Definition:Box-Jenkins Model/ARIMA

Definition
Let $S$ be a stochastic process based on an equispaced time series.

Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$

Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shock at timestamps $t, t - 1, t - 2, \dotsc$,

Let:
 * $w_t = \nabla^d z_t$

where $\nabla^d$ denotes the $d$th iteration of the backward difference operator.

Let $M$ be a model where the current value of $w_t$ is expressed as a combination of a finite linear aggregate of the past values along with a finite linear aggregate of the shocks:


 * $\tilde w_t = \phi_1 w_{t - 1} + \phi_2 w_{t - 2} + \dotsb + \phi_p w_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$

$M$ is known as an autoregressive integrated moving average (ARIMA) process of order $p$, $d$, $q$.

In practice, $d$ is usually $0$ or $1$, or at most $2$.

Also see

 * Definition:Autoregressive Model
 * Definition:Moving Average Model
 * Definition:ARMA Model