library(forecast)
## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo
library(fpp2)
## -- Attaching packages ---------------------------------------------- fpp2 2.4 --
## v ggplot2 3.3.3 v expsmooth 2.3
## v fma 2.4
##
# SARIMA(1,1,1)(1,1,1)4
set.seed(123)
ggtsdisplay(euretail)
ggtsdisplay(diff(euretail))
ggtsdisplay(diff(diff(euretail),lag=4)) # (1-B^4)*(1-B)
print(euretail,calendar = TRUE)
## Qtr1 Qtr2 Qtr3 Qtr4
## 1996 89.13 89.52 89.88 90.12
## 1997 89.19 89.78 90.03 90.38
## 1998 90.27 90.77 91.85 92.51
## 1999 92.21 92.52 93.62 94.15
## 2000 94.69 95.34 96.04 96.30
## 2001 94.83 95.14 95.86 95.83
## 2002 95.73 96.36 96.89 97.01
## 2003 96.66 97.76 97.83 97.76
## 2004 98.17 98.55 99.31 99.44
## 2005 99.43 99.84 100.32 100.40
## 2006 99.88 100.19 100.75 101.01
## 2007 100.84 101.34 101.94 102.10
## 2008 101.56 101.48 101.13 100.34
## 2009 98.93 98.31 97.67 97.44
## 2010 96.53 96.56 96.51 96.70
## 2011 95.88 95.84 95.79 95.94
fit <- Arima(euretail,order=c(1,1,1),seasonal=c(1,1,1))
fit
## Series: euretail
## ARIMA(1,1,1)(1,1,1)[4]
##
## Coefficients:
## ar1 ma1 sar1 sma1
## 0.8836 -0.5202 -0.0286 -0.9389
## s.e. 0.1376 0.1715 0.1630 0.3874
##
## sigma^2 estimated as 0.1549: log likelihood=-30.1
## AIC=70.2 AICc=71.33 BIC=80.59
summary(fit)
## Series: euretail
## ARIMA(1,1,1)(1,1,1)[4]
##
## Coefficients:
## ar1 ma1 sar1 sma1
## 0.8836 -0.5202 -0.0286 -0.9389
## s.e. 0.1376 0.1715 0.1630 0.3874
##
## sigma^2 estimated as 0.1549: log likelihood=-30.1
## AIC=70.2 AICc=71.33 BIC=80.59
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.02600165 0.3648673 0.2753599 -0.02429513 0.285308 0.2239913
## ACF1
## Training set -0.04063608
fit.aa <- auto.arima(euretail,trace=TRUE)
##
## ARIMA(2,1,2)(1,1,1)[4] : 74.62219
## ARIMA(0,1,0)(0,1,0)[4] : 95.27908
## ARIMA(1,1,0)(1,1,0)[4] : 77.00764
## ARIMA(0,1,1)(0,1,1)[4] : 75.71724
## ARIMA(2,1,2)(0,1,1)[4] : 72.20713
## ARIMA(2,1,2)(0,1,0)[4] : Inf
## ARIMA(2,1,2)(0,1,2)[4] : 74.77941
## ARIMA(2,1,2)(1,1,0)[4] : 77.25877
## ARIMA(2,1,2)(1,1,2)[4] : Inf
## ARIMA(1,1,2)(0,1,1)[4] : 70.37525
## ARIMA(1,1,2)(0,1,0)[4] : 83.82329
## ARIMA(1,1,2)(1,1,1)[4] : 72.83265
## ARIMA(1,1,2)(0,1,2)[4] : 72.84282
## ARIMA(1,1,2)(1,1,0)[4] : 76.71941
## ARIMA(1,1,2)(1,1,2)[4] : Inf
## ARIMA(0,1,2)(0,1,1)[4] : 74.26992
## ARIMA(1,1,1)(0,1,1)[4] : Inf
## ARIMA(1,1,3)(0,1,1)[4] : 70.75227
## ARIMA(0,1,3)(0,1,1)[4] : 68.39031
## ARIMA(0,1,3)(0,1,0)[4] : 77.10225
## ARIMA(0,1,3)(1,1,1)[4] : 69.89516
## ARIMA(0,1,3)(0,1,2)[4] : 70.40051
## ARIMA(0,1,3)(1,1,0)[4] : 72.30399
## ARIMA(0,1,3)(1,1,2)[4] : Inf
##
## Best model: ARIMA(0,1,3)(0,1,1)[4]
autoplot(euretail,series="Data") +
autolayer(fit$fitted,series="SARIMA(1,1,1)(1,1,1)[4]") +
autolayer(fit.aa$fitted,series="Auto arima")
cor(data.frame(euretail,fit$fitted,fit.aa$fitted))
## euretail fit.fitted fit.aa.fitted
## euretail 1.0000000 0.9955354 0.9955344
## fit.fitted 0.9955354 1.0000000 0.9996528
## fit.aa.fitted 0.9955344 0.9996528 1.0000000
summary(fit)
## Series: euretail
## ARIMA(1,1,1)(1,1,1)[4]
##
## Coefficients:
## ar1 ma1 sar1 sma1
## 0.8836 -0.5202 -0.0286 -0.9389
## s.e. 0.1376 0.1715 0.1630 0.3874
##
## sigma^2 estimated as 0.1549: log likelihood=-30.1
## AIC=70.2 AICc=71.33 BIC=80.59
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.02600165 0.3648673 0.2753599 -0.02429513 0.285308 0.2239913
## ACF1
## Training set -0.04063608
summary(fit.aa)
## Series: euretail
## ARIMA(0,1,3)(0,1,1)[4]
##
## Coefficients:
## ma1 ma2 ma3 sma1
## 0.2630 0.3694 0.4200 -0.6636
## s.e. 0.1237 0.1255 0.1294 0.1545
##
## sigma^2 estimated as 0.156: log likelihood=-28.63
## AIC=67.26 AICc=68.39 BIC=77.65
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.02965298 0.3661147 0.2787802 -0.02795377 0.2885545 0.2267735
## ACF1
## Training set 0.006455781
checkresiduals(fit)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(1,1,1)(1,1,1)[4]
## Q* = 4.4884, df = 4, p-value = 0.3439
##
## Model df: 4. Total lags used: 8
checkresiduals(fit.aa)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(0,1,3)(0,1,1)[4]
## Q* = 0.51128, df = 4, p-value = 0.9724
##
## Model df: 4. Total lags used: 8
library(gridExtra)
grid.arrange(autoplot(fit),autoplot(fit.aa))
# Forecasting
grid.arrange(autoplot(forecast(fit.aa)), autoplot(forecast(fit)))
