> [!tldr] > An ARIMA (integrated ARMA) model $\mathrm{ARIMA}(p,d,q)$ is a stochastic process $(X_{t})$ where the $d$th difference $W_{t}:=(\nabla^{d}X_{t})$ is an ARMA model: $\begin{align*} W_{t}&:= \underbrace{\alpha_{1}W_{t-1}+ \cdots + \alpha_{p}W_{t-p}}_{\text{AR}(p)}\\[0.2em] &\hphantom{blaj}+\underbrace{Z_{t}+\beta_{1}Z_{t-1}+ \cdots + \beta_{q}Z_{t-q}}_{\text{MA}(q)}\\ &~= \alpha(B)W_{t} + \beta(B)Z_{t} \end{align*}$where $(Z_{t})$ is a [[Purely Random Processes|PRP]]. ```R fold arima_params <- list(order = c(2,1,2), ar = c(0.7,-0.9), ma = c(-0.4,0.4)) arima_data <- arima.sim(n = 500, arima_params) dev.off() par(mfrow=c(2,1), mai=c(1,4,1,1)/4, oma=c(2,0,0,0), bg=NA) plot(arima_data, type="l", ylab="Series Value", main="ARIMA(2,1,2) Process") acf(diff(arima_data), ylab="Autocorr. of diff.", main="") ``` ![[ARIMA212ACF.png#invert]] In terms of backshift operators, the ARIMA model can be written as $\phi(B)W_{t}=\phi(B)(1-B)^{d}X_{t}=\theta(B)Z_{t}.$