Autoregressive integrated moving average

In statistics, an autoregressive integrated moving average (ARIMA) model is a generalisation of an autoregressive moving average or (ARMA) model. These models are fitted to time series data either in order to better understand the data or to predict future points in the series. The model is generally referred to as an ARIMA(p,d,q) model where the p, d and q are integers greater than or equal to zero and refer to the order of the autoregressive, integrated and moving average parts of the model respectively.

Given a time series of data X_t (where t is integer valued and the X_t are real numbers) then an ARMA(p,q) model is given by

$\left(1 - \sum_{i=1}^p \phi_i L^i\right) X_t = \left(1 + \sum_{i=1}^q \theta_i L^i\right) \varepsilon_t\,$

where L is the lag operator, the φ_i are the parameters of the autoregressive part of the model, the θ_i are the parameters of the moving average part and the ε_t are error terms. The error terms ε_t are generally assumed to be iid variables sampled from a normal distribution with zero mean: ε_t ~ N(0,σ²) where σ² is the variance.

The ARMA model is generalised by adding a d parameter to form the ARIMA (p, d, q) model

$\left(1 - \sum_{i=1}^p \phi_i L^i\right) (1-L)^d X_t = \left(1 + \sum_{i=1}^q \theta_i L^i\right) \varepsilon_t\,$

where d is a positive integer (if d is zero then this model is equivalent to an ARMA model). It should be noted that not all choices of parameters produce well-behaved models. In particular, if the model is required to be stationary then conditions on these parameters must be met.

Some well-known special cases arise naturally. For example, an ARIMA(0,1,0) model is given by:

$X_t = X_{t-1} + \varepsilon$

which is simply a random walk.

A number of variations on the ARIMA model are commonly used. For example, if multiple time series are used then the X_t can be thought of as vectors and a VARIMA model may be appropriate. Sometimes a seasonal effect is suspected in the model. For example, consider a model of daily road traffic volumes. Weekends clearly exhibit different behaviour from weekdays. In this case it is often considered better to use a SARIMA (seasonal ARIMA) model than to increase the order of the AR or MA parts of the model. If the time-series is suspected to exhibit long-range dependence then the d parameter may be replaced by certain non-integer values in a Fractional ARIMA (FARIMA also sometimes called ARFIMA) modal.

Autoregressive integrated moving average

See also: Autoregressive integrated moving average, Autoregressive moving average, Iid, Lag operator, Long-range dependence, Normal distribution, Random walk, Stationary process, Statistics, Time series