Integrated Autoregressive Moving Average Process

This is also known as ARIMA.

We define as follows

$$X_n=X_{n-1}+Z_n$$ $$Z_n=\varphi Z_{n-1}+{\theta }_1{ϵ}_{n-1}+{ϵ}_n$$

This is where $ϵ$ are normally distributed and this process has a mean of 0. It has a variance of ${\sigma }^2$ and ${\theta }_2$ and ${\theta }_1$ are constant.

We day an ARIMA(1,1,1) process with a drift $\mu$ is $X_n=X_{n-1}+Y_n$ where $Y_n=Z_n+\mu$.

Here we are simulating an ARMA(1,1,1) process with ${\varphi }_1=0.5$ and ${\theta }_2=0.7$.

arima11 <- arima.sim(1000, model=list(order=c(1, 1, 1),ar=c(.5), ma=c(.7)))
ts.plot(arima11)

Pasted image 20260126195708.png Differencing is used to remove trends and transform a non-stationary series into a stationary one.

We use ARIMA(p, 1, q) as an example.

Suppose $Z_n$ follows ARIMA(p, 1, q), them $Y_n=Z_n-Z_{n-1}$ will be a 0-mean ARIMA(p, q) process.

Y <- diff(arima11)
ts.plot(Y)