# Definition:Finite Difference Operator/Backward Difference

## Definition

Let $f: \R \to \R$ be a real function.

The backward difference operator on $f$ is defined as:

$\map {\nabla f} x := \map f x - \map f {x - 1}$

## Also presented as

The backward difference operator, when applied to a time series, can be written in terms of the backward shift operator as:

$\map \nabla {z_t} = z_t - z_{t - 1} = \map {\paren {1 - B} } {z_t}$

## Sources

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