# 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**