AS2.4
October 16, 2019
Data import
require(fpp2)
set.seed(123)
setwd("C:/Users/Ryu Uezato/Desktop/Monash/Forecasting/AS2")
dt = read.csv("Visitor Arrivals.csv", header = FALSE)
dt.1 = ts(dt[,-1], start = c(1978,1), end = c(2019,7), frequency = 12)Phase1 Model Identification
Plotting Full Set
autoplot(dt.1)+
stat_smooth(color = "#FC4E07", fill = "#FC4E07", method = "loess")+
ylab("Number of Visitor Arrivals")+
xlab("Years")+
ggtitle("Number of Visitors Arrivals in Singapore from 1978 to 2019")+
theme_light()+
theme(plot.title = element_text(size = 13))
Correcting outliers
dt = dt[,-1]
dt[12*25+4] = round(mean(dt[c(12*24+4, 12*26+4)]), digits = 0)
dt[12*25+5] = round(mean(dt[c(12*24+5, 12*26+5)]), digits = 0)
dt[12*25+6] = round(mean(dt[c(12*24+6, 12*26+6)]), digits = 0) #taking average of the same month of a year before and later of outliers month ie mean(2002,4 and 2004,4) etc
dt = ts(dt, start = c(1978,1), end = c(2019,7), frequency = 12)
autoplot(dt)+
stat_smooth(color = "#FC4E07", fill = "#FC4E07", method = "loess")+
ylab("Number of Visitor Arrivals")+
xlab("Years")+
ggtitle("Number of Visitors Arrivals in Singapore from 1978 to 2019")+
theme_light()+
theme(plot.title = element_text(size = 20))
Chopping Data into train and test
train = head(dt, round(length(dt)*0.8)) %>% as.ts()
h = length(dt) -length(train)
test = tail(dt, h) %>% as.ts()
autoplot(train, series = "Training Set")+autolayer(test, series = "Test Set")
Plotting trainig set to determine if transformation is needed
autoplot(train)+
theme_light()+
stat_smooth(color = "lightgreen", fill = "lightgreen", method = "loess")+
ylab("Number of Visitor Arrivals")+
xlab("Years")+
ggtitle("Number of Visitors Arrivals in Singapore from 1978 to 2011(Training Set)")+
theme_light()+
theme(plot.title = element_text(size = 20))
Plotting Seasonal Plot of trainig set
ggseasonplot(train, year.labels = TRUE, year.labels.left = TRUE)+
ylab("Number of Visitor Arrivals ") +
xlab("Months") +
ggtitle("Seasonal plot : Number of Visitors Arrivals in Singapore") 
ggsubseriesplot(train)+
ylab("Number of Visitor Arrivals")+
xlab("Years")+
ggtitle("Subseries plot : Number of Visitors Arrivals in Singapore")
ggAcf(train, lag.max = 48)+
ggtitle("ACF plot : Number of Visitors Arrivals in Singapore")
Variance and Mean stabilization
lambda.dt = BoxCox.lambda(train)
train.v = BoxCox(train, lambda = lambda.dt)
train.v %>% nsdiffs()[1] 1train.v %>% diff(lag = 12) %>% ndiffs()[1] 0train.v %>% diff(lag = 12) %>% autoplot()
train.v %>% diff(lag = 12) %>% ggtsdisplay(main = "Seasonally differenced Visitors Arrivals in Singapore ")
Possible Model List
Based on the ggtsdisplay above we may consider
- ARIMA(3,0,0)(3,1,0)[12] with constant - ARIMA(4,0,0)(3,1,0)[12] with constant - ARIMA(3,0,0)(2,1,0)[12] with constant - ARIMA(2,0,0)(2,1,0)[12] with contant
Phase2 Estimation and Testing
Estimating Potential Model
fit1 = Arima(train, order = c(3,0,0), seasonal = c(3,1,0), include.constant = TRUE, lambda=lambda.dt)
summary(fit1)Series: train
ARIMA(3,0,0)(3,1,0)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 sar1 sar2 sar3 drift
0.5335 0.2545 0.1306 -0.5219 -0.3968 -0.2349 0.0032
s.e. 0.0505 0.0559 0.0508 0.0508 0.0533 0.0501 0.0008
sigma^2 estimated as 0.001073: log likelihood=774.73
AIC=-1533.47 AICc=-1533.09 BIC=-1501.8
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -282.8799 27287.52 17644.32 -0.07659795 3.296094 0.4049379 -0.04304853fit2 = Arima(train, order = c(4,0,0), seasonal = c(3,1,0), include.constant = TRUE, lambda=lambda.dt)
summary(fit2)Series: train
ARIMA(4,0,0)(3,1,0)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 ar4 sar1 sar2 sar3 drift
0.5113 0.2094 0.0414 0.1728 -0.5499 -0.4045 -0.2426 0.0033
s.e. 0.0502 0.0566 0.0564 0.0506 0.0510 0.0538 0.0500 0.0009
sigma^2 estimated as 0.001044: log likelihood=780.47
AIC=-1542.93 AICc=-1542.45 BIC=-1507.31
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -234.7073 26988.03 17538.64 -0.08664446 3.258615 0.4025127 -0.01952831AutoARIMA
arima.auto = auto.arima(train, stepwise = FALSE, lambda = lambda.dt)
summary(arima.auto)Series: train
ARIMA(1,0,1)(1,1,2)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ma1 sar1 sma1 sma2 drift
0.9772 -0.466 -0.0497 -0.6165 -0.2078 0.0032
s.e. 0.0123 0.049 0.2772 0.2724 0.2131 0.0006
sigma^2 estimated as 0.0009618: log likelihood=792.12
AIC=-1570.24 AICc=-1569.95 BIC=-1542.53
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -164.44 25788.03 16733.45 -0.03250302 3.150999 0.3840334 0.0098187fitness.table = cbind(rbind(fit1[["aicc"]],fit2[["aicc"]], arima.auto[["aicc"]]),
rbind(summary(fit1)[,"MAPE"], summary(fit2)[,"MAPE"], summary(arima.auto)[,"MAPE"]),
rbind(summary(fit1)[,"MASE"], summary(fit2)[,"MASE"], summary(arima.auto)[,"MASE"]),
rbind(summary(fit1)[,"RMSE"], summary(fit2)[,"RMSE"], summary(arima.auto)[,"RMSE"]),
rbind("3", "2", "1"))Series: train
ARIMA(3,0,0)(3,1,0)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 sar1 sar2 sar3 drift
0.5335 0.2545 0.1306 -0.5219 -0.3968 -0.2349 0.0032
s.e. 0.0505 0.0559 0.0508 0.0508 0.0533 0.0501 0.0008
sigma^2 estimated as 0.001073: log likelihood=774.73
AIC=-1533.47 AICc=-1533.09 BIC=-1501.8
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -282.8799 27287.52 17644.32 -0.07659795 3.296094 0.4049379 -0.04304853
Series: train
ARIMA(4,0,0)(3,1,0)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 ar4 sar1 sar2 sar3 drift
0.5113 0.2094 0.0414 0.1728 -0.5499 -0.4045 -0.2426 0.0033
s.e. 0.0502 0.0566 0.0564 0.0506 0.0510 0.0538 0.0500 0.0009
sigma^2 estimated as 0.001044: log likelihood=780.47
AIC=-1542.93 AICc=-1542.45 BIC=-1507.31
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -234.7073 26988.03 17538.64 -0.08664446 3.258615 0.4025127 -0.01952831
Series: train
ARIMA(1,0,1)(1,1,2)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ma1 sar1 sma1 sma2 drift
0.9772 -0.466 -0.0497 -0.6165 -0.2078 0.0032
s.e. 0.0123 0.049 0.2772 0.2724 0.2131 0.0006
sigma^2 estimated as 0.0009618: log likelihood=792.12
AIC=-1570.24 AICc=-1569.95 BIC=-1542.53
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -164.44 25788.03 16733.45 -0.03250302 3.150999 0.3840334 0.0098187
Series: train
ARIMA(3,0,0)(3,1,0)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 sar1 sar2 sar3 drift
0.5335 0.2545 0.1306 -0.5219 -0.3968 -0.2349 0.0032
s.e. 0.0505 0.0559 0.0508 0.0508 0.0533 0.0501 0.0008
sigma^2 estimated as 0.001073: log likelihood=774.73
AIC=-1533.47 AICc=-1533.09 BIC=-1501.8
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -282.8799 27287.52 17644.32 -0.07659795 3.296094 0.4049379 -0.04304853
Series: train
ARIMA(4,0,0)(3,1,0)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 ar4 sar1 sar2 sar3 drift
0.5113 0.2094 0.0414 0.1728 -0.5499 -0.4045 -0.2426 0.0033
s.e. 0.0502 0.0566 0.0564 0.0506 0.0510 0.0538 0.0500 0.0009
sigma^2 estimated as 0.001044: log likelihood=780.47
AIC=-1542.93 AICc=-1542.45 BIC=-1507.31
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -234.7073 26988.03 17538.64 -0.08664446 3.258615 0.4025127 -0.01952831
Series: train
ARIMA(1,0,1)(1,1,2)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ma1 sar1 sma1 sma2 drift
0.9772 -0.466 -0.0497 -0.6165 -0.2078 0.0032
s.e. 0.0123 0.049 0.2772 0.2724 0.2131 0.0006
sigma^2 estimated as 0.0009618: log likelihood=792.12
AIC=-1570.24 AICc=-1569.95 BIC=-1542.53
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -164.44 25788.03 16733.45 -0.03250302 3.150999 0.3840334 0.0098187
Series: train
ARIMA(3,0,0)(3,1,0)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 sar1 sar2 sar3 drift
0.5335 0.2545 0.1306 -0.5219 -0.3968 -0.2349 0.0032
s.e. 0.0505 0.0559 0.0508 0.0508 0.0533 0.0501 0.0008
sigma^2 estimated as 0.001073: log likelihood=774.73
AIC=-1533.47 AICc=-1533.09 BIC=-1501.8
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -282.8799 27287.52 17644.32 -0.07659795 3.296094 0.4049379 -0.04304853
Series: train
ARIMA(4,0,0)(3,1,0)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 ar4 sar1 sar2 sar3 drift
0.5113 0.2094 0.0414 0.1728 -0.5499 -0.4045 -0.2426 0.0033
s.e. 0.0502 0.0566 0.0564 0.0506 0.0510 0.0538 0.0500 0.0009
sigma^2 estimated as 0.001044: log likelihood=780.47
AIC=-1542.93 AICc=-1542.45 BIC=-1507.31
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -234.7073 26988.03 17538.64 -0.08664446 3.258615 0.4025127 -0.01952831
Series: train
ARIMA(1,0,1)(1,1,2)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ma1 sar1 sma1 sma2 drift
0.9772 -0.466 -0.0497 -0.6165 -0.2078 0.0032
s.e. 0.0123 0.049 0.2772 0.2724 0.2131 0.0006
sigma^2 estimated as 0.0009618: log likelihood=792.12
AIC=-1570.24 AICc=-1569.95 BIC=-1542.53
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -164.44 25788.03 16733.45 -0.03250302 3.150999 0.3840334 0.0098187colnames(fitness.table) = c("AICc", "MAPE", "MASE","RMSE", "Ranking") ; rownames(fitness.table) = c("ARIMA(3,0,0)(3,1,0)[12] with constant","ARIMA(4,0,0)(3,1,0)[12] with constant","ARIMA(1,0,1)(1,1,2)[12] with constant")
fitness.table AICc MAPE MASE
ARIMA(3,0,0)(3,1,0)[12] with constant "-1533.08885564579" "3.29609407994998" "0.404937873265623"
ARIMA(4,0,0)(3,1,0)[12] with constant "-1542.45459694574" "3.25861482846603" "0.402512659770671"
ARIMA(1,0,1)(1,1,2)[12] with constant "-1569.94713575454" "3.15099869411926" "0.384033387016071"
RMSE Ranking
ARIMA(3,0,0)(3,1,0)[12] with constant "27287.5191750791" "3"
ARIMA(4,0,0)(3,1,0)[12] with constant "26988.0313059203" "2"
ARIMA(1,0,1)(1,1,2)[12] with constant "25788.0262804677" "1" Residual diagnosis
checkresiduals(fit2)
Ljung-Box test
data: Residuals from ARIMA(4,0,0)(3,1,0)[12] with drift
Q* = 29.649, df = 16, p-value = 0.01991
Model df: 8. Total lags used: 24checkresiduals(arima.auto)
Ljung-Box test
data: Residuals from ARIMA(1,0,1)(1,1,2)[12] with drift
Q* = 30.113, df = 18, p-value = 0.03637
Model df: 6. Total lags used: 24Re-indentification
fit2 %>% resid() %>% ggtsdisplay(main = "Residuals from ARIMA(4,0,0)(3,1,0)[12] with constant")
arima.auto %>% residuals() %>% ggtsdisplay(main = "Residuals from ARIMA(1,0,1)(1,1,2)[12] with constant")
Based on ggtsdisplay of resid, increasing MA non
#ARIMA(4,0,0)(3,1,0)[12] with constant
fit2.2 = Arima(train, order = c(4,0,0), seasonal = c(3,1,3), include.constant = TRUE, lambda=lambda.dt)
checkresiduals(fit2.2)
Ljung-Box test
data: Residuals from ARIMA(4,0,0)(3,1,3)[12] with drift
Q* = 27.52, df = 13, p-value = 0.01055
Model df: 11. Total lags used: 24ggtsdisplay(resid(fit2.2), main = "Residuals from ARIMA(4,0,0)(3,1,3)[12] with constant") 
summary(fit2.2)Series: train
ARIMA(4,0,0)(3,1,3)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 ar4 sar1 sar2 sar3 sma1 sma2 sma3 drift
0.5289 0.2011 0.0568 0.1701 -0.4740 -0.7426 0.0966 -0.1981 0.2757 -0.6989 0.0032
s.e. 0.0511 0.0579 0.0573 0.0507 0.4604 0.2849 0.1157 0.4554 0.4907 0.2419 0.0006
sigma^2 estimated as 0.0009604: log likelihood=794.98
AIC=-1565.95 AICc=-1565.12 BIC=-1518.45
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -148.5519 25564.33 16820.18 -0.03497417 3.149243 0.386024 -0.02552786fit2.3 = Arima(train, order = c(4,0,5), seasonal = c(3,1,3), include.constant = TRUE, lambda=lambda.dt)
checkresiduals(fit2.3)
Ljung-Box test
data: Residuals from ARIMA(4,0,5)(3,1,3)[12] with drift
Q* = 21.844, df = 8, p-value = 0.005213
Model df: 16. Total lags used: 24ggtsdisplay(resid(fit2.3), main = "Residuals from ARIMA(4,0,5)(3,1,3)[12] with constant")
summary(fit2.3)Series: train
ARIMA(4,0,5)(3,1,3)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 ar4 ma1 ma2 ma3 ma4 ma5 sar1 sar2
-0.3883 0.2151 0.3876 0.6888 0.9290 0.4842 0.1130 -0.3640 0.0781 -0.6467 -0.7309
s.e. 0.2087 0.1161 0.1986 0.1160 0.2164 0.1875 0.1298 0.0906 0.0648 0.5396 0.2061
sar3 sma1 sma2 sma3 drift
0.1020 -0.0136 0.1755 -0.7216 0.0032
s.e. 0.1202 0.5296 0.2699 0.2854 0.0006
sigma^2 estimated as 0.0009435: log likelihood=799.51
AIC=-1565.03 AICc=-1563.37 BIC=-1497.73
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -127.131 25050.02 16393.95 -0.03786575 3.087644 0.3762419 -0.007796833fit2.3.2 = Arima(train, order = c(6,0,5), seasonal = c(3,1,3), include.constant = TRUE, lambda=lambda.dt)
checkresiduals(fit2.3.2)
Ljung-Box test
data: Residuals from ARIMA(6,0,5)(3,1,3)[12] with drift
Q* = 16.457, df = 6, p-value = 0.0115
Model df: 18. Total lags used: 24ggtsdisplay(resid(fit2.3.2), main = "Residuals from ARIMA(6,0,5)(3,1,3)[12] with constant")
summary(fit2.3.2)Series: train
ARIMA(6,0,5)(3,1,3)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ar2 ar3 ar4 ar5 ar6 ma1 ma2 ma3 ma4 ma5
0.2823 0.5319 0.4120 0.1175 -0.6516 0.2486 0.2559 -0.2444 -0.4140 -0.2310 0.6826
s.e. 0.1342 0.1152 0.1656 0.1280 0.1413 0.1029 0.1163 0.1357 0.0901 0.1125 0.1113
sar1 sar2 sar3 sma1 sma2 sma3 drift
-0.4843 -0.7138 0.0780 -0.1673 0.2239 -0.6452 0.0031
s.e. 0.5420 0.1825 0.1127 0.5403 0.3854 0.2154 0.0005
sigma^2 estimated as 0.0009264: log likelihood=803.7
AIC=-1569.41 AICc=-1567.34 BIC=-1494.2
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -211.9534 24866.09 16204.16 -0.01465611 3.037107 0.3718862 -0.02837137fit2.4 = Arima(train, order = c(4,0,5), seasonal = c(3,1,0), include.constant = TRUE, lambda=lambda.dt)
checkresiduals(fit2.4)
Ljung-Box test
data: Residuals from ARIMA(4,0,5)(3,1,0)[12] with drift
Q* = 18.187, df = 11, p-value = 0.07735
Model df: 13. Total lags used: 24ggtsdisplay(resid(fit2.4), main = "Residuals from ARIMA(4,0,5)(3,1,0)[12] with constant")## Would be the best
summary(2.4) Min. 1st Qu. Median Mean 3rd Qu. Max.
2.4 2.4 2.4 2.4 2.4 2.4 Residuals Check
checkresiduals(fit2.4, lag = 24)
Ljung-Box test
data: Residuals from ARIMA(4,0,5)(3,1,0)[12] with drift
Q* = 18.187, df = 11, p-value = 0.07735
Model df: 13. Total lags used: 24checkresiduals(fit2.4, lag = 36)
Ljung-Box test
data: Residuals from ARIMA(4,0,5)(3,1,0)[12] with drift
Q* = 23.828, df = 23, p-value = 0.4134
Model df: 13. Total lags used: 36checkresiduals(arima.auto, lag = 24)
Ljung-Box test
data: Residuals from ARIMA(1,0,1)(1,1,2)[12] with drift
Q* = 30.113, df = 18, p-value = 0.03637
Model df: 6. Total lags used: 24checkresiduals(arima.auto, lag = 36)
Ljung-Box test
data: Residuals from ARIMA(1,0,1)(1,1,2)[12] with drift
Q* = 34.717, df = 30, p-value = 0.2531
Model df: 6. Total lags used: 36We may select ARIMA(4,0,5)(3,1,0)[12] with drift and ARIMA(1,0,1)(1,1,2)[12] with drift
Information and AICc
in.sample.table = rbind(cbind(accuracy(fit2.4)[,"MASE"],accuracy(arima.auto)[,"MASE"]),cbind(accuracy(fit2.4)[,"MAPE"],accuracy(arima.auto)[,"MAPE"]),cbind(fit2.4[["aicc"]],arima.auto[["aicc"]]))
rownames(in.sample.table) = c("MASE", "MAPE", "AICc") ; colnames(in.sample.table) = c("ARIMA(4,0,5)(3,1,0)[12] with constant","ARIMA(1,0,1)(1,1,2)[12] with constant")
in.sample.table ARIMA(4,0,5)(3,1,0)[12] with constant ARIMA(1,0,1)(1,1,2)[12] with constant
MASE 0.393111 0.3840334
MAPE 3.197522 3.1509987
AICc -1548.023305 -1569.9471358Out-sample accuracy
arima.f = forecast(fit2.4, h = length(test))
arima.auto.f = forecast(arima.auto, h = length(test))
ac.out.table = rbind(cbind(accuracy(arima.f, test)[2,"MASE"], accuracy(arima.auto.f, test)[2,"MASE"]),
cbind(accuracy(arima.f, test)[2,"MAPE"], accuracy(arima.auto.f, test)[2,"MAPE"]),
cbind(accuracy(arima.f, test)[2,"RMSE"], accuracy(arima.auto.f, test)[2,"RMSE"]))
rownames(ac.out.table) = c("MASE", "MAPE", "RMSE"); colnames(ac.out.table) = c("ARIMA(4,0,5)(3,1,0)[12] with constant", "ARIMA(1,0,1)(1,1,2)[12] with constant")
ac.out.table ARIMA(4,0,5)(3,1,0)[12] with constant ARIMA(1,0,1)(1,1,2)[12] with constant
MASE 1.713150 1.561988
MAPE 5.674618 5.194949
RMSE 92675.919367 85807.839740Estimating with full data set
arima.full = Arima(dt, order = c(1,0,1), seasonal = c(1,1,2), include.constant = TRUE, lambda=lambda.dt)
summary(arima.full)Series: dt
ARIMA(1,0,1)(1,1,2)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ma1 sar1 sma1 sma2 drift
0.9713 -0.4539 0.0985 -0.7296 -0.1151 0.0031
s.e. 0.0125 0.0458 0.2213 0.2207 0.1706 0.0004
sigma^2 estimated as 0.0008879: log likelihood=1017.37
AIC=-2020.73 AICc=-2020.5 BIC=-1991.41
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -116.3212 31300.3 21016.25 -0.02771271 3.097895 0.4053521 0.02570216H=24
arima.fcast = forecast(arima.full, h = 24)
summary(arima.fcast)
Forecast method: ARIMA(1,0,1)(1,1,2)[12] with drift
Model Information:
Series: dt
ARIMA(1,0,1)(1,1,2)[12] with drift
Box Cox transformation: lambda= -0.02807425
Coefficients:
ar1 ma1 sar1 sma1 sma2 drift
0.9713 -0.4539 0.0985 -0.7296 -0.1151 0.0031
s.e. 0.0125 0.0458 0.2213 0.2207 0.1706 0.0004
sigma^2 estimated as 0.0008879: log likelihood=1017.37
AIC=-2020.73 AICc=-2020.5 BIC=-1991.41
Error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -116.3212 31300.3 21016.25 -0.02771271 3.097895 0.4053521 0.02570216
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
Aug 2019 1733765 1637522 1835833 1588815 1892345
Sep 2019 1449643 1359807 1545592 1314588 1599004
Oct 2019 1541912 1437420 1654229 1385080 1717058
Nov 2019 1500204 1391235 1617967 1336878 1684114
Dec 2019 1728199 1594694 1873221 1528360 1954999
Jan 2020 1685566 1548951 1834599 1481293 1918912
Feb 2020 1571014 1438511 1716098 1373074 1798408
Mar 2020 1674959 1528228 1836213 1455970 1927953
Apr 2020 1665982 1515350 1832051 1441351 1926760
May 2020 1596291 1448017 1760218 1375333 1853906
Jun 2020 1624929 1470095 1796576 1394355 1894881
Jul 2020 1897680 1711989 2104140 1621358 2222647
Aug 2020 1828882 1636223 2044939 1542809 2169769
Sep 2020 1527538 1361367 1714633 1281041 1823058
Oct 2020 1635466 1451382 1843637 1362697 1964679
Nov 2020 1598244 1413262 1808210 1324396 1930640
Dec 2020 1844850 1625016 2095374 1519732 2241898
Jan 2021 1781932 1564923 2029997 1461238 2175424
Feb 2021 1666372 1459619 1903353 1361037 2042568
Mar 2021 1786135 1559988 2046123 1452407 2199201
Apr 2021 1763130 1536207 2024656 1428462 2178930
May 2021 1701513 1479366 1958098 1374066 2109711
Jun 2021 1722773 1494625 1986875 1386667 2143195
Jul 2021 2012332 1741328 2326885 1613351 2513446autoplot(arima.fcast, level = 95)+
xlab("Years")+
ylab("Number of Visitor Arrivals")+
theme_light()+
theme(legend.title = element_text(colour = "black", size = 25),
legend.text = element_text(colour = "black", size = 25))+
theme(legend.position = c(0.8,0.2))+
ggtitle("Forecasts of Number of Visitor Arrivals in Singapore for the Next 24 Months")+
theme(plot.title = element_text(size = 30))
Comparison with an ETS model
ets.model = ets(dt, model = "ZZZ", lambda = lambda.dt)
summary(ets.model)ETS(A,Ad,A)
Call:
ets(y = dt, model = "ZZZ", lambda = lambda.dt)
Box-Cox transformation: lambda= -0.0281
Smoothing parameters:
alpha = 0.4465
beta = 0.0121
gamma = 0.1769
phi = 0.98
Initial states:
l = 10.1692
b = 0.0083
s = 0.0572 0.0113 5e-04 -0.0413 0.0885 0.0388
-0.0401 -0.0249 -0.0297 -0.0111 -0.0544 0.0052
sigma: 0.0307
AIC AICc BIC
-359.1399 -357.7149 -283.3130
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set 2757.32 33397.09 22515.4 0.2546606 3.224471 0.4342672 0.08679948ets.fcast = forecast(ets.model, h = 24)
summary(ets.fcast)
Forecast method: ETS(A,Ad,A)
Model Information:
ETS(A,Ad,A)
Call:
ets(y = dt, model = "ZZZ", lambda = lambda.dt)
Box-Cox transformation: lambda= -0.0281
Smoothing parameters:
alpha = 0.4465
beta = 0.0121
gamma = 0.1769
phi = 0.98
Initial states:
l = 10.1692
b = 0.0083
s = 0.0572 0.0113 5e-04 -0.0413 0.0885 0.0388
-0.0401 -0.0249 -0.0297 -0.0111 -0.0544 0.0052
sigma: 0.0307
AIC AICc BIC
-359.1399 -357.7149 -283.3130
Error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set 2757.32 33397.09 22515.4 0.2546606 3.224471 0.4342672 0.08679948
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
Aug 2019 1711062 1613457 1814747 1564124 1872228
Sep 2019 1413012 1325068 1506968 1280811 1559282
Oct 2019 1524473 1421379 1635271 1369733 1697242
Nov 2019 1503998 1394708 1622113 1340192 1688458
Dec 2019 1755231 1618500 1903865 1550601 1987726
Jan 2020 1706815 1565772 1860954 1496015 1948273
Feb 2020 1572266 1435277 1722732 1367788 1808301
Mar 2020 1659466 1507061 1827762 1432284 1923859
Apr 2020 1633242 1475959 1807807 1399089 1907871
May 2020 1559453 1402567 1734436 1326180 1835116
Jun 2020 1577367 1411762 1763009 1331440 1870233
Jul 2020 1840986 1638975 2068682 1541407 2200746
Aug 2020 1753137 1548459 1985731 1450221 2121478
Sep 2020 1446870 1272931 1645337 1189706 1761525
Oct 2020 1560340 1366107 1783077 1273530 1913971
Nov 2020 1538654 1341058 1766306 1247213 1900564
Dec 2020 1795032 1556463 2071354 1443622 2234978
Jan 2021 1744705 1506145 2022284 1393702 2187224
Feb 2021 1606383 1380965 1869803 1275068 2026842
Mar 2021 1694802 1450106 1982149 1335583 2154090
Apr 2021 1667301 1420299 1958678 1305101 2133636
May 2021 1591274 1349788 1877397 1237542 2049774
Jun 2021 1608913 1358631 1906839 1242711 2086961
Jul 2021 1877219 1577011 2236491 1438539 2454589autoplot(dt)+
autolayer(arima.fcast$mean, lwd = 1.0, series = "ARIMA Forecast")+
autolayer(ets.fcast[["mean"]], lwd = 1.0, lty = 2, series = "ETS Forecast")+
theme_light()+
theme(legend.title = element_text(colour = "black", size = 25),
legend.text = element_text(colour = "black", size = 25))+
theme(legend.position = c(0.8,0.2))+
xlab("Years")+
ylab("Number of Visitor Arrivals")+
ggtitle("Forecasts from 2 models")+
theme(plot.title = element_text(size = 30))
16/10/2019 consultation
- autoARIMA better goodness of fit not white noise at lag = 24,
- autoARIMA ggtsdisplay may not able to updata
- Manual ARIMA sightly worse goodness of fit, but whitenoise(but complex model)