Synthesis of Time Series Forecasting Scheme Based on Forecasting Models System

Transcription

1 Syhess of Tme Seres Forecasg Scheme Based o Forecasg Models Sysem Fedr Geche, Vladysla Kososky, Aaoly Bayuk 3, Sadra Geche 4, ad Mykhaylo Vashkeba Uzhhorod Naoal Uersy, Deparme of Cyberecs ad Appled Mahemacs, Uzhhorod, Ukrae (fgeche@homal.com, ashkebam99@gmal.com) Uzhhorod Naoal Uersy, Deparme of Iformao Maageme Sysems, Uzhhorod, Ukrae kosalad@gmal.com 3 L Polyechc Naoal Uersy, Deparme of Auomac Corol Sysems, L, Ukrae abayuk@gmal.com 4 Uzhhorod Naoal Uersy, Deparme of Ecoomc Theory, Uzhhorod, Ukrae sadra.geche@gmal.com Absrac. Ths arcle s dedcaed o he deelopme of me seres forecasg scheme. I s creaed based o he forecasg models sysem ha deermes he red of me seres ad s eral rules. The deeloped scheme s syheszed wh he help of basc forecasg models "compeo" o a cera me eral. As a resul of hs "compeo", for each basc predce model here s deermed he correspodg weghg coeffce, wh whch s cluded he forecasg scheme. Creaed forecasg scheme allows smple mplemeao eural bass. The deeloped flexble scheme of forecasg of ecoomc, socal, eromeal, egeerg ad echologcal parameers ca be successfully used he deelopme of subsaaed sraegc plas ad decsos he correspodg areas of huma acy. Keywords. Tred, forecasg model, me seres, fucoal, sep of forecas, auoregresso, eural eleme, eural ework. Key Terms. MacheIellgece, DecsoSuppor, MahemacalModel

2 Iroduco A he prese sage, for effece maageme of eerprses s ecessary o be able o predc he major reds socal ad ecoomc sysems, he ma ecoomc dcaors characerzg facal poso ad effcecy of he use of compaes produco resources. Esmaes ad forecass of he facal codo of he compay make possble o fd addoal resources, o crease s profably ad solecy. Problems of he aalyss ad he forecas of facal codo of he compay by meas of correspodg dcaors are a acual ask, because o he oe had hs s he resul of he compay, o he oher defes he precodos for he deelopme of he compay. Qualae forecas ges us a opporuy o deelop reasoable sraegc plas for ecoomc acy of eerprses. Uder marke codos, he adequae forecasg ad capacy plag of eerprses are mpossble whou workg ou ecoomc ad mahemacal models ha descrbe he use of aalable resources durg he operao of eerprses. To deerme sraeges for eerprse deelopme, calculao of forecass of ecoomc dcaors ad facors of orgazaos plays a mpora role. If here s relable formao abou he compay he pas, mahemacal mehods ca be appled o oba ecessary forecass. These mehods deped o he objeces ad dealed forecas facors; hey also deped o he erome. Varous aspecs of he heory, pracce, ad forecas of facal codo of a compay hae bee he subjec of research of may domesc ad foreg scess, such as Blak I.A [], Heyes V.M. [], Zaycheko Y.P. [3], Iakheko V.M. [4], Iakheko O.G. [5], Yarka N.M. [6], Tymashoa L. [7], Sepaeko O.P. [8], Tkacheko R.O. [9], Machuk A.V. [0], Hake J.E. [], Lews C.D. [], Box G.E. [3]. Whe forecasg he dcaors by whch he facal poso or effcecy of he compay s produco resources use are deermed, s mpossble o po ou a sgle "he bes" mehod of predco because he eral laws (reds) of arous dcaor sysems are dffere ad here arses he problem of choosg he mehod of forecasg he suded dcaor sysem. Therefore, he deelopme of ew forecasg models of correspodg sysems of dcaors s a acual ad mpora problem. The am of he sudy s o deelop a effce scheme of me seres predco ha auomacally ( he course of s rag) adjuss o he approprae sysem of ecoomc, socal, eromeal, ad egeerg parameers, ad ca be successfully used he deelopme of hgh-qualy sraegc plas he brach of ecoomy, erome, ad for forecas of dffere aural processes. The research mehodology cludes he mehod of leas squares, expoeal smoohg mehod, erae echques of mmzao of fucoals, ad mehods of syhess of eural-ework schemes.

3 Syhess of Forecasg Schemes of Tme Seres Le,,...,,..., be a me seres. Progosc alue he sa of me ca be wre as follows [4-6] of he eleme f ( a,..., a,,...,, ), () r k where a,..., a r are he model parameers, k s he deph of prehsory. To fd he parameers a,...,a r, we cosruced he fucoal a L ( a,..., a ) r, () whch s usually o be mmzed. Le for whch he fucoal L of he model f a,..., ar are he alues of parameers akes s mmum alue. The he progosc alue wh opmal parameers a,..., ar s deermed as follows a,..., ar f ( a,..., a,,...,, ), (3) r k where s he sep of he forecas. Depedg o he ype of he fuco f wh he parameers a,..., a r, we hae dffere opmal forecasg models of me seres. To buld a predce scheme, a he begg le us cosder he auoregresso mehod by meas of whch we defe he opmal sep of he prehsory ge me seres wh he fxed sep of he forecas model, s assumed ha he dcaor alue a he sa of me k for he. I he auoregresso depeds o,,..., k, where k s he parameer of he prehsory wh fxed. The progosc alue by he auoregresso mehod s foud accordg o he followg model ( ) ( ) ( ) a a... a. (4) k k To deerme he opmal alues of he parameers ( ) a (,,..., k ) for a fxed ( = 0 ), we mmze he fucoal ( ) ( ) a... a, ( ) ( ) L( a,..., ak ) k k (5) k.e. we sole he sysem of equaos

4 L a 0,,,..., k. ( ) (6) Le ( ) ( ) a a k be a soluo of he sysem (6). The, accordg o (4) we hae,..., ( ) ( ) ( ) a a... a, (7) k k where k. 0 I s obous ha he arable for a fxed alue of depeds o he parameer k ( k ). To deerme he opmal alue of he prehsory parameer k for 0 for he ge me seres, le us cosder he arables a a ( ), ( ) a ( ), ( ) ( ) a... a Thus we oba m δ,δ,...,δ τ δ k τ. The arable k deermes he opmal alue of he prehsory parameer he auoregresso model for a fxed. 0 Afer deermg he k for a fxed 0, cosder he ma base forecasg models M, M,... Mq of me seres wh he fxed sep of he forecas,.e. models o he bases of whch a ew forecasg scheme are syheszed. Usg he resuls of he forecasg models meoed aboe o he me eral k, k,,, we draw he followg able

5 Forecasg Models M M M q Table. The Progosc Values of Tme Seres k () k () k Elemes of Tme Seres k () k () k ( q) ) k () () ( q ) k ( q I each colum,,..., k k of Table, we ca fd he leas squared dfferece of he progosc ad he acual alues of he correspodg me seres erms. Mahemacally hs ca be wre as followg: le j k ad () () m ( ),( ),...,( ( q) ), j j j j j j j k ad, () () ( q ) m ( j ),( j ),...,( j ) j j j k Defe he ses I, I,..., I as follows j k ad () () m ( ),( ),...,( ( q) ). k

6 I I,,..., q (,,..., q ( j j ( ) j ) ( ) j ),, ad draw he able I k,,..., q k ( ( ) ) Forecasg Models Table. Parameers for Deermg he Weghg Coeffces of he Model M M j a a j a a j k a a k k Resula Colum M q a q a q a qk S S S q where a ps k s, f s Is, 0, f s I, s S p k a pj j,0,( p,,..., q, s,,..., k Wh he help of S p S p () ad S( ) S p ( ) we deerme he weghg coeffces of he forecasg models M p ( p q), wh whch hese models are cluded he followg forecasg scheme q p ). S ( ) ( ) () S ( ) () Sq... ( q). S( ) S( ) S( ) (8)

7 The coeffces of he forecasg models he scheme (8) deped o he parameer ha deermes he fluece of he eleme upo he progosc alue. The more remoe eleme s fluece o he progosc alue me seres he progosc po s from he progosc po I he case of ( 0 )., he less s, all pos of are equale,.e. he model (8) he dsace of he eleme s o ake o accou. from Syhess of he predce scheme (8) wll be compleed he course of rag s cocerg. For hs purpose, we cosruc he fucoal k L( ) ( j S ( ) ( ) () Sqr ( q)... ), ( j k S( ) j S( ) j ), ad mmze by aryg he alue. The eral (0,] we dde o m equal suberals ad fd he alue a he pos (,,..., m). I s obous m ha m ges he accuracy of he fdg he mmum of he fucoal. Le L( ) L() m m L( ). The he forecas of me seres we coduc accordg o he scheme (8), subsug m for. 3 Implemeao of Forecasg Schemes of Tme Seres Arfcal Neural Bass The bass of all forecasg mehods s a dea of exrapolao of paers of he deelopme of he process, whch was formed by he me whe he forecas came rue for fuure perod of me. Le,,...,,..., s me seres. For he syhess of arfcal eural-ework forecasg scheme, here mus exs a mehod (mehods) of syhess of eural elemes ha mpleme approprae forecasg models, o whose bass a eural scheme should be cosruced. For example, he followg arfcal eural eleme wh lear acao fuco mplemes he auoregresso model ( ) ( ) ( ) w w w k, wh he k

8 opmal sep w ( ) a ( ) k,..., w ( k k Fg.. Neuro of he Opmal Auoregresse Model of he prehsory ad he sep of he forecas f ) a are opmal alues of parameers of he a ( ),..., a ( ) k auoregresse model). Afer he deelopme of mehods for he syhess of eural elemes ha mpleme he opmal forecasg models he correspodg classes of models, o predc he alues (,,..., ) a sas of me, le us desg he followg eural- ework scheme Fg.. Neuro-scheme for Tme Seres Predco All he blocks of he s layer coa he same umber of euros, where each euro mplemes oe of he forecasg models (auoregresse model, polyomal, expoeal, lear oes, Brow s lear model, ec.). Neuros ha mpleme he same model dffere blocks of hs layer hae he same seral umber. s

9 k k Each Block. m ( m,,..., k ; ) of he d layer coas as much euros as Block. m. I Block. m each euro has wo pus ad a wegh ecor (,), where he alue ( ) km, km s ge o he frs pu, ad he progosc alue s ge o he d pu, whch s he oupu sgal of he і h euro of Block.m. Acao fuco of he і h euro of Block. m s se as follows ( ) exp( ( ) ). The euro of he seral umber of Block. m s km km, h relaed o euro of he 3 rd h layer he followg way: from he euro of h h Block. m o he m pu of he euro of he 3 rd layer here s ge he sgal ( ) f m,, where f ( ) m,, f 0, ( ) arg max(exp( ( k m k m, ) oherwse. ), Neuros of he 3 rd layer hae he lear acao fuco, ad each of he weghg coeffces of each euro s equal o. A he oupu of he h euro of he 3 rd layer for he fxed we oba he umber. The 3 rd layer, excep for ( ) w euros wh lear acao fuco, has oe more BlokPROG coag exacly as may euros as a Block of he s layer coas. Neuros of hs block mpleme correspodg forecasg model wh he deph ad her seral umbers cocde wh he umbers of euros of Blocks of Layer. The 4 h layer coas wo lear euros. The frs euro has s pus, all s weghg coeffces are equal o, ad has acao fuco ( ) ( ) ( ) w s w w.... The secod euro of hs layer has weghg coeffces he forecas resul of he h model of BlockPROG s deoed by oupu of he secod euro of Layer 4 we hae w ( ) ( ) w, w,..., ( ) () ( ) ( s)... ws. ( ) w s. If, he a he The 5 h layer coas oe euro ha has wo pus, a wegh ecor (.), ad he ( ) () ( )... ( s) acao fuco w ws. ( ) ( ) ( ) w w... w Blocks. m ( m,,..., k s ) deerme he mos effece basc forecasg models. A he oupu of he scheme we hae a coex lear combao of he bes forecasg models. 4 Effeceess of he Cosruced Forecasg Scheme Followg ypes of errors are ofe used he mplemeao of forecasg me seres forecasg

10 where МАЕ Mea Absolue Error MAE s he alues of he me seres a me ; predcable alue. (9) The aerage absolue error of predco (9) s a absolue measure of he qualy of forecas, esmag depedely of he oher predcos. I's eough o se a leel of absolue error ad compare he alue of he specfed error calculaed by he formula (9). To compare he qualy of forecasg, s ofe used he aerage relae error (MRE - Mea Relae Error) s ofe used MRE, (0) ad he aerage square error (RMSE - Roo Mea Square Error) s also used RMRE () where are he erms of he me seres, are he progosc alues of. RMSE ad MRE are relae errors,.e. hey ca be used o compare wo (or more) dffere me seres predco he bes s he forecas whose alue of MRE (0) or RMSE () s less. Accordg o he aerage relae error crero, he qualy of he forecas of he cosruced predcg scheme s esmaed by comparg s resuls wh he resuls of ma forecasg models o base of whch s syheszed. To perform hs, we use daa from he followg Table 3 [7]., Table 3. The Orgal ad Forecased Volumes of Passeger Traffc Year Ralway Sea Rer Auomoble (coaches) Arcraf Udergroud ralway

11

12 Table 4. Forecas Errors of Passeger Traffc accordg o MRE crero Forecasg mehods Kds of passeger raffc Ralway Rer Auomoble Auoregresso mehod Sep of he forecas The mehod of leas squares wh weghs Brow s lear model Brow s quadrac model Forecasg scheme Auoregresso mehod Sep of he forecas The mehod of leas squares wh weghs Brow s lear model Brow s quadrac model Forecasg scheme Hag aalyzed he daa Table 4, we see ha he leas aerage relae error occurs he cosruced forecasg scheme. I he wo cases (for ), he error of he scheme cocdes wh he error of auoregresso mehod. Thus, geeral, he scheme deeloped hs work s he mos effece amog he mehods o whch s based. To oba he aerage error (%) of he predco mehods for he ge me seres perceage, oe should mulply by 00% he correspodg alues of qualy from Table 4. The qualy of he predco mehods of passeger raffc for he forecas perod (04-08) wh he seps of he forecas ad 5 s show he followg chars

13 Fg. 3. Forecasg errors of predco mehods wh he sep ( %) Fg. 4. Forecasg errors of predco mehods wh he sep 5 ( %)

14 Noe. The cosruced forecasg scheme s flexble. Ths meas ha a ew model ca be added o or excluded from basc models (o bass of whch he predce scheme s cosruced) a ay me. I should be oed ha he mehod of syhess of he ery predce scheme does o chage. Here are some resuls of he program mplemeao of deeloped forecasg scheme for deermg he share of road passeger raspor Ukrae o all oher ypes of rasporao durg me spa sce 980 o 03. Table 3 coas prmary daa of passeger raffc olume (perod ) ad projecos of passeger raffc (forecas perod 04-08). O he base of hs able s ede ha he aerage share of road passeger raspor Ukrae was 5.85% oer he aboe meoed perod. Accordgly o he forecas hs share wll aerage 45.56% durg he predco perod Thus, he role of road passeger raspor Ukrae oer he obserable forecas perod s leadg. Aual share of road passeger raspor Ukrae durg he predco perod s show o he followg dagram: Fg.5. The share of road passeger raspor Ukrae oer he perod (04-08) To compare he dyamcs of chages of he olume of passeger raffc Ukrae for dffere ypes of ehcles (ral, rer, road) we cosruc he followg dagram.

15 Fg.6. Dyamcs of passeger raffc Ukrae (04-08) 5 Coclusos A flexble scheme for forecasg of ecoomc, socal, eromeal, egeerg ad echologcal dcaors ha ca be successfully used he deelopme of reasoable sraegc plas ad decsos he correspodg felds of huma acy s worked ou. Ths forecasg scheme allows us o clude ew forecasg models of me seres or o exclude a model or groups of models from a ay sa of me. As for he models whch rema he scheme, he compeo bewee hem s made oer a ge perod of me, ad he fal forecasg scheme represes a coex lear combao of models -wers wh correspodg weghg coeffces. Refereces. Blak, І.A. Sraegy ad Taccs of Facal Maageme. The Iem LTD, Ky (996) ( Ukraa)

16 . Heyes, V.M. Isably ad Ecoomc Growh. Isue of Ecoomc Forecasg of Naoal Academy of Sceces of Ukrae, Ky (00) ( Ukraa) 3. Zaycheko, Y.P., Moamed, M., Shapoaleko, N.V. Fuzzy Neural Neworks ad Geec Algorhms Problems of Macroecoomc Forecasg. Scece ews of "Ky Polyechc Isue", 4, Ky (00) ( Ukraa) 4. Iakheko, V. Course of Ecoomc Aalyss. Zaya Press, Ky (000). ( Ukraa) 5. Iakheko, O.H., Lapa, V.G. Predco of Rradom Processes. Naukoa Dumka, Ky (969) ( Ukraa) 6. Yarka, N.M. Ecoomerc Modelg he Maageme of Busess Rsks. Face of Ukrae,, Ky (003) ( Ukraa) 7. Tmashoa, L., Sepaeko O. Ecoomc-mahemacal Ealuao Model of Eerprse Marke Ecoomy. Joural of he Academy of Labour ad Socal Affars Federao of Trade Uos of Ukrae, 3 (7), Ky (004) ( Ukraa) 8. Sepaeko, A.P. Moder Сompuer Tools ad Techologes for he Iformao of he Facal Sysem. New Compuer Tools, Compuers ad Neworks, Vol., 5-3. Ky, Isue of Cyberecs by V.Glushko of Naoal Academy of Sceces of Ukrae (00) ( Ukraa) 9. Tkacheko, R., Palyuk, O. Approaches o forecas elecrcy cosumpo power dsrbuo compaes // Bulle "L Polyechc": Compuer Egeerg ad Iformao Techology Pp (00) ( Ukraa) 0. Machuk, A.V. Modelg of Ecoomc Processes Usg Fuzzy Logc Mehods. Ky Naoal Ecoomc Uersy, Ky (007) ( Ukraa). Hake, Joh E., Arhur, G. Resch, ad Dea W. Wcher. Busess forecasg. Up per Saddle Rer, NJ: Prece Hall, (00). Lews, Col Dad. Idusral ad busess forecasg mehods: A praccal gude o expoeal smoohg ad cure fg. Buerworh-Heema, (98) 3. Box, George EP, ad Gwlym, M. Jeks. Tme seres aalyss: forecasg ad corol, resed ed. Holde-Day, (976) 4. Tal G. Ecoomc forecass ad decso makg / G. Tal. - M.: Sascs p. (97) ( Russa) 5. Kukhare, V.N., Sally V.N., Erper A.M. Ecoomc-mahemacal Mehods ad Models he Plag ad Maageme. Vyshcha shcola, Ky (99) ( Russa) 6. Hol, Charles C. "Forecasg seasoals ad reds by expoeally weghed mog aerages." Ieraoal Joural of Forecasg 0.: 5-0. (004) 7. Wers, Peer R. "Forecasg sales by expoeally weghed mog aerages." Maageme Scece 6.3: (960) 8. Traspor ad Commucao Ukrae - 03 [Tex] / Sae Sascs Serce. Sascal Yearbook, Ky (03) ( Ukraa).