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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 yt , the value at time t, is given (for t=1, 2, 3,…) by yt=a + bt. Then, for each value of t>1, the first difference, Δyt, is given by

Δyt=yt-yt-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, Δ2yt (=Δyt−Δyt−1), will remove any quadratic dependence on time.

Subjects: Probability and Statistics.

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