Projet
2024-10-30
library(tseries)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(caschrono)## Le chargement a nécessité le package : zoo##
## Attachement du package : 'zoo'## Les objets suivants sont masqués depuis 'package:base':
##
## as.Date, as.Date.numericlibrary(forecast)Chargement des données :
data=read.csv2(file="4MA_ST_2024_Projet.csv", header=TRUE, sep=";", dec=",")
sum(is.na(data)) #vérification de la présence ou non de NA ## [1] 0head(data)## Year Quarter ASK_NB
## 1 1993 1 2.99201e+11
## 2 1993 2 3.18298e+11
## 3 1993 3 3.36073e+11
## 4 1993 4 3.22796e+11
## 5 1994 1 3.19586e+11
## 6 1994 2 3.33927e+11tail(data)## Year Quarter ASK_NB
## 119 2022 3 1.30407e+12
## 120 2022 4 1.14489e+12
## 121 2023 1 1.24400e+12
## 122 2023 2 1.42893e+12
## 123 2023 3 1.59119e+12
## 124 2023 4 1.43466e+12On passe les données en log car beaucoup trop grandes
data.log=data
data.log$ASK_NB=log(data.log$ASK_NB)
data.log.ts=ts(data=data.log$ASK_NB, start=c(1993,1), end=c(2023,4), frequency=4)
head(data.log)## Year Quarter ASK_NB
## 1 1993 1 26.42438
## 2 1993 2 26.48625
## 3 1993 3 26.54059
## 4 1993 4 26.50029
## 5 1994 1 26.49029
## 6 1994 2 26.53419tail(data.log)## Year Quarter ASK_NB
## 119 2022 3 27.89651
## 120 2022 4 27.76633
## 121 2023 1 27.84935
## 122 2023 2 27.98795
## 123 2023 3 28.09550
## 124 2023 4 27.99195On parlera désormais toujours des données en log (si non précisé) lorsque que l’on parlera des ASK.
plot(data.log.ts,ylab="ASK_NB",main="ASK en fonction du temps")
Décomposition de la série à l’aide d’un modèle de régression linéaire
trimestre1 <- rep(c(1,0,0,0),31)
trimestre2 <- rep(c(0,1,0,0),31)
trimestre3 <- rep(c(0,0,1,0),31)
#on met que 3 trimestres sur 4 pour avoir la matrice X inversible
tendance <- seq(1993, 2023 + 3/4, 1/4)
mod = lm(data.log$ASK_NB ~ tendance + trimestre1 + trimestre2 + trimestre3)
summary(mod)##
## Call:
## lm(formula = data.log$ASK_NB ~ tendance + trimestre1 + trimestre2 +
## trimestre3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.43690 -0.03689 0.01112 0.08112 0.25896
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -63.513622 3.894717 -16.308 <2e-16 ***
## tendance 0.045176 0.001939 23.301 <2e-16 ***
## trimestre1 -0.030542 0.049070 -0.622 0.535
## trimestre2 -0.006310 0.049058 -0.129 0.898
## trimestre3 0.078732 0.049051 1.605 0.111
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1931 on 119 degrees of freedom
## Multiple R-squared: 0.8223, Adjusted R-squared: 0.8163
## F-statistic: 137.7 on 4 and 119 DF, p-value: < 2.2e-16Tendance significative car petite p-value
On enlève le trimestre 2 car il n’est pas significatif (grande p-value)
mod <- lm(data.log$ASK_NB ~ tendance + trimestre1 + trimestre3)
summary(mod)##
## Call:
## lm(formula = data.log$ASK_NB ~ tendance + trimestre1 + trimestre3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.44011 -0.03771 0.01279 0.08245 0.25576
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -63.52667 3.87741 -16.384 <2e-16 ***
## tendance 0.04518 0.00193 23.404 <2e-16 ***
## trimestre1 -0.02738 0.04231 -0.647 0.5187
## trimestre3 0.08189 0.04230 1.936 0.0553 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1923 on 120 degrees of freedom
## Multiple R-squared: 0.8223, Adjusted R-squared: 0.8178
## F-statistic: 185.1 on 3 and 120 DF, p-value: < 2.2e-16On enlève le trimestre 1 car il n’est pas significatif
mod <- lm(data.log$ASK_NB ~ tendance + trimestre3)
summary(mod)##
## Call:
## lm(formula = data.log$ASK_NB ~ tendance + trimestre3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.43132 -0.03498 0.01411 0.08391 0.26460
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -63.593040 3.866733 -16.446 <2e-16 ***
## tendance 0.045209 0.001925 23.481 <2e-16 ***
## trimestre3 0.091010 0.039789 2.287 0.0239 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1918 on 121 degrees of freedom
## Multiple R-squared: 0.8217, Adjusted R-squared: 0.8187
## F-statistic: 278.8 on 2 and 121 DF, p-value: < 2.2e-16par(mfrow = c(2,2))
plot(mod)
checkresiduals(mod)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 56.65, df = 10, p-value = 1.551e-08plot(mod$fitted.values,mod$residuals,ylab="residus",xlab="valeurs ajustées")
abline(h=0)
plot(mod$fitted.values,rstudent(mod),ylab="residus",xlab="valeurs ajustées",ylim=c(-12,3))
abline(h=c(-2,2),col="chartreuse4")
plot(tendance, data.log$ASK_NB, type = "l",col="chartreuse4",ylab="time",xlab="ASK_NB",main="Values and fitted values")
lines(tendance, mod$fitted.values, col = "red")
Lissage Exponentiel
Simple
data.log.ts.decompose=decompose(data.log.ts)
plot(diff(diff(data.log.ts,lag=4),lag=1),col="chartreuse4")
lines(data.log.ts-data.log.ts.decompose$seasonal-data.log.ts.decompose$trend,col="red")
legend("topleft", legend=c("lag 1 et 12","decompose"),col=c("chartreuse4", "red"), lty=1, cex=0.8)
data.log.ts.notrend=na.omit(data.log.ts-data.log.ts.decompose$trend)
data.log.ts.notrend.noseason=data.log.ts.notrend-data.log.ts.decompose$seasonal
data.log.ts.noseason=data.log.ts-data.log.ts.decompose$seasonaldata.log.ts.notrend.noseason.ses=HoltWinters(x=data.log.ts.notrend.noseason,beta=FALSE,gamma=FALSE)
data.log.ts.notrend.noseason.ses$alpha## [1] 0.01393039data.log.ts.notrend.noseason.ses$SSE## [1] 1.097775plot(data.log.ts.notrend.noseason.ses)
data.log.ts.notrend.noseason.ses=HoltWinters(x=data.log.ts.notrend.noseason,alpha=0.7,beta=FALSE,gamma=FALSE)
data.log.ts.notrend.noseason.ses$alpha## [1] 0.7data.log.ts.notrend.noseason.ses$SSE## [1] 2.073741plot(data.log.ts.notrend.noseason.ses)
data.log.ts.notrend.noseason=diff(diff(data.log.ts,lag=4),lag=1)
data.log.ts.notrend.noseason.ses=HoltWinters(x=data.log.ts.notrend.noseason,beta=FALSE,gamma=FALSE)
data.log.ts.notrend.noseason.ses$alpha## [1] 6.610696e-05data.log.ts.notrend.noseason.ses$SSE## [1] 6.140676plot(data.log.ts.notrend.noseason.ses)
data.log.ts.notrend.noseason=diff(diff(data.log.ts,lag=4),lag=1)
data.log.ts.notrend.noseason.ses=HoltWinters(x=data.log.ts.notrend.noseason,alpha=0.7,beta=FALSE,gamma=FALSE)
data.log.ts.notrend.noseason.ses$alpha## [1] 0.7data.log.ts.notrend.noseason.ses$SSE## [1] 11.18091plot(data.log.ts.notrend.noseason.ses)
Double
data.log.ts.notrend=na.omit(data.log.ts-data.log.ts.decompose$trend)
data.log.ts.notrend.noseason=data.log.ts.notrend-data.log.ts.decompose$seasonal
data.log.ts.noseason=data.log.ts-data.log.ts.decompose$seasonaldata.log.ts.noseason.des=HoltWinters(x=data.log.ts.noseason,gamma=FALSE)## Warning in HoltWinters(x = data.log.ts.noseason, gamma = FALSE): difficultés
## d'optimisation : ERROR: ABNORMAL_TERMINATION_IN_LNSRCHdata.log.ts.noseason.des$alpha## alpha
## 0.6710605data.log.ts.noseason.des$beta## beta
## 0.01777311data.log.ts.noseason.des$SSE## [1] 2.804286plot(data.log.ts.noseason.des)
data.log.ts.noseason.des=HoltWinters(x=data.log.ts.noseason,gamma=FALSE,alpha=0.6,beta=0.6)
data.log.ts.noseason.des$alpha## [1] 0.6data.log.ts.noseason.des$beta## [1] 0.6data.log.ts.noseason.des$SSE## [1] 3.700657plot(data.log.ts.noseason.des)
Holt-Winters
La saisonnalité a l’air globalement constante donc on part plus sur un modèle additif
data.log.ts.HoltWinters.additive=HoltWinters(data.log.ts,seasonal = "additive")
data.log.ts.HoltWinters.additive$alpha## alpha
## 0.630329data.log.ts.HoltWinters.additive$beta## beta
## 0data.log.ts.HoltWinters.additive$gamma## gamma
## 0data.log.ts.HoltWinters.additive$SSE## [1] 2.761871plot(data.log.ts.HoltWinters.additive)
Modèles (S)AR(I)MA
Modèle ARMA
Série résiduelle
data.log.ts.notrend=na.omit(data.log.ts-data.log.ts.decompose$trend)
data.log.ts.notrend.noseason=data.log.ts.notrend-data.log.ts.decompose$seasonal
data.log.ts.noseason=data.log.ts-data.log.ts.decompose$seasonalTest de la stationnarité à l’aide du test de Dickey-Fuller
#on utilise la serie residuelle donc on sait deja que la tendance est nulle, on passe direct a l'etape suivante
delta_data.log.ts.notrend.noseason=diff(data.log.ts.notrend.noseason)
serie_moins_1=data.log.ts.notrend.noseason[-length(data.log.ts.notrend.noseason)]
summary(lm(formula = delta_data.log.ts.notrend.noseason ~ serie_moins_1))##
## Call:
## lm(formula = delta_data.log.ts.notrend.noseason ~ serie_moins_1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.77335 -0.02263 -0.00102 0.02456 0.41846
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.0002495 0.0083520 -0.03 0.976
## serie_moins_1 -1.2976025 0.0883621 -14.69 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09111 on 117 degrees of freedom
## Multiple R-squared: 0.6483, Adjusted R-squared: 0.6453
## F-statistic: 215.7 on 1 and 117 DF, p-value: < 2.2e-16adf.test(data.log.ts.notrend.noseason)## Warning in adf.test(data.log.ts.notrend.noseason): p-value smaller than printed
## p-value##
## Augmented Dickey-Fuller Test
##
## data: data.log.ts.notrend.noseason
## Dickey-Fuller = -7.812, Lag order = 4, p-value = 0.01
## alternative hypothesis: stationaryDétermination des ordres p et q du modèle
On peut ainsi tracer les graphes des acf et pacf
par(mfrow=c(1,2))
acf(data.log.ts.notrend.noseason)
pacf(data.log.ts.notrend.noseason)
Grâce à l’ACF on trouve q=2 et grâce au PACF, on trouve p=6
On se retrouve donc avec un modèle ARMA(6,2)
Suppression des coefficients non significatifs
modele1=arima(data.log.ts.notrend.noseason, order=c(6,0,2))
t_stat(modele1)## ar1 ar2 ar3 ar4 ar5 ar6 ma1
## t.stat -0.179096 -0.571911 -0.039604 -0.343206 -0.729837 -0.780751 -0.518654
## p.val 0.857862 0.567382 0.968409 0.731443 0.465490 0.434949 0.604002
## ma2 intercept
## t.stat -0.366685 -0.031670
## p.val 0.713854 0.974735On supprime les coefficients non significatifs
modele2=arima(data.log.ts.notrend.noseason, order=c(6,0,2),fixed=c(NA,NA,NA,NA,NA,NA,NA,NA,0))
t_stat(modele2)## ar1 ar2 ar3 ar4 ar5 ar6 ma1
## t.stat 11.88989 -3.171445 2.128154 -1.120168 0.022555 0.005831 -9.284231
## p.val 0.00000 0.001517 0.033324 0.262642 0.982005 0.995348 0.000000
## ma2
## t.stat 4.639807
## p.val 0.000003modele3=arima(data.log.ts.notrend.noseason, order=c(5,0,2),fixed=c(NA,NA,NA,NA,NA,NA,NA,0))
t_stat(modele3)## ar1 ar2 ar3 ar4 ar5 ma1 ma2
## t.stat -1.143104 -0.189301 -0.696197 0.097244 -0.868919 -0.115717 -1.514863
## p.val 0.252996 0.849857 0.486305 0.922532 0.384891 0.907876 0.129807modele4=arima(data.log.ts.notrend.noseason, order=c(5,0,2),fixed=c(NA,NA,NA,NA,NA,0,NA,0))
t_stat(modele4)## ar1 ar2 ar3 ar4 ar5 ma2
## t.stat -8.369831 -0.173379 -1.172925 0.139257 -0.821698 -34.27537
## p.val 0.000000 0.862354 0.240826 0.889247 0.411249 0.00000modele5=arima(data.log.ts.notrend.noseason, order=c(5,0,2),fixed=c(NA,NA,NA,0,NA,0,NA,0),method="CSS")## Warning in arima(data.log.ts.notrend.noseason, order = c(5, 0, 2), fixed =
## c(NA, : certains des paramètres AR etaient fixés : transform.pars = FALSEt_stat(modele5)## ar1 ar2 ar3 ar5 ma2
## t.stat -12.94807 -1.872890 -2.152993 -1.535810 -145.6863
## p.val 0.00000 0.061084 0.031319 0.124585 0.0000modele6=arima(data.log.ts.notrend.noseason, order=c(3,0,2),fixed=c(NA,NA,NA,0,NA,0),method="CSS")
t_stat(modele6)## ar1 ar2 ar3 ma2
## t.stat -23.07748 -0.574012 -4.885643 -101.4561
## p.val 0.00000 0.565960 0.000001 0.0000modele7=arima(data.log.ts.notrend.noseason, order=c(3,0,2),fixed=c(NA,0,NA,0,NA,0),method="CSS")## Warning in arima(data.log.ts.notrend.noseason, order = c(3, 0, 2), fixed =
## c(NA, : certains des paramètres AR etaient fixés : transform.pars = FALSEt_stat(modele7)## ar1 ar3 ma2
## t.stat -45.65908 -30.54278 -83.33558
## p.val 0.00000 0.00000 0.00000summary(modele7)##
## Call:
## arima(x = data.log.ts.notrend.noseason, order = c(3, 0, 2), fixed = c(NA, 0,
## NA, 0, NA, 0), method = "CSS")
##
## Coefficients:
## ar1 ar2 ar3 ma1 ma2 intercept
## -0.8013 0 -0.276 0 -1.0995 0
## s.e. 0.0175 0 0.009 0 0.0132 0
##
## sigma^2 estimated as 0.004727: part log likelihood = 150.99
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.0005078177 0.06788911 0.0257532 28.15948 108.9935 0.4779648
## ACF1
## Training set -0.02816611checkresiduals(modele7)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(3,0,2) with non-zero mean
## Q* = 6.1685, df = 3, p-value = 0.1037
##
## Model df: 5. Total lags used: 8pacf(modele7$residuals)
C’est bien décorélé (pas de process ma ou ar dans les résidus donc ok) et ca suit une espece de loi normale
plot(data.log.ts.notrend.noseason,col="chartreuse4",ylab="Série résiduelle")
lines(fitted(modele7),col="blue")
legend("bottomleft", legend=c("Série", "Série ajustée"),
col=c("chartreuse4", "blue"), lty=1)
plot(data.log.ts,ylab="ASK_NB",col="chartreuse4")
lines(data.log.ts.decompose$trend+data.log.ts.decompose$seasonal+fitted(modele7),col="blue")
legend("topleft", legend=c("Série", "Série ajustée"),
col=c("chartreuse4", "blue"), lty=1)
plot(forecast(modele7))## Warning in predict.Arima(object, n.ahead = h): La partie MA du modèle est non
## inversible
Modèle ARIMA
Détermination des coefficients du modèle ARIMA
par(mfrow=c(1,2))
acf(diff(data.log.ts.noseason))
pacf(diff(data.log.ts.noseason))
Suppression des coefficients non significatifs
modarima=arima(data.log.ts.noseason, order=c(2,1,1))
t_stat(modarima)## ar1 ar2 ma1
## t.stat -0.663126 -1.224379 0.056284
## p.val 0.507250 0.220809 0.955115arima2=arima(data.log.ts.noseason, order=c(2,1,0))
t_stat(arima2)## ar1 ar2
## t.stat -3.418936 -2.023382
## p.val 0.000629 0.043034summary(arima2)##
## Call:
## arima(x = data.log.ts.noseason, order = c(2, 1, 0))
##
## Coefficients:
## ar1 ar2
## -0.3024 -0.178
## s.e. 0.0884 0.088
##
## sigma^2 estimated as 0.02218: log likelihood = 59.63, aic = -113.26
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.01864005 0.1483493 0.05574108 0.0664706 0.2058271 0.9759373
## ACF1
## Training set -0.01710958checkresiduals(arima2)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(2,1,0)
## Q* = 2.3897, df = 6, p-value = 0.8806
##
## Model df: 2. Total lags used: 8pacf(arima2$residuals)
plot(data.log.ts.noseason,col="chartreuse4",ylab="Série sans saisonnalité")
lines(fitted(arima2),col="blue")
legend("bottomleft", legend=c("Série", "Série ajustée"),
col=c("chartreuse4", "blue"), lty=1)
plot(data.log.ts,ylab="ASK_NB",col="chartreuse4")
lines(data.log.ts.decompose$seasonal+fitted(arima2),col="blue")
legend("topleft", legend=c("Série", "Série ajustée"),
col=c("chartreuse4", "blue"), lty=1)
Modèle SARIMA
diff_season <- diff(data.log.ts, lag = 4) # Différenciation saisonnière
plot(diff_season)
adf.test(diff_season)## Warning in adf.test(diff_season): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: diff_season
## Dickey-Fuller = -4.5577, Lag order = 4, p-value = 0.01
## alternative hypothesis: stationaryserie stationnaire donc D=1
acf(diff_season)
pacf(diff_season)
On regarde les pics aux lags multiples de la période (d’après p183 du poly)
acf –> Q=0
pacf –> P=1
sarima=arima(data.log.ts, order = c(2,1,1), seasonal = list(order = c(0,1,1), period = 4))
t_stat(sarima)## ar1 ar2 ma1 sma1
## t.stat 6.526795 1.405316 -17.21847 -14.15189
## p.val 0.000000 0.159927 0.00000 0.00000sarima2=arima(data.log.ts, order = c(1,1,1), seasonal = list(order = c(0,1,1), period = 4))
t_stat(sarima2)## ar1 ma1 sma1
## t.stat 9.600516 -4.944707 -14.09764
## p.val 0.000000 0.000001 0.00000checkresiduals(sarima2)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(1,1,1)(0,1,1)[4]
## Q* = 5.1633, df = 5, p-value = 0.3963
##
## Model df: 3. Total lags used: 8pacf(sarima2$residuals)
pas de ma ni ar dans les residus donc ok
plot(data.log.ts,ylab="ASK_NB",col="chartreuse4")
lines(fitted(sarima2),col="blue")
legend("topleft", legend=c("Série", "Série fitted avec SARIMA"),
col=c("chartreuse4", "blue"), lty=1)
Calcul des RMSE
summary(sarima2) #SARIMA##
## Call:
## arima(x = data.log.ts, order = c(1, 1, 1), seasonal = list(order = c(0, 1, 1),
## period = 4))
##
## Coefficients:
## ar1 ma1 sma1
## 0.6791 -1.0000 -0.9101
## s.e. 0.0707 0.2022 0.0646
##
## sigma^2 estimated as 0.02214: log likelihood = 51, aic = -94.01
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.008523232 0.1458693 0.04008316 -0.03350983 0.1482445 0.506924
## ACF1
## Training set -0.0923017summary(arima2) #ARIMA##
## Call:
## arima(x = data.log.ts.noseason, order = c(2, 1, 0))
##
## Coefficients:
## ar1 ar2
## -0.3024 -0.178
## s.e. 0.0884 0.088
##
## sigma^2 estimated as 0.02218: log likelihood = 59.63, aic = -113.26
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.01864005 0.1483493 0.05574108 0.0664706 0.2058271 0.9759373
## ACF1
## Training set -0.01710958summary(modele7) #ARMA##
## Call:
## arima(x = data.log.ts.notrend.noseason, order = c(3, 0, 2), fixed = c(NA, 0,
## NA, 0, NA, 0), method = "CSS")
##
## Coefficients:
## ar1 ar2 ar3 ma1 ma2 intercept
## -0.8013 0 -0.276 0 -1.0995 0
## s.e. 0.0175 0 0.009 0 0.0132 0
##
## sigma^2 estimated as 0.004727: part log likelihood = 150.99
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.0005078177 0.06788911 0.0257532 28.15948 108.9935 0.4779648
## ACF1
## Training set -0.02816611n=length(data.log.ts)
sqrt(data.log.ts.notrend.noseason.ses$SSE / n) #SES## [1] 0.3002809sqrt(data.log.ts.noseason.des$SSE / n) #DES## [1] 0.1727542sqrt(data.log.ts.HoltWinters.additive$SSE /n) #HW## [1] 0.1492419Modèles sans covid + Prévisions
data.log.ts.nocovid=ts(data=data.log$ASK_NB, start=c(1993,1), end=c(2019,4), frequency=4)Séparation en set d’entrainement et validation pour vérifier nos résultats
data.log.ts.nocovid.training=window(x=data.log.ts.nocovid, start=c(1993,1), end=c(2014,4), frequency=4)
data.log.ts.nocovid.validation=window(x=data.log.ts.nocovid, start=c(2015,1), end=c(2019,4), frequency=4)HW
data.log.ts.nocovid.HoltWinters.additive.training=HoltWinters(data.log.ts.nocovid.training,seasonal = "additive")plot(forecast(data.log.ts.nocovid.HoltWinters.additive.training,h=5*4))
lines(data.log.ts.nocovid.validation,col="green",lwd=2)
SARIMA
data.log.ts.nocovid.sarima.training=auto.arima(data.log.ts.nocovid.training,d=1,D=1,seasonal=TRUE)checkresiduals(data.log.ts.nocovid.sarima.training)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(0,1,1)(0,1,1)[4]
## Q* = 2.9615, df = 6, p-value = 0.8137
##
## Model df: 2. Total lags used: 8pacf(data.log.ts.nocovid.sarima.training$residuals)
plot(forecast(data.log.ts.nocovid.sarima.training,h=5*4))
lines(data.log.ts.nocovid.validation,col="green",lwd=2)
plot(forecast(data.log.ts.nocovid.sarima.training,h=10*4))
lines(data.log.ts,col="green",lwd=2)
Prévisions
data.log.ts.nocovid.HoltWinters.additive=HoltWinters(data.log.ts.nocovid,seasonal = "additive")
checkresiduals(data.log.ts.nocovid.HoltWinters.additive)
##
## Ljung-Box test
##
## data: Residuals from HoltWinters
## Q* = 24.953, df = 8, p-value = 0.001583
##
## Model df: 0. Total lags used: 8plot(forecast(data.log.ts.nocovid.HoltWinters.additive,h=23*4+1))
f=forecast(data.log.ts.nocovid.HoltWinters.additive,h=23*4+1)
tail(f$lower,1)## 80% 95%
## 2043 Q1 28.85268 28.74097tail(f$upper,1)## 80% 95%
## 2043 Q1 29.27474 29.38645