library(tseries)
## Warning: package 'tseries' was built under R version 4.4.3
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
library(TSA)
## Warning: package 'TSA' was built under R version 4.4.3
## 
## Attaching package: 'TSA'
## The following objects are masked from 'package:stats':
## 
##     acf, arima
## The following object is masked from 'package:utils':
## 
##     tar
library(forecast)
## Warning: package 'forecast' was built under R version 4.4.3
## Registered S3 methods overwritten by 'forecast':
##   method       from
##   fitted.Arima TSA 
##   plot.Arima   TSA

#Pembangkitan Data Time Series coba bangkitkan data time series dengan model ARIMA (1,1,1). Tentukan nlai AR dan MA

set.seed(123)

n = 200

ar = 0.7
ma = -0.5

ts_arima = arima.sim(model=list(order=c(1,1,1), ar=ar, ma=ma), n=n)

ts.plot(ts_arima, main = "Simulasi Data ARIMA(1,1,1)")

#Melakukan Pemodelan 1. Buat Plot ACF dan PACF 2. Cek kestasioneritas dengan dengan ADF Test 3. Melakukan differencing 4. Buat Plot ACF daan PACF 5. Cek kestasioneritas dengan ADF Test 6. Ubah ke data ts 7. Buat Kandidat Model melalui ACF, PACF, dan EACF 8. Bandingkan dengan hasil auto.arima 9. Cek AIC Terkecil

acf(ts_arima)

pacf(ts_arima)

library(tseries)
adf.test(ts_arima)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  ts_arima
## Dickey-Fuller = -2.449, Lag order = 5, p-value = 0.388
## alternative hypothesis: stationary

Dikarenakan p-value = 0.388 > 0.05 sehingga disimpulkan data tidak stasioner dan harus dilakukan prosesdifferencing.

diff1 <- diff(ts_arima)
acf(diff1)

pacf(diff1)

adf.test(diff1)
## Warning in adf.test(diff1): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  diff1
## Dickey-Fuller = -5.4572, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary

Karena nilai p-value = 0.01 < 0.05 sehingga disimpulkan data sudah stasioner

data.ts<-ts(diff1)
head(data.ts)
## Time Series:
## Start = 1 
## End = 6 
## Frequency = 1 
## [1] -0.4362295 -1.1367886 -0.4798151 -1.2528876 -1.0929103 -1.0256309

#Kandidat Model

acf(data.ts)

pacf(data.ts)

library(TSA)
eacf(data.ts)
## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x o o o o o o o o o o  o  o  o 
## 1 x o o o o o o o o o o  o  o  o 
## 2 x x o o o o o o o o o  o  o  o 
## 3 x x o o o o o o o o o  o  o  o 
## 4 x x o o o o o o o o o  o  o  o 
## 5 x o o o o o o o o o o  o  o  o 
## 6 x o o x o o o o o o o  o  o  o 
## 7 o x x x x o o o o o o  o  o  o
library(forecast)
auto.arima(data.ts)
## Series: data.ts 
## ARIMA(2,0,2) with zero mean 
## 
## Coefficients:
##           ar1     ar2     ma1      ma2
##       -0.1116  0.6336  0.3108  -0.6250
## s.e.   0.2175  0.1701  0.2294   0.2122
## 
## sigma^2 = 0.8631:  log likelihood = -267.28
## AIC=544.57   AICc=544.88   BIC=561.06

Kandidat Model ARIMA(1,1,1) ARIMA(1,1,3) ARIMA(0,1,1) ARIMA(2,0,2)

auto.arima(ts_arima)
## Series: ts_arima 
## ARIMA(2,1,2) 
## 
## Coefficients:
##           ar1     ar2     ma1      ma2
##       -0.1116  0.6336  0.3108  -0.6250
## s.e.   0.2175  0.1701  0.2294   0.2122
## 
## sigma^2 = 0.8631:  log likelihood = -267.28
## AIC=544.57   AICc=544.88   BIC=561.06

#Penentuan Model Terbaik berdasar AIC

arima(data.ts, order=c(1,1,1), method="ML")
## 
## Call:
## arima(x = data.ts, order = c(1, 1, 1), method = "ML")
## 
## Coefficients:
##          ar1      ma1
##       0.1488  -1.0000
## s.e.  0.0706   0.0164
## 
## sigma^2 estimated as 0.8926:  log likelihood = -273.56,  aic = 551.13
arima(data.ts, order=c(1,1,3), method="ML")
## 
## Call:
## arima(x = data.ts, order = c(1, 1, 3), method = "ML")
## 
## Coefficients:
##           ar1     ma1      ma2      ma3
##       -0.8559  0.0335  -0.9642  -0.0693
## s.e.   0.0800  0.1018   0.0443   0.0772
## 
## sigma^2 estimated as 0.8611:  log likelihood = -270.25,  aic = 548.49
arima(data.ts, order=c(0,1,1), method="ML")
## 
## Call:
## arima(x = data.ts, order = c(0, 1, 1), method = "ML")
## 
## Coefficients:
##           ma1
##       -0.9294
## s.e.   0.1078
## 
## sigma^2 estimated as 0.93:  log likelihood = -276.14,  aic = 554.28
arima(data.ts, order=c(2,0,2), method="ML")
## 
## Call:
## arima(x = data.ts, order = c(2, 0, 2), method = "ML")
## 
## Coefficients:
##           ar1     ar2     ma1      ma2  intercept
##       -0.1096  0.6350  0.3087  -0.6269    -0.0214
## s.e.   0.2164  0.1692  0.2283   0.2112     0.0931
## 
## sigma^2 estimated as 0.8456:  log likelihood = -267.26,  aic = 544.51

Berdasarkan hasil pemodelan, diperoleh model dengan nilai AIC terkecil adalah ARIMA(2,0,2), hal ini dikarenakan data yang terbaca adalah data hasil differencing. Diperoleh pembelajaran bahwa, data time seriesyang dibangkitkan dengan model tertentu, belum tentu akan sama dengan hasil pemodelan terbaiknya. Hal inididuga karena adanya faktor-faktor lain yang mempengaruhi proses pemodelan.