The **autoregressive integrated moving average
model**, or ARIMA model, is a broadening of the class of ARMA
models in order to include differencing. The word *integrated* is
due to the fact that the stationary model is fitted 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.

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 and

\(q-\) 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.

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

**Citation**

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: *Hartmann,
K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis
using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.*