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4.3 Time-Invariant Linear Filters 127
of weight functions and to select the one that appears to strike a satisfactory balance
between bias and variance.
The option Spectrum>Smoothed Periodogram in the program ITSM allows
the user to apply up to 50 successive discrete spectral average filters with weights
W(j) 1/(2m + 1), j −m, −m + 1,...,m, to the periodogram. The value of m
for each filter can be specified arbitrarily, and the weights of the filter corresponding
to the combined effect (the convolution of the component filters) is displayed by
the program. The program computes the corresponding discrete spectral average
estimators f(λ),0≤ ˆ λ ≤ π.
Example 4.2.2 The sunspot numbers, 1770–1869
4.3 Time-Invariant Linear Filters
Figure 4.9 displays a plot of (2π) −1 times the periodogram of the annual sunspot
numbers (obtained by opening the project SUNSPOTS.TSM in ITSM and select-
ing Spectrum>Periodogram). Figure 4.10 shows the result of applying the discrete
spectral weights
1 1 1
, , (corresponding to m 1, W(j) 1/(2m+1), |j| ≤m). It
3 3 3
is obtained from ITSM by selecting Spectrum>Smoothed Periodogram, entering
1 for the number of Daniell filters, 1 for the order m, and clicking on Apply. As
expected, with such a small value of m, not much smoothing of the periodogram
occurs. If we change the number of Daniell filters to 2 and set the order of the first
filter to 1 and the order of the second filter to 2, we obtain a combined filter with a
more dispersed set of weights, W(0) W(1) 3
2
1
, W(2) , W(3) . Click-
15 15 15
ing on Apply will then give the smoother spectral estimate shown in Figure 4.11.
When you are satisfied with the smoothed estimate click OK, and the dialog box will
close. All three spectral density estimates show a well-defined peak at the frequency
ω10 2π/10 radians per year, in keeping with the suggestion from the graph of the
data itself that the sunspot series contains an approximate cycle with period around
10 or 11 years.
In Section 1.5 we saw the utility of time-invariant linear filters for smoothing the data,
estimating the trend, eliminating the seasonal and/or trend components of the data,
etc. A linear process is the output of a time-invariant linear filter (TLF) applied to a
white noise input series. More generally, we say that the process {Yt} is the output of
a linear filter C {ct,k,t,k 0 ± 1,...} applied to an input process {Xt} if
∞
Yt ct,kXk, t 0, ±1,.... (4.3.1)
k−∞
The filter is said to be time-invariant if the weights ct,t−k are independent of t, i.e., if
ct,t−k ψk.