library(tseries)
## 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)
## Registered S3 methods overwritten by 'forecast':
##   method       from
##   fitted.Arima TSA 
##   plot.Arima   TSA

Pembangkitan Data Time Series

Coba bangkitkan data tie series model ARIMA(1, 1, 1). Tentukan nilai AR dan MA secara acak.

# Set seed untuk reprodusibilitas
set.seed(123)

# Pnjang Data
n <- 200

# Parameter ARIMA(p=1, d=1, q=1)
ar <- 0.7   # AR(1)
ma <- -0.5  # MA(1)

# Simulasi data
ts_arima <- arima.sim(model = list(order = c(1, 1, 1), ar = ar, ma = ma), n = n)

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

Melakukan Pemodelan

  1. Buat Plot ACF dan PACF
  2. Cek Kestasioneran dengan ADF Test
  3. Melakukan differencing
  4. Buat Plot ACF dan PACF
  5. Cek Kestasioneran dengan ADF Test
  6. Ubah ke data ts
  7. Buat Kandidat Model melalui ACF,PACF, dan EACF
  8. Bandingkan dengan hasil aoto.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

Dikarenakan p-value = 0.388 > 0.05 sehingga disimpulkan data tidaj stasioner dan harus dilakukan proses 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(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 series yang dibangkitkan dengan model tertentu, belum tentu akan sama dengan hasil pemodelan terbaiknya.Hal ini diduga karena adanya faktor-faktor lain yang mempengaruhi proses pemodelan.