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.