1. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mod`les de prvision e e Partie 2 - sries temporelles e Arthur Charpentier charpentier.arthur@uqam.ca http ://freakonometrics.blog.free.fr/ Automne 2012

2. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Plan du cours Motivation et introduction aux sries temporelles e Mthodes de lissage e Mod`les de rgression (Buys-Ballot) e e Lissage(s) exponentiel(s) (Holt-Winters) Notions gnrales sur les processus stationnaires e e Les processus SARIM A Les mod`les autorgressifs, AR(p), (L)Xt = t e e Les mod`les moyennes mobiles, M A(q) (moving average), Xt = (L)t e Les mod`les autorgtressifs et moyenne mobiles, ARM A(p, q), e e (L)Xt = (L)t Les mod`les autorgtressifs, ARIM A(p, d, q), (1 L)d (L)Xt = (L)t e e Les mod`les autorgtressifs, SARIM A(p, d, q), e e (1 L)d (1 Ls )(L)Xt = (L)t Prvision avec un SARIM A, T XT +h e

3. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mthode de Buys Balot - la tendance eNotons (Xt ) une srie temporelle, observe jusqu` la date T . e e aSi on a une tendance linaire, on cherche ` rsoudre e a e T = (0 , 1 ) argmin (Xt (0 + 1 t))2 t=1> autoroute=read.table(+ "http://freakonometrics.blog.free.fr/public/data/autoroute.csv",+ header=TRUE,sep=";")> a7=autoroute$a007> X=ts(a7,start = c(1989, 9), frequency = 12)> T=time(X)> S=cycle(X)> B=data.frame(x=as.vector(X),T=as.vector(T),S=as.vector(S))> regT=lm(x~T,data=B)> plot(X)> abline(regT,col="red")

4. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mthode de Buys Balot - la tendance e 70000 60000 50000 X 40000 30000 20000 1990 1991 1992 1993 1994 1995 1996 Time

5. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mthode de Buys Balot - la tendance ePosons Yt = Xt (0 + 1 t)> X1=predict(regT)> B$X1=X1> Y=B$X1> plot(X,xlab="",ylab="")> YU=apply(cbind(X,Y),1,max)> YL=apply(cbind(X,Y),1,min)> i=which(is.na(Y)==FALSE)> polygon(c(T[i],rev(T[i])),c(Y[i],rev(YU[i])),col="blue")> polygon(c(T[i],rev(T[i])),c(Y[i],rev(YL[i])),col="red")> lines(B$T,Y,lwd=3)

6. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mthode de Buys Balot - la tendance e 70000 60000 50000 40000 30000 20000 1990 1991 1992 1993 1994 1995 1996

7. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mthode de Buys Balot - le cycle eOn cherche un cycle mensuel (de priode s = 12). Supposons e s1 Yt = k 1(t = k mod. s) + Zt k=0> B$res1=X-X1> regS=lm(res1~0+as.factor(S),data=B)> B$X2=predict(regS)> plot(B$S,B$res1,xlab="saisonnalite")> lines(B$S[1:4],B$X2[1:4],col="red",lwd=2)> lines(B$S[5:13],B$X2[5:13],col="red",lwd=2)Remarque mod. est loprateur de calcul du reste de la division euclidienne, ex : e 27 mod. 12 = 27 (12 2) = 3

8. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mthode de Buys Balot - le cycle e q q 30000 q q q q q q 20000 q 10000 q B$res1 q q q q q q q q q q q q q q q q q q q 0 q q q q q q q q q 10000 q q q q q q q q q q q q q q q q q q q q q q q q q q q 20000 q q 2 4 6 8 10 12 saisonnalit

9. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mthode de Buys Balot - le cycle eAlors s1 Xt = 0 + 1 t + k 1(t = k mod. s) +Zt k=0 tendance linaire e cycle> plot(X,xlab="",ylab="")> YU=apply(cbind(X,Y),1,max)> YL=apply(cbind(X,Y),1,min)> i=which(is.na(Y)==FALSE)> polygon(c(T[i],rev(T[i])),c(Y[i],rev(YU[i])),col="blue")> polygon(c(T[i],rev(T[i])),c(Y[i],rev(YL[i])),col="red")> lines(B$T,Y,lwd=3)

10. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mthode de Buys Balot - tendance et cycle e 70000 60000 50000 40000 30000 20000 1990 1991 1992 1993 1994 1995 1996

11. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mthode de Buys Balot - prvision e eOn peut alors faire de la prvision, en supposant le bruit Zt i.i.d., i.e. e s1 T XT +h = E(XT +h |X1 , XT ) = 0 + 1 [T + h] + k 1(T + h = k mod. s) k=0> n=length(T)> T0=T[(n-23):n]+2> plot(X,xlim=range(c(T,T0)))> X1p=predict(regT,newdata=data.frame(T=T0))> Yp=X1p+B$X2[(n-23):n]> se=sd(X-Y)> YpU=Yp+1.96*se> YpL=Yp-1.96*se> polygon(c(T0,rev(T0)),c(YpU,rev(YpL)),col="yellow",border=NA)> lines(T0,Yp,col="red",lwd=3)> lines(B$T,Y,lwd=3,col="red")

12. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Mthode de Buys Balot - prvision e e 70000 60000 50000 X 40000 30000 20000 1990 1992 1994 1996 1998 Time

13. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Formalisation de Buys-Ballot (1847)> sncf=read.table(+ "http://freakonometrics.blog.free.fr/public/data/sncf.csv",+ header=TRUE,sep=";")> SNCF=ts(as.vector(t(as.matrix(sncf[,2:13]))),+ ,start = c(1963, 1), frequency = 12)> plot(SNCF,lwd=2,col="purple")

14. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Formalisation de Buys-Ballot (1847)La srie Xt est la somme de 2 composantes dterministes : une tendance Zt , e edune saisonnalit St et dune composante alatoire t e e Xt = Zt + St + t .On suppose que Zt et St sont des combinaisons linaires de fonctions connues e i jdans le temps, Zt et St , i.e. Z = + Z 1 + Z 2 + ... + Z m t 0 t 1 t 2 t m (ex : 0 + 1 t) St = S 1 1 + S 2 2 + ... + S s n . t t tLe but est destimer les 1 , ..., m et 1 , ..., n ` partir des T observations. a m s i j Xt = Z t i + St j + t pour t = 1, ..., T. i=1 j=1

15. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Formalisation de Buys-Ballot (1847)La forme de St dpend du type de donnes, et de la forme de la saisonnalit. On e e e iconsid`rera ici des fonctions St indicatrices, e 0 si t = mois i 0 si t = 0 [modulo i] i i St = ou St = 1 si t = mois i 1 si t = 0 [modulo i] .Example : Considrons des donnes trimestrielles, e e 1 2 3 4 Xt = + t + 1 St + 2 St + 3 St + 4 St + t , Zt St

16. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012)Que lon peut crire de faon matricielle, e c 5130 1 1 1 0 0 0 1 6410 1 2 0 1 0 0 2 8080 1 3 0 0 1 0 3 5900 1 4 0 0 0 1 4 5110 1 5 1 0 0 0 5 1 6680 = 1 6 0 1 0 0 + 6 2 8350 1 7 0 0 1 0 7 3 5910 1 8 0 0 0 1 8 4 5080 1 9 1 0 0 0 9 . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 4 Xt 1 t St St St St t

17. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Formalisation de Buys-Ballot (1847)Remark : pour transformer les donnes en donnes trimestrielles (et non plus e emensuelles)> SNCFQ= ts(apply(matrix(as.numeric(SNCF),3,length(SNCF)/3),2,sum),+ start = c(1963, 1), frequency = 4)> plot(SNCFQ,col="red")> SNCFQ Qtr1 Qtr2 Qtr3 Qtr41963 5130 6410 8080 59001964 5110 6680 8350 59101965 5080 6820 8190 59901966 5310 6600 8090 6020... etc.

18. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Formalisation de Buys-Ballot (1847)Le graphique des donnes trimestrielles est le suivant e

19. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Formalisation de Buys-Ballot (1847)On consid`re un mod`le de la forme e e m s i j Xt = Z t i + St j + t pour t = 1, ..., T. i=1 j=1La mthode des moindres carrs ordinaires consiste ` choisir les i et j de faon e e a ca` minimiser le carr des erreurs e T i , j = arg min 2 t t=1 2 T m s Xt i j = arg min Zt i + St j . t=1 i=1 j=1

20. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) Formalisation de Buys-Ballot (1847)> T = seq(from=1963,to=1980.75,by=.25)> Q = rep(1:4,18)> reg=lm(SNCFQ~0+T+as.factor(Q))> summary(reg)Coefficients: Estimate Std. Error t value Pr(>|t|)T 231.87 12.55 18.47 plot(D,T,col="light blue",xlab="Temperature minimalejournaliere Paris",ylab="",cex=.5)1. Estimer la tendance linaire Xt = [0 + 1 t] + Yt e> abline(h=0,lty=2,col="grey")> abline(lm(T~D),lwd=2,col="red")

25. ` Arthur CHARPENTIER - Modeles de previsions (ACT6420 - Automne 2012) q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q