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Difference Operator

The difference operator maps a function or sequence to the change between values at separated arguments. For a step h, the forward difference and backward difference operators are

Delta_hf(x)=f(x+h)-f(x)
(1)
del _hf(x)=f(x)-f(x-h).
(2)

In conventional time-series notation, the backward-shift operator B maps X_t to its value at time series lag one, so BX_t=X_(t-1). The first-difference operator is therefore 1-B, with (1-B)X_t=X_t-X_(t-1). Thus (1-B)X_(t-j)=X_(t-j)-X_(t-j-1) is a difference evaluated at lag j. Repeated differencing can remove polynomial trends and is used in the analysis of nonstationary time series.

See also

Backward Difference, Difference Equation, Finite Difference, Forward Difference, Nonstationary Time Series, Time Series, Unit Root

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References

Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, 1965.

Cite this as:

Weisstein, Eric W. "Difference Operator." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DifferenceOperator.html

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