# Definition:ARIMA Model/Linguistic Note

## Linguistic Note on ARIMA Model

The I in the acronym ARIMA stands for integrated.

This arises because the relationship which is the inverse of the backward difference operator:

$w_t = \map {\nabla^d} {z_t}$

is:

$z_t = S^d w_T$

where $S$ is the summation operator, defined as:

$S = \nabla^{-1} = \paren {1 - B}^{-1}$

so that:

$\map S {w_t} = \ds \sum_{j \mathop = 0}^\infty w_{t - j} = w_t + w_{t - 1} + w_{t - 2} + \dotsb$

Hence the autoregressive integrated moving average (ARIMA) process can be generated by summing or integrating the ARMA process a total of $d$ times.

## Sources

$1$: Introduction:
$1.2$ Stochastic and Deterministic Dynamic Mathematical Models
$1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Nonstationary models