The autoregressive integrated moving average model, or ARIMA model, is a broadening of the class of ARMA models to include differencing. The word integrated comes from the fact that the stationary model that was fitted is based on the differenced data.

$y'_t = \phi_1y'_{t-1}+...+\phi_p y'_{t-p}+\theta_iw_{t-1}+...+\theta_qw_{t-q}$ where $$y'_t$$ is the differenced series.

The notation for an ARIMA process of order $$p$$ for the AR part, order $$q$$ for the MA part and differences $$d$$ is denoted as ARIMA($$p, d, q$$), where

$$p=$$ order of the autoregressive part,
$$d=$$ degree of first differencing involved,
$$q=$$ order of the moving average part.

The basic idea is that if differencing the data at some order $$d$$ produces an ARMA process, then the original process is said to be ARIMA. Often, a single difference will suffice to yield a stationary series.

Many of the models we have already discussed are special cases of the ARIMA model.

$\begin{array}{ll} \hline \text{White noise} & \text{ARIMA(0,0,0)} \\ \text{Random walk} & \text{ARIMA(0,1,0) with no constant}\\ \text{Random walk with drift} & \text{ARIMA(0,1,0) with a constant} \\ \text{Autoregression} & \text{ARIMA(}p\text{,0,0)} \\ \text{Moving average} & \text{ARIMA(0,0,}q\text{)} \end{array}$