Tugas Seasonal ARIMA Simulation

Identitas Mahasiswa
Nama : Jasmine Sabrina Rahmaira Saputra
NRP : 5003231114
Kelas : Pengantar Analisis Deret Waktu (A)

Model ARIMA(0,0,0)(4,1,4)¹² \[ Y_t = Y_{t-12}+\Phi_1(Y_{t-12}-Y_{t-24})+\Phi_2(Y_{t-24}-Y_{t-36})+\Phi_3(Y_{t-36}-Y_{t-48})+\Phi_4(Y_{t-48}-Y_{t-60})+a_t+\Theta_1a_{t-12}+\Theta_2a_{t-24}+\Theta_3a_{t-36}+\Theta_4a_{t-48} \]

set.seed(123)

# Parameters
phi_seasonal <- c(0.4, -0.3, 0.2, -0.1)  # Seasonal AR coefficients
theta_seasonal <- c(0.3, -0.2, 0.1, -0.05) # Seasonal MA coefficients
sigma <- 1
n <- 500                                  # Final desired observations
burn_in <- 200                            # Burn-in period = untuk menghilangkan dependensi dari nilai inisial
seasonal_period <- 12                     # Seasonal period
max_lag <- seasonal_period * 4 + seasonal_period # Butuh hingga t-60 untuk differencing

# Total length including burn-in and initialization
total_length <- n + burn_in + max_lag

# Initialize
a <- rnorm(total_length, 0, sigma)       # White noise terms
y <- numeric(total_length)               # Time series values

# Initial values (set to mean or zero)
y[1:max_lag] <- 0  # Initialize up to t-48 for seasonal AR(4)

# Generate the series
for (t in (max_lag + 1):total_length) {
  # Calculate indices safely
  lag_indices <- t - seasonal_period * (1:4)
  diff_indices <- lag_indices - seasonal_period
  
  # SAR component (seasonal AR(4))
  sar_part <- sum(phi_seasonal * (y[lag_indices] - y[diff_indices]))
  
  # SMA component (seasonal MA(4))
  sma_part <- sum(theta_seasonal * a[lag_indices])
  
  # Seasonal differencing (D=1)
  y[t] <- y[t - seasonal_period] + sar_part + a[t] + sma_part
}

# Remove burn-in and initialization period
start_index <- burn_in + max_lag + 1
y_clean <- ts(y[start_index:total_length], frequency = seasonal_period)

# Plot the results
plot(y_clean, main = "Manual SARIMA(0,0,0)(4,1,4)12 Simulation")

Plot ACF dan PACF
sebelum differencing

acf(y_clean, main = "ACF Data Asli", lag.max=48)

pacf(y_clean, main = "PACF Data Asli", lag.max=48)

Differencing Data
Karena model memiliki D=1 (seasonal differencing orde 1) perlu dilakukan differencing musiman 12

# Seasonal differencing
data_diff <- diff(y_clean, lag = 12)
# Plot setelah differencing musiman
plot(data_diff, main = "Time Series Setelah Seasonal Differencing", ylab = "Nilai")

Plot ACF dan PACF setelah differencing

acf(data_diff, main = "ACF Setelah Seasonal Differencing", lag.max = 48)

pacf(data_diff, main = "PACF Setelah Seasonal Differencing", lag.max = 48)