Dari data yang kalian miliki dan sudah melakukan plot data serta uji stasioneritasnya, selanjutnya adalah a. Transformasi awal dan identifikasi model. (jika data kalian tidak stasioner) b. Estimasi parameter dari model yang memungkinkan.

library(readxl)
library(ggplot2)
library(tseries)
## Warning: package 'tseries' was built under R version 3.6.3
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
library(forecast)
## Warning: package 'forecast' was built under R version 3.6.3

#Import Data

JPP <- read_xlsx("Jumlah Penumpang Pesawat.xlsx")

Data saya tidak bersifat stasioner maka saya melakukan proses differencing

#Diff

JPP_diff <- diff(JPP$Jumlah.Penumpang.Pesawat,
                        differences = 1)

ts.plot(JPP_diff,
        col = "Blue",
        xlab = "Time",
        ylab = "Differencing",
        main = "Time Series Plot (Differencing)")

Estimasi Parameter

AR(1)

JPP_log <- log(JPP$Jumlah.Penumpang.Pesawat)
AR1 <- arima(JPP_log,
             order = c(1,2,0),
             seasonal = list(order = c(0,0,0),
                             period = NA),
             include.mean = F)
summary(AR1)
## 
## Call:
## arima(x = JPP_log, order = c(1, 2, 0), seasonal = list(order = c(0, 0, 0), period = NA), 
##     include.mean = F)
## 
## Coefficients:
##           ar1
##       -0.0410
## s.e.   0.1828
## 
## sigma^2 estimated as 0.4056:  log likelihood = -30,  aic = 64
## 
## Training set error measures:
##                      ME      RMSE       MAE       MPE     MAPE     MASE
## Training set 0.01575419 0.6172644 0.3797839 0.6000282 4.892401 1.160034
##                     ACF1
## Training set -0.01756661

Dari summary(arima) terdapat bahwa nilai koefisien AR1 = -0.0410 dan s.e. = 0.1828 dan sigma^2 = 0.4056

MA(1)

MA1 <- arima(JPP_log,
             order = c(0,2,1),
             seasonal = list(order = c(0,0,0),
                             period = NA),
             include.mean = F)
summary(MA1)
## 
## Call:
## arima(x = JPP_log, order = c(0, 2, 1), seasonal = list(order = c(0, 0, 0), period = NA), 
##     include.mean = F)
## 
## Coefficients:
##           ma1
##       -0.2535
## s.e.   0.4199
## 
## sigma^2 estimated as 0.4012:  log likelihood = -29.86,  aic = 63.73
## 
## Training set error measures:
##                      ME      RMSE       MAE       MPE     MAPE     MASE
## Training set 0.01407424 0.6139113 0.3798848 0.5807833 4.900348 1.160342
##                    ACF1
## Training set 0.09823616

Dari summary(arima) terdapat bahwa nilai koefisien ma1 = -0.2535 dan s.e. = 0.4199 serta sigma^2 = 0.4199

ARMA(1,1)

ARMA1 <- arima(JPP_log,
               order = c(1,2,1),
               seasonal = list(order = c(0,0,0),
                               period = NA),
               include.mean = F)
summary(ARMA1)
## 
## Call:
## arima(x = JPP_log, order = c(1, 2, 1), seasonal = list(order = c(0, 0, 0), period = NA), 
##     include.mean = F)
## 
## Coefficients:
##          ar1      ma1
##       0.5203  -1.0000
## s.e.  0.1621   0.1233
## 
## sigma^2 estimated as 0.3048:  log likelihood = -26.76,  aic = 59.52
## 
## Training set error measures:
##                       ME      RMSE       MAE        MPE     MAPE      MASE
## Training set -0.04073504 0.5351042 0.3156023 -0.3817704 3.905818 0.9639938
##                   ACF1
## Training set 0.1046594

Dari summary(arima) terdapat bahwa nilai koefisien AR1 = 0.5203, ma1 = -1 dan s.e.ar = 0.1621, s.e.ma = 0.1233 serta sigma^2 = 0.1233

AR(2)

AR2 <- arima(JPP_log,
             order = c(2,2,0),
             seasonal = list(order = c(0,0,0),
                             period = NA),
             include.mean = F)
summary(AR2)
## 
## Call:
## arima(x = JPP_log, order = c(2, 2, 0), seasonal = list(order = c(0, 0, 0), period = NA), 
##     include.mean = F)
## 
## Coefficients:
##           ar1      ar2
##       -0.0347  -0.4254
## s.e.   0.1668   0.1641
## 
## sigma^2 estimated as 0.331:  log likelihood = -27.05,  aic = 60.1
## 
## Training set error measures:
##                      ME     RMSE       MAE       MPE     MAPE     MASE
## Training set 0.02056356 0.557616 0.3405879 0.6770012 4.323875 1.040311
##                    ACF1
## Training set -0.0446033

Dari summary(arima) terdapat bahwa nilai koefisien AR1 = -0.0347, AR2 = -0.4254 dan s.e.1 = 0.1668, s.e.2 = 0.1641 serta sigma^2 = 0.331

Model MA(2)

MA2 <- arima(JPP_log,
             order = c(0,2,2),
             seasonal = list(order = c(0,0,0),
                             period = NA),
             include.mean = F)
summary(MA2)
## 
## Call:
## arima(x = JPP_log, order = c(0, 2, 2), seasonal = list(order = c(0, 0, 0), period = NA), 
##     include.mean = F)
## 
## Coefficients:
##           ma1      ma2
##       -0.3773  -0.5517
## s.e.   0.1801   0.1745
## 
## sigma^2 estimated as 0.2912:  log likelihood = -25.82,  aic = 57.64
## 
## Training set error measures:
##                       ME      RMSE       MAE        MPE     MAPE      MASE
## Training set -0.05082296 0.5230585 0.2931629 -0.5687564 3.597235 0.8954537
##                    ACF1
## Training set 0.06107055

Dari summary(arima) terdapat bahwa nilai koefisien MA1 = -0.3773, MA2 = -0.5517 dan s.e.1 = 0.1801, s.e.2 = 0.1745 serta sigma^2 = 0.2912

ARMA(2,2)

ARMA2 <- arima(JPP_log,
               order = c(2,2,2),
               seasonal = list(order = c(0,0,0),
                               period = NA),
               include.mean = F)
summary(ARMA2)
## 
## Call:
## arima(x = JPP_log, order = c(2, 2, 2), seasonal = list(order = c(0, 0, 0), period = NA), 
##     include.mean = F)
## 
## Coefficients:
##          ar1      ar2      ma1      ma2
##       0.3238  -0.1072  -0.6312  -0.3688
## s.e.  0.3689   0.2645   0.4294   0.3548
## 
## sigma^2 estimated as 0.2748:  log likelihood = -25.41,  aic = 60.82
## 
## Training set error measures:
##                       ME      RMSE       MAE        MPE     MAPE      MASE
## Training set -0.04689238 0.5081291 0.2876889 -0.5236586 3.501176 0.8787336
##                     ACF1
## Training set -0.01872531

Dari summary(arima) didapatkan nilai koefisien AR1 = 0.3238, AR2 = -0.1072, MA1 = -0.6312, MA2 = -0.3688 dan s.e.AR1 = 0.3689, s.e.AR2 = 0.2645, s.e.MA1 = 0.4294, s.e.MA2 = 0.3548 serta sigma^2 = 0.2748