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Figure 1-22
Strike data smoothed
by elimination of high
frequencies with f 0.4.
1.5 Estimation and Elimination of Trend and Seasonal Components 29
(thousands)
3.5 4.0 4.5 5.0 5.5 6.0
1950 1960 1970 1980
a0,a1, and a2 to minimize the sum of squares, n
t1 (xt − mt) 2 (see Example 1.3.4).
The method of least squares estimation can also be used to estimate higher-order
polynomial trends in the same way. The Regression option of ITSM allows least
squares fitting of polynomial trends of order up to 10 (together with up to four harmonic
terms; see Example 1.3.6). It also allows generalized least squares estimation
(see Section 6.6), in which correlation between the residuals is taken into account.
Method 2: Trend Elimination by Differencing
Instead of attempting to remove the noise by smoothing as in Method 1, we now
attempt to eliminate the trend term by differencing. We define the lag-1 difference
operator ∇ by
∇Xt Xt − Xt−1 (1 − B)Xt, (1.5.9)
where B is the backward shift operator,
BXt Xt−1. (1.5.10)
Powers of the operators B and ∇ are defined in the obvious way, i.e., B j (Xt) Xt−j
and ∇ j (Xt) ∇(∇ j−1 (Xt)), j ≥ 1, with ∇ 0 (Xt) Xt. Polynomials in B and ∇ are
manipulated in precisely the same way as polynomial functions of real variables. For
example,
∇ 2 Xt ∇(∇(Xt)) (1 − B)(1 − B)Xt (1 − 2B + B 2 )Xt
Xt − 2Xt−1 + Xt−2.