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Figure 1-20
Smoothing with a
low-pass linear filter.
1.5 Estimation and Elimination of Trend and Seasonal Components 27
data {Xt} and removes from it the rapidly fluctuating (or high frequency) component
{ ˆYt} to leave the slowly varying estimated trend term {ˆmt} (see Figure 1.20).
The particular filter (1.5.5) is only one of many that could be used for smoothing.
For large q, provided that (2q + 1) −1 q
j−q Yt−j ≈ 0, it not only will attenuate
noise but at the same time will allow linear trend functions mt c0 + c1t to pass
without distortion (see Problem 1.11). However, we must beware of choosing q to
be too large, since if mt is not linear, the filtered process, although smooth, will not
be a good estimate of mt. By clever choice of the weights {aj} it is possible (see
Problems 1.12–1.14 and Section 4.3) to design a filter that will not only be effective
in attenuating noise in the data, but that will also allow a larger class of trend functions
(for example all polynomials of degree less than or equal to 3) to pass through without
distortion. The Spencer 15-point moving average is a filter that passes polynomials
of degree 3 without distortion. Its weights are
with
and
aj 0, |j| > 7,
aj a−j, |j| ≤7,
[a0,a1,...,a7] 1
[74, 67, 46, 21, 3, −5, −6, −3]. (1.5.6)
320
Applied to the process (1.5.2) with mt c0 + c1t + c2t 2 + c3t 3 ,itgives
7
7
7
7
ajXt−j ajmt−j + ajYt−j ≈ ajmt−j mt,
j−7
j−7
j−7
j−7
where the last step depends on the assumed form of mt (Problem 1.12). Further details
regarding this and other smoothing filters can be found in Kendall and Stuart (1976),
Chapter 46.
(b) Exponential smoothing. For any fixed α ∈ [0, 1], the one-sided moving
averages ˆmt, t 1,...,n, defined by the recursions
and
ˆmt αXt + (1 − α) ˆmt−1, t 2,...,n, (1.5.7)
ˆm1 X1
(1.5.8)