Definition:ARIMA Model/ARIMA Operator

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 $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
 * $\tilde z_t = z_t - \mu$

Let $a_t$ be an independent shock at timestamp $t$.

Let $M$ be a generalized autoregressive (AR) process on $S$:


 * $\map \varphi B z_t = \map \phi B \paren {1 - B}^d z_t = \map \theta B a_t$

The generalized autoregressive operator $\map \varphi B$, is:
 * $\map \varphi B = \map \phi B \paren {1 - B}^d$

where $B$ denotes the backward shift operator.

Hence the generalized autoregressive model can be written in the following compact manner:


 * $\map \phi B w_t = \map \theta B a_t$

where:
 * $w_t = \nabla^d z_t$
 * $\nabla^d$ denotes the $d$th iteration of the backward difference operator.

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