The difference operator maps a function or sequence to the change between values at separated arguments. For a step 
, the forward difference
and backward difference operators are
 |  |  |
(1)
|
 |  |  |
(2)
|
In conventional time-series notation, the backward-shift operator 
maps 
to its value at time series
lag one, so 
.
The first-difference operator is therefore 
, with 
. Thus 
is a difference evaluated at
lag 
.
Repeated differencing can remove polynomial trends and is used in the analysis of
nonstationary time series.
