Model umum :

\[ Φ_4(B^{12})ϕ_4(B)y_t​=Θ_4(B^{12})θ_4(B)a_t \]

dengan komponen

\[ Φ_4(B^{12})=(1-Φ_1B^{12}-Φ_2B^{24}-Φ_3B^{36}-Φ_4B^{48}) \]

\[ ϕ_4(B)=(1-ϕ_1B-ϕ_2B^2-ϕ_3B^3-ϕ_4B^4) \]

\[ Θ_4(B^{12})=(1-Θ_1B^{12}-Θ_2B^{24}-Θ_3B^{36}-Θ_4B^{48}) \]

\[ θ_4(B)=(1-θ_1B-θ_2B^2-θ_3B^3-θ_4B^4) \]

Model secara keseluruhan :

\[ y_t=0.2y_{t-1}-0.2y_{t-2}+0.1y_{t-3}+0.2y_{t-4}+0.2y_{t-12}+0.1y_{t-24}-0.2y_{t-36}+0.3y_{t-48}\]

\[+0.3a_{t-1}+0.2a_{t-2}-0.2a_{t-3}+0.1a_{t-4}+0.3a_{t-12}+0.2a_{t-24}-0.1a_{t-36}+0.2a_{t-48} \] \[ +0.04y_{t-13}-0.04y_{t-14}+0.02y_{t-15}+0.04y_{t-16}+0.02y_{t-25}-0.02y_{t-26}+0.01y_{t-27}+0.02y_{t-28} \] \[ -0.04y_{t-37}+0.04y_{t-38}-0.02y_{t-39}-0.04y_{t-40} \] \[ +0.06y_{t-49}-0.06y_{t-50}+0.03y_{t-51}+0.06y_{t-52}+0.09a_{t-13}+0.06a_{t-14}-0.06a_{t-15}+0.03a_{t-16} \] \[ +0.06a_{t-25}+0.04a_{t-26}-0.04a_{t-27}+0.02a_{t-28}-0.03a_{t-37}-0.02a_{t-38}+0.02a_{t-39}-0.01a_{t-40} \] +0.06a_{t-49}+0.04a_{t-50}-0.04a_{t-51}+0.02a_{t-52} \[ \]

set.seed(123) 
at <- rnorm(1000, mean = 0, sd = 2)
#parameter non-seasonal (4,0,4)
phi1   <- 0.2
phi2   <- -0.2
phi3   <- 0.1
phi4   <- 0.2
theta1 <- 0.3
theta2 <- 0.2
theta3 <- -0.2
theta4 <- 0.1

#parameter seasonal (4,0,4)^12
Phi1 <- 0.2
Phi2 <- 0.1
Phi3 <- -0.2
Phi4 <- 0.3
Theta1 <- 0.3
Theta2 <- 0.2
Theta3 <- -0.1
Theta4 <- 0.2
#parameter interaksi ar x sar
phi13 <- phi1*Phi1
phi14 <- phi2*Phi1
phi15 <- phi3*Phi1
phi16 <- phi4*Phi1
phi25 <- phi1*Phi2
phi26 <- phi2*Phi2
phi27 <- phi3*Phi2
phi28 <- phi4*Phi2
phi37 <- phi1*Phi3
phi38 <- phi2*Phi3
phi39 <- phi3*Phi3
phi40 <- phi4*Phi3
phi49 <- phi1*Phi4
phi50 <- phi2*Phi4
phi51 <- phi3*Phi4
phi52 <- phi4*Phi4

#parameter interaksi ma x sma
theta13 <- theta1*Theta1
theta14 <- theta2*Theta1
theta15 <- theta3*Theta1
theta16 <- theta4*Theta1
theta25 <- theta1*Theta2
theta26 <- theta2*Theta2
theta27 <- theta3*Theta2
theta28 <- theta4*Theta2
theta37 <- theta1*Theta3
theta38 <- theta2*Theta3
theta39 <- theta3*Theta3
theta40 <- theta4*Theta3
theta49 <- theta1*Theta4
theta50 <- theta2*Theta4
theta51 <- theta3*Theta4
theta52 <- theta4*Theta4
n <- length(at); n
## [1] 1000
y <- numeric(n)
y[1:48] <- 0
for (t in 49:n) {
  
  ar <- phi1*(y[t-1]) + phi2*(y[t-2]) + phi3*(y[t-3]) + phi4*(y[t-4])
  ma <- theta1*at[t-1] + theta2*at[t-2] + theta3*at[t-3] + theta4*at[t-4]
  
  sar <- Phi1*(y[t-12]) + Phi2*(y[t-24]) + Phi3*(y[t-36]) + Phi4*(y[t-48])
  sma <- Theta1*at[t-12] + Theta2*at[t-24] + Theta3*at[t-36] + Theta4*at[t-48]
  
  arsar <- phi13*(y[t-13]) + phi14*(y[t-14]) + phi15*(y[t-15]) + phi16*(y[t-16])             + phi25*(y[t-25]) + phi26*(y[t-26]) + phi27*(y[t-27]) + phi28*(y[t-28])
            + phi37*(y[t-37]) + phi38*(y[t-38]) + phi39*(y[t-39]) + phi40*(y[t-40])
            + phi49*(y[t-49]) + phi50*(y[t-50]) + phi51*(y[t-51]) + phi52*(y[t-52])
  
  masma <- theta13*at[t-13] + theta14*at[t-14] + theta15*at[t-15] + theta16*at[t-16]
            + theta25*at[t-25] + theta26*at[t-26] + theta27*at[t-27] + theta28*at[t-28]
            + theta37*at[t-37] + theta38*at[t-38] + theta39*at[t-39] + theta40*at[t-40]
            + theta49*at[t-49] + theta50*at[t-50] + theta51*at[t-51] + theta52*at[t-52]
  
  y[t] <- ar + ma + sar + sma + arsar + masma + at[t] #model dengan interaksi
}

y <- y[-(1:100)]
plot.ts(y, main = "Simulasi ARIMA(4,0,4)(4,0,4)[12]")

par(mfrow=c(1,2)) 
acf(y) 
pacf(y)