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1.5 Estimation and Elimination of Trend and Seasonal Components 31
1.5.2 Estimation and Elimination of Both Trend and Seasonality
The methods described for the estimation and elimination of trend can be adapted in
a natural way to eliminate both trend and seasonality in the general model, specified
as follows.
Classical Decomposition Model
Xt mt + st + Yt, t 1,...,n, (1.5.11)
where EYt 0, st+d st, and d
j1 sj 0.
We shall illustrate these methods with reference to the accidental deaths data of
Example 1.1.3, for which the period d of the seasonal component is clearly 12.
Method S1: Estimation of Trend and Seasonal Components
The method we are about to describe is used in the Transform>Classical option
of ITSM.
Suppose we have observations {x1,...,xn}. The trend is first estimated by applying
a moving average filter specially chosen to eliminate the seasonal component
and to dampen the noise. If the period d is even, say d 2q, then we use
ˆmt (0.5xt−q + xt−q+1 +···+xt+q−1 + 0.5xt+q)/d, q < t ≤ n − q. (1.5.12)
If the period is odd, say d 2q + 1, then we use the simple moving average (1.5.5).
The second step is to estimate the seasonal component. For each k 1,...,d,we
compute the average wk of the deviations {(xk+jd−ˆmk+jd), q < k+jd ≤ n−q}. Since
these average deviations do not necessarily sum to zero, we estimate the seasonal
component sk as
ˆsk wk − d −1
d
wi, k 1,...,d, (1.5.13)
i1
and ˆsk ˆsk−d,k >d.
The deseasonalized data is then defined to be the original series with the estimated
seasonal component removed, i.e.,
dt xt −ˆst, t 1,...,n. (1.5.14)
Finally, we reestimate the trend from the deseasonalized data {dt} using one of
the methods already described. The program ITSM allows you to fit a least squares
polynomial trend ˆm to the deseasonalized series. In terms of this reestimated trend
and the estimated seasonal component, the estimated noise series is then given by
ˆYt xt −ˆmt −ˆst, t 1,...,n.