The application research of ARMA forecasting model in prediction of medals and ranking for 2016 Olympic Games

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1 Available olie Joural of Chemical ad Pharmaceuical Research, 04, 6(7): Research Aricle ISSN : CODEN(USA) : JCPRC5 The alicaio research of ARMA forecasig model i redicio of medals ad raig for 06 Olymic Games Shibiao Dog School of Physical Educaio, uchag Uiversiy, uchag, Hea, Chia ABSTRACT The medal umber ad raig of revious Olymic Game is he focus of eole s aeio. By sudyig he ime series mehod, his aer alies i i redicig he umber of medals. I resecively redics he medal umbers of Chia, he U.S. ad Russia, arrives a he medal umber ad raig of Chia i he 06 Olymic, obais ha Chia will resecively wi 40 gold medals, 5 silver medals ad 7 broze medals wih a oal of 9 medals a he 06 Olymic Games by usig he weighed movig average mehod, uses auoregressive ( AR ) model, movig average ( AM ) model ad auoregressive average ARMA model o resecively redic he medal codiio of America ad Russia i he 06 Olymics, uses SAS sofware o es he saioary of he daa, ess he auocorrelaio coefficie model, ad ulimaely deermies he rediced ad esimaed value. I is rediced ha U.S. will ge 44 gold medals, 36 silver medals ad 3 broze medals, a oal of 03 medals; Russia will ge 3 gold medals, 36 silver medals ad 3 broze medals, a oal of 9 medals; hus Chia will be he secod. Key words: Medals amou, ime series aalysis, auoregressive model, ARMA redicio model, Olymic Games INTRODUCTION The firs moder Olymic Games i 896 were orgaized i Ahes by he Gree. Olymic aes solidariy, eace ad friedshi as he urose for a log ime [-3]. The Olymic Games is he comeiio ooruiy o show he comrehesive sregh of he aio, i rereses o oly he sor develome of a aio, bu also rereses he codiio i all asecs of a coury s humaisic ualiy, oliical sysem, ecoomic develome ad social harmoy [4-6]. The erformace of a coury a he Olymics direcly deeds o he umber of Olymic medals. Olymic medal o oly rereses he sors comeiio abiliy of idividual, bu also bears eole s love for sors, ad aiciaio of he masses [7-9]. The umber of medals direcly reflecs wheher a coury s sors, ecoomy, olicy is advaced or o. Chia s erformace a he Olymics is more romie, which is a miiaure of he raid develome afer Chia's reform ad oeig u [0-]. Esecially i he 008 Beijig Olymics, Chia raed firs i he world wih he advaage of 5 gold medals; which effecively romoes he develome of sors uderaigs i Chia a he same ime fulfills Chia s Olymic dream, bu also shows he Chiese owerful comrehesive aioal sregh ad he humaisic ualiy o he world [3]. The umber of Olymic medals i he Olymic Games is he focus of eole s aeio aroud he world. Peole ofe icororae a lo of feeligs i medal. Chia's ousadig erformace i he Olymic Games romoes he awareess uiy ad arioism of he eole. The medal umber redicio ca mae Chia develo beer sors-relaed olicies, which is coducive o he develome of sors ad ecoomic cause [4]. Therefore, he forecas for he umber of medals has imora sigificace for he ecoomic sors develome. Mos of he aers are ow redicig he umber of gold medals a home ad abroad, ad aers o redic he oal umber of medals ad Chiese raig o he whole are very lile. The mos owerful ooes for Chia o have good raig i he Olymics are he Uied Saes ad Russia. This aricle resecively redics he medals umber of Chia, he U.S. ad Russia i 06 as a whole hereby deduces Chia s raig. From he daa observaio, he daa 383

2 Shibiao Dog J. Chem. Pharm. Res., 04, 6(7): of he hree couries have heir ow characerisics. For he differe characerisics of ie daa ses, his aricle uses differe mehods of ime series o redic, which is more argeed ad more accurae. THE APPLICATION AND RESULTS ANALYSIS OF ARMA FORECASTING MODEL IN THE OLYMPICS PERFORMANCE PREDICTION Weighed movig average mehod Suose he ime seuece is y, y, L, y,l ; he weighed movig average formula: M w y + w y + L+ w N N+ w w + w + L+ wn y, N () I he formula formula: y ˆ + M w M w is a w eriod weighed movig average value; i is he weigh of y i+. The redicio () Tha is o ae he eriod weighed movig average value as he redicive value of he + eriod. Figure : Chiese medal case of all revious Figure shows ha he 008Olymic Game was orgaized i Chia, he umber of medals obaied is obvious abormal, so roud he Chiese medal daa i 008. Table : The medal raig of Chia s Olymic Games ad he weighed movig average rediced value able Years Gold The weighed movig average rediced value of hree years Relaive error (%) Silver Relaive error (%) Coer Relaive error (%) The oal umber of medals Relaive error (%) Tae w 3, w, y + y + y w 3, he redicio formula is: y Accordig o he above euaio, afer he weighed average, he resuls of he calculaed redicio value are show i Table. The redicive value for he umber of Chia wiig he 06 Olymic gold medal is (umber):

3 Shibiao Dog J. Chem. Pharm. Res., 04, 6(7): As he overall red eds o rise, he rediced values lag. I is ecessary o correc he redicive value, he mehod is: firs calculae he relaive error of he rediced value ad he acual value of each sessio, such as he % sessio: 6 The relaive error is show i Table, ad he calculaes he oal average error % 00% 6.75% y 4 From he above he average of he oal rediced value is lower ha he acual value 33.04%, hus he rediced value i 06 ca be correced as: 6.75% Similarly draw he umber of silver ad broze, ad fill i i Table. I he 06 sessio i is rediced ha Chia will wi 40 gold medals, 5 silver medals, 7 broze medals, oally 9 medals. Sochasic ime series models: () The auo-regressio ( AR ) model; () The movig average ( AM ) model; (3) The auo-regressive movig average ( ARMA) model; Iroduce bacward oeraor B ad differece oeraor : Ad deoe i as: φ ( B) ( φ B φ B L φ B Firs order differece: ) B, B C C, 0,, L; C is a cosa. (3) Seuece AR( ) : suose {, 0, ±, ±,L} is a zero mea saioary series, which mees he followig: (4) φ + φ + L + φ + (5) Parameer σ is he saioary whie oise wih zero mea ad variace. The seuece of order AR( ), which is deoed as ( φ, φ, ) T φ, L φ seuece, whereas: is he auo-regressio (6) I is more coveie o describe he formula afer he iroducio of bacward shif oeraor. Oeraor B is defied as follows: B, B (7) Comuig sub-olyomial: φ ( B) φ B φ B L φ B (8) 385

4 Shibiao Dog J. Chem. Pharm. Res., 04, 6(7): The formula (8) ca be rewrie as: ( B) φ ( ) Polyomial euaio φ λ 0 AR( ) is called he characerisic euaio of roos λ, λ,, λ (9) model. The es of he model: is L are called eigevalues of he model. Eigevalue may be real, ad also could be lural. Accordig o he codiios, if he eigevalues are ouside he ui circle, amely: λi >, i,, L, AR( ) Model is called sable or saioary. The above formula is called he sable codiio. Seuece MA ( ) : suose {, 0, ±, ±,L} is he zero mea saioary series, which mees he followig model: (0) θ θ L θ () Parameer whereas: is he saioary whie oise wih zero mea ad σ variace, which is deoed as seuece MA( ), ( θ, θ, ) T θ, L θ I is called he movig average arameer vecor, whose comoes coefficie. For he liear bacward shif oeraor B, we have: θ j, j,, L, () is called average slidig B, B (3) Re-iroduce he oeraor olyomials: θ ( B) θ B θ B L θ B (4) The formula (4) ca be wrie as: θ ( B) ( ) Polyomial euaio θ λ 0 MA( ) is called he characerisic euaio of MA( ) eigevalue of. MA( ) The model es: If he eigevalue of ARMA (, ) Seuece saioary seuece The ( ) ARMA, (5) models; is roos are called he MA( ) are ouside of he ui circle, he he model is reversible. φ ( B) θ : he defiiio of, ad E mees: 0, φ L φ θ L θ ( B) is he same wih ha of above, he wide ( ) Is called auoregressive movig average model of order,, which is referred o as model. 386

5 Shibiao Dog J. Chem. Pharm. Res., 04, 6(7): ARMA(, ) ca be wrie i he form of oeraors: φ ( B) θ ( B) (6) ARMA(, ) φ (B) (B) θ For model, we always assume ad (as a olyomial of he variable B) have o commo facor, ad resecively saisfy saioary codiios ad reversible codiio. Predic he gold medal umber of U.S. i 06 Olymic Game, as show i Figure. Figure : The U.S. s medal codiio of all revious Use SAS sofware for daa rocessig, ad es saioary of he daa: Auocorrelaios Lag Covariace Correlaio Sd Error ******************** *************** ******* * The calculaio resuls of auocorrelaio coefficies are give, i ca be see ha wih he icreasig of delay deloyme, he auocorrelaio coefficies shows a decliig red. Auocorrelaio coefficie of he seuece uicly reduces o 0; i is ow ha he seuece is he saioary ime series. As ca be see from he able he auocorrelaio P is. Coduc correlaio aalysis ad arial correlaio aalysis of he daa o deermie he model order: Parial Auocorrelaios Lag Correlaio *************** The calculaio resuls of arial correlaio coefficie are give, wherei he firs colum is he delayed deloyme, he secod colum is he arial correlaio coefficie, he hird colum uses a aseris o exress he arial correlaio coefficies. I ca be see ha afer oe se delay, he arial correlaio coefficies are all bewee wice of he sadard errors. Ad he value of correlaio coefficie is much smaller. So we ca say ha he model eds afer he oe se delay. Through he above aalysis i ca be cosidered ha he hyoheical model is alicable. Accordig o he ideified order, esimae he coefficies ha he model esablishes. I uses he esimae saeme o esablish a firs-order auoregressive model, firs-order movig regressio model, as well as a firs-order auoregressive movig average ARMA(, ) model. Model es: The fiig saisics gives he evaluaio of he model resuls. I he give saisical arameers, i icludes he esimaed value of he model, he esimaed value of he residual variace, he esimae value of sadard deviaio, he sadard of iformaio amou AIC ad SBC, ad he residual umber. Wih he rogram ruig resuls, i esablishes hree models of differe ime series. We ca sudy he models from he followig wo crieria: he higher he lielihood fucio value is, he beer; he fewer he umber of model osiio arameers is, he beer. 387

6 Shibiao Dog J. Chem. Pharm. Res., 04, 6(7): The model ha ca mae he fucio miimum is cosidered o be he bes model. For crierio SBC, he smaller he beer SBC is. By judgig AIC ad SBC, he hird model is much beer. Sigifica es: AR() MA() ARMA(, ) AIC SBC Codiioal Leas Suares Esimaio Sadard Arox Parameer Esimae Error Value Pr > Lag MU < MA, AR, The value of P is less ha 0.05, which mees he es reuiremes. I is derived from SAS sofware ha he develome fucio of he gold medal ha Uied Saes wo is: x 0.985x x I is obaied ha U.S. i he 06 Olymic Game will wi 44 gold medals, based o his model he umber of silver medal is 36, he umber of broze medal is 3, a oal of 03 medals. Predic he umber of gold medals for Russia a he 06 Olymic Games, as show i Figure 3. Figure 3: Russia's medal codiio of all revious The mehod is he same wih he redicio of he umber of gold medals he Uied Saes, i is calculaed ha is ad is. The model checig is he same wih he above model: AR() MA() ARMA(, ) AIC SBC By combiig he value of AIC ad SBC, i draws a oimal sigifica es of model oe: Codiioal Leas Suares Esimaio Sadard Arox Parameer Esimae Error Value Pr > Lag MU < AR, Value P is less ha 0.05 ad mees he es reuiremes. I is derived from SAS sofware ha he develome fucio of he gold medal ha he Russia wo is: x 0.975x x I is obaied ha Russia will wi 3 gold medals, based o his model he umber of silver medal is 36, he umber of broze medal is 3, a oal of 9 medals. 388

7 Shibiao Dog J. Chem. Pharm. Res., 04, 6(7): CONCLUSION Time series model is he commo soluio o solve he redicio roblem ha has a clear chroological order. I he revious Olymic Games, he sreghs of Chia, he U.S. ad Russia are early he same, which are all i he world o level. Aimig a he uaiy of gold silver ad broze medals ha he hree couries go i he revious Olymic Games, accordig o he differe characerisics of ie ses of daa, his aer resecively selecs he auo-regressio ( AR) model, movig average ( AM ) model, auoregressive average ARMA model o coduc redicio, obais he umber of Olymic medals obaied i 06, ad derives he raig. Thus each se of he daa is more argeed, ad he resuls are more accurae. Time series model ca be widely used o chroological redicio roblems, i addiio o sors medal redicios; i also ca be used for a variey of daa forecasig roblems, such as oulaio redicio. REFERENCES []TU Chu-jig, DU He-ig, WANG-Wei. Joural of Jiggagsha Uiversiy, 0, 3(). [] WU i. Sors Culure Guide, 0, (5). [3] ZHANG Zheg-mi. Joural of Physical Educaio, 0, 8(4), -4. [4] LIN Dehua. Joural of Caial College of Physical Educaio, 0, 3(6), ,549. [5] HE Jiag-hai. Joural of Wuha Isiue of Physical Educaio, 007, 4(6), [6] JIANG Yi-eg. Joural of Tiaji Isiue of Physical Educaio, 004, 9(), [7]WU i-li,li Jia-che. Joural of Wuha Isiue of Physical Educaio, 005, 39(6), [8]WANG Guo-fa, ZHAO Wu, LIU u-ju, FENG Shu-hui, UE Er-jia, CHEN Li, WANG Bo. Chia Sor Sciece ad Techology, 0, 47(). [9]FAN We jie, ZHOU Ai jie, LIU Jia mi. Joural of Beijig Sor Uiversiy, 00, 5(6), [0] iaomi Zhag. Joural of Chemical ad Pharmaceuical Research, 03, 5(), 8-4. [] Wag Bo; Zhao Yuli. Joural of Chemical ad Pharmaceuical Research, 03, 5(), -6. [] Migmig Guo. Joural of Chemical ad Pharmaceuical Research, 03, 5(), [3] Big Zhag; Zhag S.; Lu G.. Joural of Chemical ad Pharmaceuical Research, 03, 5(9), [4] Big Zhag. Joural of Chemical ad Pharmaceuical Research, 04, 5(),