Definition:Box-Jenkins Model
Definition
The Box-Jenkins model is a very general mathematical model for time series analysis in forecasting and prediction.
ARMA (Autoregressive Moving Average)
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 $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
- $\tilde z_t = z_t - \mu$
Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$
Let $M$ be a model where the current value of $\tilde z_t$ is expressed as a combination of a finite linear aggregate of the past values along with a finite linear aggregate of the shocks:
- $\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$
$M$ is known as a mixed autoregressive (order $p$), moving average (order $q$) process, usually referred as an ARMA process.
ARIMA (Autoregressive Integrated Moving Average)
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 shocks 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:
- $w_t = \phi_1 w_{t - 1} + \phi_2 w_{t - 2} + \dotsb + \phi_p w_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$
$M$ is known as an autoregressive integrated moving average (ARIMA) process of order $p$, $d$, $q$.
Also see
- Results about Box-Jenkins models can be found here.
Source of Name
This entry was named for George Edward Pelham Box and Gwilym Meirion Jenkins.
Historical Note
The Box-Jenkins model was first proposed by George Edward Pelham Box and Gwilym Meirion Jenkins in $1967$.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Box-Jenkins model (G.E.P. Box and G.M. Jenkins, 1967)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Box-Jenkins model (G.E.P. Box and G.M. Jenkins, 1967)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Box-Jenkins model