## Quick Reference

The process of subtracting successive terms in a series of equi-spaced observations. Used in numerical analysis and as a simple method for removing trends in time series. Suppose, for example, that *y** _{t}* , the value at time

*t*, is given (for

*t*=1, 2, 3,…) by

*y*

*=*

_{t}*a*+

*bt*. Then, for each value of

*t*>1, the first difference, Δ

*y*

*, is given by*

_{t}Δ*y** _{t}*=

*y*

*-*

_{t}*y*

*}=(*

_{t-1}*a+bt*)-{

*a+b*(

*t*-1)}=b,

which is no longer dependent on *t*. In the same way, further differences will remove higher polynomial dependencies on time. For example, the second difference, Δ^{2}*y** _{t}* (=Δ

*y*

*−Δ*

_{t}*y*

*), will remove any quadratic dependence on time.*

_{t−1}**From:**
differencing
in
A Dictionary of Statistics »

*Subjects:*
Probability and Statistics.