DEVELOPMENT OF EFFECTIVE TIME SERIES FORECASTING MODEL. Fedir Geche, Anatoliy Batyuk, Oksana Mulesa, Mykhaylo Vashkeba.

Transcription

1 Ieraoal Joural of Adaced Research Compuer Egeerg & Techology (IJARCET Volume 4 Issue, December 05 DEVELOPMENT OF EFFECTIVE TIME SERIES FORECASTING MODEL Fedr Geche, Aaoly Bayuk, Oksaa Mulesa, Mykhaylo Vashkeba. 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. I. INTRODUCTION Aalyss of problems ha are soled by specalss arous scefc ad appled felds he course of hem of her professoal aces dcae he expedecy of specaless use mahemacal ools for solg a arey of applcaos. I our days, s ery mpora, o be able o forecas he ma dcaors, such as: ecoomcal, socal, medcal, echcal ad so o. 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. 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 obeces 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 subec 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], Machuk A.V. [9]. 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. Forecasg he medcal daa s also a mpora ask as make possble o ge accurae dagoses ad predc ad pree dsease. Oe of more dffcul ad serous problems medce s quaae predco characerscs of offcally regsered HIV-feced persos he rego. 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 aduss 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, medce 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. II. Le SYNTHESIS OF FORECASTING SCHEMES OF TIME SERIES BASED ON CLASSIC FORECASTING METHODS.,...,,,... be a me seres. Progosc alue of he eleme a he sa of me ca be wre as follows [0] f ( a,..., a,,...,,, ( r k where a,..., a r are he model parameers, k s he deph of prehsory. To fd he parameers a,...,ar, we cosruced he fucoal L ( a,..., ar, ( whch s usually o be mmzed. Le a,..., a r are he alues of parameers a,..., a r for whch he fucoal L akes s mmum alue. The he progosc alue of he model f wh opmal parameers a,..., a r s deermed as follows f ( a,...,a r,,..., k,, (3 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 k for he ge me seres wh he fxed sep of he forecas. I he auoregresso model, s assumed ha he dcaor alue a he sa of me depeds o,,..., k, where k s he parameer of he prehsory wh fxed. The ISSN: All Rghs Resered 05 IJARCET 4377

2 ISSN: All Rghs Resered 05 IJARCET 4378 Ieraoal Joural of Adaced Research Compuer Egeerg & Techology (IJARCET Volume 4 Issue, December 05 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 Table. The Progosc Values of Tme Seres Forecasg Models k M ( k M ( k Elemes of Tme Seres k ( k ( ( ( k M ( q q ( q ( q k k.e. we sole he sysem of equaos L a 0,,,..., k. ( (6 ( ( a a k be a soluo of he sysem (6. The, Le,..., accordg o (4 we hae where k. ( ( ( a a... ak k, (7 I s obous ha he arable for a fxed alue of 0 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 (, (,,..., τ a k τ ( Thus we oba m. The arable deermes he opmal alue of he prehsory parameer he auoregresso model for a fxed. Afer deermg he ma base forecasg models 0 k for a fxed M,... 0, k, cosder he, M M q 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 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 k ad ( ( m (,(,...,( ( q, k ad ( (,...,( ( q m (,(, k ad ( ( m (,(,...,( ( q. k Defe he ses I, I,..., I as follows I I I k ad draw he able k,,..., q (,,..., q (,,..., q k ( ( ( (,, Table. Parameers for Deermg he Weghg Coeffces of he Model where a S ps p Forecasg Models M a M a Resula k Colum a S a a k a S k M q a q q k s, f s Is, 0, f s Is, k a qk a S q a,0, ( p,,..., q, s,,..., k. p

3 ISSN: All Rghs Resered 05 IJARCET 4379 Ieraoal Joural of Adaced Research Compuer Egeerg & Techology (IJARCET Volume 4 Issue, December 05 Wh he help of S p S p ( ad S( S p ( we q p deerme he weghg coeffces of he forecasg models M p ( p q, wh whch hese models are cluded he followg forecasg scheme S ( S ( S ( S ( ( ( S ( ( Sq... ( q. (8 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 from he progosc po, he less s s fluece o he progosc alue ( 0. I he case of, all pos of me seres are equale,.e. he model (8 he dsace of he eleme from he progosc po s o ake o accou. Syhess of he predce scheme (8 wll be compleed he course of rag s cocerg. For hs purpose, we cosruc he fucoal Fg.. Neuro of he opmal 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 L( k ( ( ( ( Sqr... ( q, ( k, S S( S ( ad mmze by aryg he alue. The eral (0,] we dde o m equal suberals ad fd he alue L( a he pos (,,...,m. I s obous ha m ges he m accuracy of he fdg he mmum of he fucoal L (. Le L m L m. The he forecas of me seres we coduc accordg o he scheme (8, subsug m for. 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, wh k k he opmal sep k of he prehsory ad he sep of he ( ( ( forecas f w a,..., w k a k ( ( a,..., a k are opmal alues of parameers of he auoregresse model. Fg.. Neuro-scheme for Tme Seres Predco All he blocks of he s layer coa he same umber s 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. Each Block. m ( m,,..., k ; k k! Referece source o foud. 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 k m s ge o he frs pu! Referece source o foud., ad he progosc alue ( km, s ge o he d pu,! Referece source o foud. 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( ( km km,! Referece source o foud.. The euro of he seral umber of Block. m s relaed o h euro of he 3 rd layer he followg way: from he h euro of Block. m o he m h pu of he h euro of he 3 rd layer here s ge he sgal f ( m, f ( m,! Referece source o foud., where, f 0, arg max(exp( ( oherwse. k m ( k m, 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 w. The 3 rd layer, excep for,

4 ISSN: All Rghs Resered 05 IJARCET 4380 Ieraoal Joural of Adaced Research Compuer Egeerg & Techology (IJARCET Volume 4 Issue, December 05 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 w... w s. The secod euro of hs layer has weghg coeffces ( ( ( w s w, w,...,! Referece source o foud.. If! Referece source o foud. he forecas resul of he h ( model of BlockPROG s deoed by, he a he oupu of he secod euro of Layer 4 we hae ( ( ( ( s w... ws! Referece source o foud.. The 5 h layer coas oe euro ha has wo pus, a wegh ecor (., ad he acao fuco ( ( (... ( s w ws. ( ( ( w w... w s Blocks. m ( m,,..., k! Referece source o foud. deerme he mos effece basc forecasg models. A he oupu of he scheme we hae a coex lear combao of he bes forecasg models. ( ( III. FORECASTING THE ECONOMICAL INDICATORS. To compare he qualy of forecasg, s ofe used he aerage relae error (MRE - Mea Relae s ofe used MRE, (9 ad he aerage square error (RMSE -Roo Mea Square s also used RMRE, (0 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 (9 or RMSE (0 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[]. Table 3.The Orgal ad Forecased Volumes of Passeger Traffc Year Ralway Sea Rer Auomoble (coaches Arcraf Udegroud ralway

5 ISSN: All Rghs Resered 05 IJARCET 438 Ieraoal Joural of Adaced Research Compuer Egeerg & Techology (IJARCET Volume 4 Issue, December Table 4. Forecas s of Passeger Traffc accordg o MRE crero Forecasg mehods Kds of passeger raffc Ralway Rer Auomoble Sep of he forecas Auoregresso mehod The mehod of leas squares wh weghs Brow s lear model Brow squadrac model Forecasg scheme Sep of he forecas 5 Auoregresso mehod The mehod of leas squares wh weghs Brow s lear model Brow squadrac 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

6 ISSN: All Rghs Resered 05 IJARCET 438 Ieraoal Joural of Adaced Research Compuer Egeerg & Techology (IJARCET Volume 4 Issue, December 05 Fg. 3. Forecasg errors of predco mehods wh he sep Fg. 4. Forecasg errors of predco mehods wh he sep 5 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. IV. FORECASTING MEDICAL DATA. The obec of sudy s he ask of forecasg quaae characerscs of offcally regsered HIV-feced persos he rego. For progosg quaae characerscs of HIV-feced perso, we were clude our scheme hese mehod: auoregresso, leas square mehod wh wegh, -s order Brau s mehod, -d order Brau s mehod ad Wer s mehod, whch o bee cluded scheme of foresg ecoomcal daa. Wer s mehod apply o accou seasoal compoes of progosg me seres. Wer s model s a hree-paramercal model of expoeal smoohg. The sysem of equaos of Wer s mehod:

7 ISSN: All Rghs Resered 05 IJARCET 4383 z sk s s z z s where,, 0, z z z k s Ieraoal Joural of Adaced Research Compuer Egeerg & Techology (IJARCET Volume 4 Issue, December 05! Referece source o foud.. Frs equao, he sysem aboe, clude seasoaly wh parameer! Referece source o foud.. I he case of forecasg medcal daa, we were progosed he umber of HIV-feced perso ad umber of AIDS pae he Trascarpaa rego of Ukrae from 987 ll 04 years. Forecasg daa we ca see he Table 5. Table 5. Number of HIV-feced ad AIDS perso rego. Number of Trascarpao Number of HIV-feced rego AIDS pae by perso by he (year he year year Table 6. Forecasg umber of HIV-feced perso Sep (05 Auoregresso Wers mehod mehod (М (M4 Number of HIV-feced perso Leas square mehod wh wegh (M -s order Brau s mehod (M3 -d order Brau s mehod (M5 Forecasg alues 47,785 36,838 47,833 34,675 8, ,536 Roo Mea Square 3,77 5,8654, ,946 7,486,0878 Mea Relae 0,009 0,034 0,0044 0,087 0,350 0,0043 Sep 3( ( ( 04 Auoregresso Wers mehod mehod (М (M4 0 (3 04 Leas square mehod wh wegh (M 0. ( s order Brau s mehod (M3 (5 04 -d order Brau s mehod (M5 Forecasg alues 550, ,43 504,480 6,66 98, ,9736 Roo Mea Square 9,34 7,0439 5,90 55,845 67,4809 3,439 Mea Relae 0,07 0,037 0,00 0,04 0,4 0,005 Sep 5( ( ( 04 3 Auoregresso Wers mehod mehod (М (M4 0 ( Leas square mehod wh wegh (M 0.9 ( s order Brau s mehod (M3 0 ( d order Brau s mehod (M5 Forecasg alues 73,449 45,8 64, ,70 55, ,6653 Roo Mea Square,4666 6,8865, ,843 54,476,4689 Mea Relae 0,080 0,07 0,003 0,069 0,084 0, ( ( ( ( (5 04 5

8 ISSN: All Rghs Resered 05 IJARCET 4384 Ieraoal Joural of Adaced Research Compuer Egeerg & Techology (IJARCET Volume 4 Issue, December 05 Table 7. Forecasg umber of AIDS pae Number of AIDS pae Sep (05 Auoregresso mehod (M4 Wers mehod (М Leas square mehod wh wegh (M -s order Brau s mehod (M3 -d order Brau s mehod (M5 Forecasg alues 9,04 6, ,5556 8,5667,673 73,63 Roo Mea Square,73,667 9,078 5,6766 7,57,3099 Mea Relae 0,00 0,00 0,076 0,046 0,5 0,039 Sep 3( Auoregresso mehod (M4 ( 04 0 ( 04 Wers mehod (М 0 (3 04 Leas square mehod wh wegh (M 0.38 (4 04 -s order Brau s mehod (M3 0 (5 04 -d order Brau s mehod (M5 Forecasg alues 00,936 97,076 8,5659 8,007, ,9085 Roo Mea Square 5,990,660 8,800 6,576 7,55,660 Mea Relae 0,0665 0,08 0,44 0,060 0,07 0, ( ( ( ( ( Sep 5(09 Auoregresso mehod (M4 Wers mehod (М Leas square mehod wh wegh (M -s order Brau s mehod (M3 -d order Brau s mehod (M5 Forecasg alues 9,5008 3,603 39,47 6,074,784 9,545 Roo Mea Square 0,8784,660 4,943 7,5899 7,960 0,8785 Mea Relae 0,045 0,080 0,0989 0,087 0,36 0, ( ( ( ( (5 04 5

9 Ieraoal Joural of Adaced Research Compuer Egeerg & Techology (IJARCET Volume 4 Issue, December Progosc alues Real alues Fg. 5. Number of HIV-feced perso years Progosc alues Real alues Fg. 6. Number of AIDS pae years Hag aalyzed he daa Table 5, we ca see ha each case of sep (, 3, 5 wo he Leas square mehod wh wegh. Bu he Table 6 we see, ha each case preferece receed Wers mehod. I he process of forecasg medcal daa we obaed resuls, whch show o he fg. 5, 6. V. CONCLUSIONS 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. The forecasg mehods ca be dyamcally cluded o he progosg scheme, ad also ca be excluded of he scheme he process of forecasg. Tha makes our algorhm ery flexble, ca adap o dffere suaos, so we ca use o forecass may kds of daa: ecoomcal, medcal, socal, echcal ad so o. REFERENCES. Blak, І.A. Sraegy ad Taccs of Facal Maageme. The Iem LTD, Ky (996 ( Ukraa. 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. Machuk, A.V. Modelg of Ecoomc Processes Usg Fuzzy Logc Mehods. Ky Naoal Ecoomc Uersy, Ky (007 ( Ukraa 0.Kukhare, V.N., Sally V.N., Erper A.M. Ecoomcmahemacal Mehods ad Models he Plag ad Maageme. Vyshcha shcola, Ky (99 ( Russa.Traspor ad Commucao Ukrae - 03 [Tex] / Sae Sascs Serce. Sascal Yearbook, Ky(03 ( Ukraa ISSN: All Rghs Resered 05 IJARCET 4385

10 ISSN: All Rghs Resered 05 IJARCET 4386 Ieraoal Joural of Adaced Research Compuer Egeerg & Techology (IJARCET Volume 4 Issue, December 05 Dr. Fedr Geche head of deparme of cyberec ad appled mahemac Uzhgorod Naoal Uersy, Ukrae. He has more ha 80 papers eraoal ad aoal scefc oural, moograph, eo. Research eress: processg, preseao ad recogo of dscree sgals ad mages eurobases, syhess of eural ad geeralzed eural elemes ad eworks, me seres forecasg, mahemacal modelg. Maages he scefc work of posgraduae sudes. Ph.D. Aaoly Bayuk s a lecurer a L Polyechc Naoal Uersy, Ukrae. He has M.Sc. degree Auomaed Corol Sysems, ad Phd. degree s Sysem ad Tools Arfcal Iellgece. He has publshed seeral papers scefc ourals ad has parcpaed arous eraoal ad aoal cofereces. Hs research eress are he sysems of arfcal ellgece. Ph.D. Oksaa Mulesa s a lecurer a Uzhgorod Naoal Uersy, Ukrae. She has M.Sc. degree Appled Mahemacs ad Phd degree s Iformao Techologes. She has publshed seeral papers scefc ourals ad has parcpaed arous eraoal ad aoal cofereces. Her research eress are he IT appled problems. Mr. Mykhalo Vashkeba posgraduae sude Uzhgorod Naoal Uersy, Ukrae. He has M.Sc. degree Appled Mahemacs. He has publshed seeral papers scefc ourals ad has parcpaed arous eraoal ad aoal cofereces. Hs research eress: me seres forecasg, daa processg arfcal eural ework.