CompSysch 06 Rcursiv Estimation of Dynamic im-varying Dmand Modls Alxandr Efrmov Abstract: h papr prsnts an implmntation of a st of rcursiv algorithms and thir modifications for stimation of th dynamic part of a pric modl h rasons for ral tim modl stimation of th sals dynamics ar considrd As th mart bhaviour varis a numbr of modifications ar applid to p th stimations snsitivity with rspct to th currnt mart bhaviour Also problms with applying th considrd procdurs for modl updating ar discussd and solutions ar prsntd y words: Rcursiv Estimation, Dmand Modl, im-varying Systms, Sals Forcasts INRODUCION h considrd dmand modl is a mathmatical rlation btwn th prics, discounts, ads and displays of a givn product and of its cross rlatd products h updatd modls ar dsignd to forcast product sals from a slctd group of products Usually th products ar collctd into product catgoris In a givn stor or a group of stors such division of th whol st of products is ncssary for dcrasing th modl s paramtrs dimnsion in rasonabl limits h modls discussd in th papr comput th forcastd sals into two stps First stp is to stimat th futur sals basd on static modls h obtaind stimats ar usd as an initial forcast of th unit sals h static part handls th main factors gathrd by th rtailrs (such as product pric, ads, displays, discounts) For th initial sals forcast is usd a modification of th modl of Blattbrg-Wisniwsi h scond stp of th forcast provids an improvd rprsntation of th mart bhaviour, taing into account th dynamic aspct of th mart his stp is a corrction of th stimatd sals using dynamic modls For th scond stp rgrssion modls ar usd with input th diffrnc btwn th forcastd sals, computd by th corrsponding static modls and th baslin product sals h output of th rgrssion modls is th corrctd unit sals forcast his papr is focusd on th ral tim stimation of th scond part of th dmand modls h rcursiv modl updating is ncssary bcaus ordinarily ar obsrvd thousands of products and it mas th idntification procdur too slow Also aftr th modl stimation could b applid othr actions, such as optimal sals forcast and pric optimization, so th tim for modl updating has to b dcrasd h rcursiv procdurs can not b adjustd manually for a grat numbr of products his is a problm bcaus th modls quality strongly dpnds on th choic of procdurs paramtrs maintaining th stimators snsitivity for tim-varying systms dynamics As th considrd systm is naturally tim-varying (th mart bhaviour dpnds on a non-stationary nvironmnt and th sts of products and cross ffcts ar not fixd too), th problm with stimators snsitivity is invstigatd and appropriat solutions ar prsntd RECURSIVE ESIMAORS AND HEIR MODIFICAIONS FOR DYNAMIC DEMAND MODELS UPDAING h prsntd algorithms us information about th modl structur and paramtrs that ar obtaind by an automatic systm idntification cycl [3] his procdur is basd on running of a st of bloc-mthods for a st of modls with diffrnt ordrs A validation critrion is usd for th bst modl dtrmination h rcursiv approach prsntd blow is a continuation of this bloc-idntification procdur - V7- -
Intrnational Confrnc on Computr Systms and chnologis - CompSysch 06 Rcursiv stimators For updating of th dynamic part of dmand modls th following algorithms ar applid [, 2]: Rcursiv last squars (RLS) Rcursiv gnral last squars (RGLS) Rcursiv xtndd last squars (RELS) Modification of th rcursiv xtndd last squars mthod basd on a postrior rror (MRELS) Rcursiv xtndd matrix last squars (REMLS) Rcursiv forcasting rror (RFE) for ARMAX modls Part of th procdurs stimats th sam typ of rgrssion modls hs stimators ar compard and a st of th most appropriat stimators is dtrmind In th last subsction ARMAX modls ar stimatd by RFE mthod Rcursiv Last Squars RLS stimats th paramtrs of ARX modl d A ( q ) y B ( q ) q u +, () whr u, y and ar th input, output and th rsidual in momnt ; d is tim dlay and polynomials A ( q ) and B ( q ) ar of ordrs na and nb rspctivly h vctor form of () is y ϕ +, (2) whr [ a a a b b b ],, 2,, 2, [ y y u u u ] ϕ y 2 na d d d nb+ ar th vctor of modl paramtrs and th rgrssion vctor rspctivly h algorithm consists of th following stps: Computing th rror basd on ˆ : ε ˆ y ϕ 2 alman gain dtrmination: G P ϕ ( + ϕ P ϕ ) 3 Updating th paramtr vctor: ˆ ˆ + Gε 4 Updating th covarianc matrix: P ( I Gϕ ) P h initial conditions of th rcursiv procdur ar: * Initial paramtrs stimation is chosn to b ˆ 0 0, whr obtaind by th idntification cycl for data sts with maximum lngth N 0 Initial rgrssion vctor is constructd basd on th last availabl data N Initial valu of th covarianc matrix is [ ] 0 P ϕ ϕ 0 i 0 Rcursiv Gnral Last Squars If th rsidual is a colour nois (in this cas it will b dnotd by i i * 0 is an stimation, ), th abov algorithm provids biasd stimations o avoid this problm, diffrnt modification of RLS can b applid In RGLS, additional forming filtr is applid to ta into account th colour dynamics of, h systm bhaviour is prsntd by ARARX modl c A ( q ) y d B ( q ) q u + (3) D ( q ) c - V7-2 -
Hr with Intrnational Confrnc on Computr Systms and chnologis - CompSysch 06 h vctor form of th rgrssion quation is c, D ( q ) y + c, ϕ, [ a a a b b b ],, 2,, 2, [ y y u u u ] y 2 na d d d nb+ ϕ h abov quation can b writtn as with f, ϕ f,, (4) y + ϕ [ y y u u u ] f, y f, f, 2 f, na f, f, f, nb+ f, D ( q y, u f, D ( q ) u y ) h vctor form of filtr s rgrssion quation is c, d, d, ϕ + (5) h filtr s paramtr vctor and th corrspondd rgrssion vctor ar [ d d ] and [ ] d, d, 2, nd, ϕ d, c, c, 2 c, nd RGLS is rducd to two RLS stimators applid to modls (4) and (5) h initial conditions ar chosn in th sam way as in RLS Rcursiv Extndd Last Squars RELS is a mthod that stimats ARMAX modls d ) y B ( q ) q u + C ( q A ( q ) (6) Hr th vctor of modl paramtrs and th rgrssion vctor ar xtndd with th filtr paramtrs and th valus of rspctivly, i ϕ [ a a a b b b c c c ],, 2,, 2, [ y y y u u u ] 2 na d d, 2, d nb+ h obtaind modl form is th sam as (2), but and ϕ hav diffrnt structurs h procss rprsnts all uncrtaintis such as masurmnt nois and unmodlld systm dynamics If ˆ is not optimal, th systm dynamics, which is not accountd in th currnt modl lads to colour rsidual o gnrat th rgrssion vctor an stimation of is ncssary In RELS is rplacd by ε y ϕ Modification of RELS using a postrior rror Othr variant to construct ϕ is to us th postrior rror ε p y ϕ ˆ, his modification has bttr prformanc as ε p, is a mor prcis stimation of than ε usd in th standard RELS Rcursiv Extndd Matrix Last Squars REMLS is a mthod that stimats ARARMAX modls d C ( q ) A ( q ) y B ( q ) q u + (7) D ( q ) o obtain th vctor form of th rgrssion quation, th vctor of optimal modl ˆ nc,, 2 nc - V7-3 -
Intrnational Confrnc on Computr Systms and chnologis - CompSysch 06 paramtrs and th rgrssion vctor ar xtndd with th filtr paramtrs and th valus of c, and rspctivly, i ϕ [ [ a, c, y a c 2, 2, y 2 2 a c nc, b d y,, nc na b d 2, 2, u d c, u b d nd, d c, 2 ] [ ], ab, h obtaind modl form is th sam as (2), but and again Hr c, and ar rplacd by thir stimats ε and ε c, y ϕ ab, ˆ ab, and ε ε c, ϕ ˆ cd, cd, u cd, d nb ] [ ϕ ϕ ] c, nd ab, cd, ϕ hav diffrnt structurs ε c, computd as Rcursiv Forcasting Error Mthod h applid RFE is dductd for ARMAX modl stimation h diffrnc btwn RFE and RELS is in th rgrssion vctor, which (for ARMAX modls) is filtrd with a forming filtr, i ϕ f, ϕ C ( q ) For constructing th rgrssion vctor ϕ again th postrior rror ε p, is usd as an stimation of o avoid obtaining of unstabl modls during th rcursiv stimation, an additional procdur [2] for ach algorithm is applid At ach tim instant a modl stability chc is applid If th currnt modl is unstabl, th roots of th charactristic polynomial ar shrind in th unit circl Modifications of th Rcursiv Estimators h stimators snsitivity dpnds on th covarianc matrix P If th lmnts of P ar too small, th algorithm bcoms inrtial with rspct to th nw data and th rat of paramtrs updating dcrass h following modifications ar rlatd to changs in th updating rul of th covarianc matrix in such a way that th lmnts of P ar big nough to maintain th stimators snsitivity For P can b writtn P ϕ ϕ P P P (8) + ϕ P ϕ h following modifications ar applid for ach stimator [2]: Modification with adding of a positiv dfinit matrix to th covarianc matrix Modification with a constant forgtting factor Modification with a variabl forgtting factor Modification with a constant trac of th covarianc matrix Modification with a constant covarianc matrix Modification with adding of a constant positiv dfinit matrix to th covarianc matrix h updating quation of th covarianc matrix in this modification is P ϕ ϕ P P P + + ϕ P ϕ S, if tr( P ) trpmin, whr trp min is th lowst limit of th tr (P) If tr( P ) > trpmin, th covarianc matrix is updatd by (8) - V7-4 -
Intrnational Confrnc on Computr Systms and chnologis - CompSysch 06 Modification with a constant forgtting factor h quation (8) in this modification has th form P ϕ ϕ P P P ρc ρc + ϕ P ϕ h factor ρ c is a positiv numbr, but lss thn If ρ c, th abov quation bcoms th sam as (8) With dcrasing of ρ c, th stimations snsitivity with rspct to data incrass, but in this cas ˆ is mor snsitiv to th nois Also in som cass th procdur can bcom unstabl, what is a disadvantag of this modification Modification with a variabl forgtting factor h disadvantag of th abov modification can b avoidd with th variabl forgtting factor h quation (8) in this modification has th nxt form P ϕ ϕ P P P ρ v, ρv, + ϕ P ϕ h variabl forgtting factor ρ ν, can b dtrmind in diffrnt ways An appropriat updating rul is ϕ G 2 ρv, 2 (9) σ N E 2 N E is th ffctiv obsrvation intrval and σ is th varianc of On disadvantag of th procdur is that an incras of can b causd by othr factors that hav an influnc on th rsidual, but not by changs in th systm dynamics In this cas th stimation rror will incras in spit of th tim-invariant charactr of th systm bhaviour h concrt ralization uss a floating window for rcursiv dtrmination of th varianc of Modification with a constant trac of th covarianc matrix his modification provids a constant stimators snsitivity P is updating from P ϕ ϕ P P P, ρ + ϕ P ϕ whr ρ is 2 ϕ P ϕ ρ ( + ϕ P ϕ ) tr( P ) Modification with a constant covarianc matrix In this cas P P P0 his modification is simpl and usful, but it is snsitiv to th choic of P 0 If th lmnts of P 0 ar too small th procdur will bcom inrtial ES DESCRIPION AND RESULS h prsntd abov algorithms ar tstd with a ral datast containing data for a numbr of 2 products h information about th prics, discounts, ads, displays and sals is collctd wly h obsrvation intrval is 2 yars h first 84 ws ar usd for an initial idntification, ralisd by th mntiond abov idntification cycl h maximum modls ordr was 4, but th maximum ordr of D ( q ) polynomial in ARARX - V7-5 -
Intrnational Confrnc on Computr Systms and chnologis - CompSysch 06 modl was 8 h rmaining 20 ws ar usd for th ral-tim modls stimation h modification with a variabl forgtting factor was usd to maintain th stimators snsitivity For both initial and wly updatd modls ar dtrmind th valus of th validation critrion Varianc Accountd For (VAF) [4] by using of th last 20 ws VAFs for both modls of ach product ar compard in abl A numbr of xprimnts (not discussd hr) wr undrtan to dtrmin th most appropriat stimators and modification As a rsult for stimation of ARMAX modls RFE is chosn and th modification with variabl forgtting factor is applid abl VAF /Initial Modls/ VAF/Updatd Modls/ 0000 9997 0000 9943 9875 9905 7552 757 6978 7057 6400 622 6396 620 637 6526 680 6598 64 5892 6049 5862 5892 5849 5795 5767 5450 5304 5204 575 509 5087 4986 492 4240 4855 4067 4082 4058 4866 3678 4266 Fig Sortd valus of VAF for th initial modls (a) and for th updatd modls (b) CONCLUSIONS AND FUURE WOR In [3] wr mad important conclusions xplaining som spcific cass hr wr considrd th rasons for low modl accuracy obtaind for crtain products h main advantags of th rcursiv approach ar th ability to assss th tim varying systm dynamics and also th drastic dcras of th computation tim hr ar cass, whr th rcursiv procdurs do not provid modl improvmnt A rason for that could b inappropriat choic of th paramtrs taing part in (9) It is also possibl that optimal modl typ and structur can b diffrnt for that priod h dvlopmnt of an automatic idntification cycl for dmand modls whr th static part is sippd is an objct of a futur wor In ths rprsntations th modl input will b xtndd with th main significant factors gathrd by th rtailrs A disadvantag of th currnt dmand modls is that on rgrssion modl is usd for rprsntation of th whol mart dynamics Significant improvmnt is xpctd if apply multipl inputs singl output (MISO) modls as th dynamics btwn ach factor and th unit sals will b prsntd by diffrnt dynamic modl h MISO dynamic dmand modl is a mor gnral and closr to th natur of th considrd systm REFERENCES [] Garipov, E Systm Idntification U Sofia, 997 [2] Vlv, Adaptiv Systms Sofia, 995 [3] Efrmov, A An Automatic Idntification Cycl for Dynamic Dmand Modl Gnration Intrnational Confrnc on Computr Systms and chnologis CompSysch 2005, Var Bulgaria [4] Vrhagn, M, Vrdult, V Filtring and Systm Idntification: An Introduction U Dlft, 2003 ABOU HE AUHOR Alxandr Efrmov, Scintific Dvlopr at Rtail Analytics Bulgaria Ltd Phon: +359 02 863-485, -mail: alfrmov@yahoocom - V7-6 -