library(tseries)
## Warning: package 'tseries' was built under R version 4.3.3
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
library(TSA)
## Warning: package 'TSA' was built under R version 4.3.3
## 
## 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 4.3.3
## Registered S3 methods overwritten by 'forecast':
##   method       from
##   fitted.Arima TSA 
##   plot.Arima   TSA

Pembangkitan Data Time Series

coba bangkiitkan 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)

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

Karena nilai p-value = 0.388 > 0.05, sehingga perlu dilakukan differencing

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)

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
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 berdasarkan 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 differencing.