Pembangkitan Data Time Series

misal: AR = 0.5 MA = -0.3 n = 200

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 4.5.3

## 
## 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

1. Pembangkitan Data

set.seed(456)

n <- 200

ar <- 0.5
ma <- -0.3

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)",
        ylab="Nilai",
        xlab="Waktu")

2. Plot ACF dan PACF Awal

acf(ts_arima,
    main="ACF Data Awal")

pacf(ts_arima,
     main="PACF Data Awal")

3. Uji Stasioneritas Awal

adf.test(ts_arima)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  ts_arima
## Dickey-Fuller = -2.4717, Lag order = 5, p-value = 0.3785
## alternative hypothesis: stationary

4. Differencing

diff1 <- diff(ts_arima)

ts.plot(diff1,
        main="Data Setelah Differencing")

5. Plot ACF dan PACF Setelah Differencing

acf(diff1,
    main="ACF Setelah Differencing")

pacf(diff1,
     main="PACF Setelah Differencing")

6. Uji Stasioneritas Setelah Differencing

adf.test(diff1)

## Warning in adf.test(diff1): p-value smaller than printed p-value

## 
##  Augmented Dickey-Fuller Test
## 
## data:  diff1
## Dickey-Fuller = -5.9035, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary

7. Ubah ke Time Series

data.ts <- ts(diff1)

head(data.ts)

## Time Series:
## Start = 1 
## End = 6 
## Frequency = 1 
## [1]  1.2453574  1.2180758  1.9663487 -0.9538095  1.9026933  2.1040759

8. 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 x o o o o o o o o o  o  o  o 
## 1 x o o x o o o o o o o  o  o  o 
## 2 o x o o o o o o o o o  o  o  o 
## 3 o x x o o o o o o o o  o  o  o 
## 4 o x x x o o o o o o o  o  o  o 
## 5 x x o x o o o o o o o  o  o  o 
## 6 x x o x o o o o o o o  o  o  o 
## 7 x x x x o x o o o o o  o  o  o

9. Auto ARIMA

auto.arima(data.ts)

## Series: data.ts 
## ARIMA(3,0,2) with zero mean 
## 
## Coefficients:
##           ar1      ar2     ar3     ma1     ma2
##       -0.1986  -0.4312  0.2401  0.4254  0.6773
## s.e.   0.2430   0.3161  0.1009  0.2429  0.3005
## 
## sigma^2 = 0.9581:  log likelihood = -277.08
## AIC=566.16   AICc=566.59   BIC=585.95

auto.arima(ts_arima)

## Series: ts_arima 
## ARIMA(3,1,2) 
## 
## Coefficients:
##           ar1      ar2     ar3     ma1     ma2
##       -0.1986  -0.4312  0.2401  0.4254  0.6773
## s.e.   0.2430   0.3161  0.1009  0.2429  0.3005
## 
## sigma^2 = 0.9581:  log likelihood = -277.08
## AIC=566.16   AICc=566.59   BIC=585.95

10. Perbandingan Beberapa Model

model1 <- arima(data.ts,
                order=c(1,0,1),
                method="ML")

model2 <- arima(data.ts,
                order=c(1,0,2),
                method="ML")

model3 <- arima(data.ts,
                order=c(2,0,1),
                method="ML")

model4 <- arima(data.ts,
                order=c(2,0,2),
                method="ML")

model1

## 
## Call:
## arima(x = data.ts, order = c(1, 0, 1), method = "ML")
## 
## Coefficients:
##          ar1      ma1  intercept
##       0.5223  -0.2791     0.0213
## s.e.  0.1605   0.1738     0.1041
## 
## sigma^2 estimated as 0.9565:  log likelihood = -279.39,  aic = 564.77

model2

## 
## Call:
## arima(x = data.ts, order = c(1, 0, 2), method = "ML")
## 
## Coefficients:
##          ar1      ma1     ma2  intercept
##       0.2981  -0.0836  0.1613     0.0215
## s.e.  0.2085   0.2020  0.0908     0.1053
## 
## sigma^2 estimated as 0.9451:  log likelihood = -278.21,  aic = 564.41

model3

## 
## Call:
## arima(x = data.ts, order = c(2, 0, 1), method = "ML")
## 
## Coefficients:
##          ar1     ar2     ma1  intercept
##       0.1982  0.1327  0.0261     0.0216
## s.e.  0.3100  0.1034  0.3068     0.1054
## 
## sigma^2 estimated as 0.9512:  log likelihood = -278.83,  aic = 565.66

model4

## 
## Call:
## arima(x = data.ts, order = c(2, 0, 2), method = "ML")
## 
## Coefficients:
##          ar1      ar2      ma1     ma2  intercept
##       1.1410  -0.7271  -0.9401  0.6950     0.0168
## s.e.  0.1805   0.1743   0.2014  0.1572     0.0874
## 
## sigma^2 estimated as 0.9217:  log likelihood = -275.79,  aic = 561.59

11. Membandingkan Nilai AIC

AIC(model1,
    model2,
    model3,
    model4)

##        df      AIC
## model1  4 566.7715
## model2  5 566.4116
## model3  5 567.6625
## model4  6 563.5852

Simulasi data time series dilakukan menggunakan model ARIMA(1,1,1) dengan parameter AR = 0.5 dan MA = -0.3. Hasil uji ADF menunjukkan bahwa data awal tidak stasioner sehingga dilakukan differencing satu kali. Setelah differencing, data menjadi stasioner dengan p-value sebesar 0.01. Berdasarkan perbandingan beberapa kandidat model menggunakan nilai AIC, diperoleh model terbaik yaitu ARIMA(2,0,2) dengan nilai AIC sebesar 563.5852. Model terbaik yang diperoleh berbeda dengan model pembangkitan awal ARIMA(1,1,1). Perbedaan ini menunjukkan bahwa proses identifikasi dan estimasi model ARIMA sangat dipengaruhi oleh variasi sampel, unsur acak dalam data simulasi, serta metode pemilihan model yang digunakan. Dengan demikian, model pembangkitan data tidak selalu sama dengan model terbaik hasil pemodelan.