γ
(0) =
E
(
x
2
t
) =
q
X
k
=1
σ
2
k
,
(4.5)
which exhibits the overall variance as a sum of variances of each of the com
ponent parts.
Example 4.1
A Periodic Series
Figure 4.1
shows an example of the mixture (
4.3
) with
q
= 3 constructed in
the following way. First, for
t
= 1
, . . . ,
100, we generated three series
x
t
1
= 2 cos(2
πt
6
/
100) + 3 sin(2
πt
6
/
100)
x
t
2
= 4 cos(2
πt
10
/
100) + 5 sin(2
πt
10
/
100)
x
t
3
= 6 cos(2
πt
40
/
100) + 7 sin(2
πt
40
/
100)
These three series are displayed in
Figure 4.1
along with the corresponding
frequencies and squared amplitudes. For example, the squared amplitude of
x
t
1
is
A
2
= 2
2
+ 3
2
= 13. Hence, the maximum and minimum values that
x
t
1
will attain are
±
√
13 =
±
3
.
61.
Finally, we constructed
x
t
=
x
t
1
+
x
t
2
+
x
t
3
and this series is also displayed in
Figure 4.1
. We note that
x
t
appears to
behave as some of the periodic series we saw in Chapters 1 and 2. The
184
4 Spectral Analysis and Filtering
systematic sorting out of the essential frequency components in a time series,
including their relative contributions, constitutes one of the main objectives
of spectral analysis.
The R code to reproduce
Figure 4.1
is
x1 = 2*cos(2*pi*1:100*6/100) + 3*sin(2*pi*1:100*6/100)
x2 = 4*cos(2*pi*1:100*10/100) + 5*sin(2*pi*1:100*10/100)
x3 = 6*cos(2*pi*1:100*40/100) + 7*sin(2*pi*1:100*40/100)
x = x1 + x2 + x3
par(mfrow=c(2,2))
plot.ts(x1, ylim=c(10,10), main=expression(omega==6/100~~~A^2==13))
plot.ts(x2, ylim=c(10,10), main=expression(omega==10/100~~~A^2==41))
plot.ts(x3, ylim=c(10,10), main=expression(omega==40/100~~~A^2==85))
plot.ts(x,
ylim=c(16,16), main="sum")
Example 4.2
The Scaled Periodogram for
Example 4.1
In
2.3,
Example 2.10
, we introduced the periodogram as a way to discover
the periodic components of a time series. Recall that the scaled periodogram
is given by
P
(
j/n
) =
2
n
n
X
t
=1
x
t
cos(2
πtj/n
)
!
2
+
2
n
n
X
t
=1
x
t
sin(2
πtj/n
)
!
2
,
(4.6)
and it may be regarded as a measure of the squared correlation of the data
with sinusoids oscillating at a frequency of
ω
j
=
j/n
, or
j
cycles in
n
time
points. Recall that we are basically computing the regression of the data
on the sinusoids varying at the fundamental frequencies,
j/n
. As discussed
in
Example 2.10
, the periodogram may be computed quickly using the fast
Fourier transform (FFT), and there is no need to run repeated regressions.
The scaled periodogram of the data,
x
t
, simulated in
Example 4.1
is shown
in
Figure 4.2
, and it clearly identifies the three components
x
t
1
, x
t
2
,
and
x
t
3
of
x
t
. Note that
P
(
j/n
) =
P
(1

j/n
)
,
j
= 0
,
1
, . . . , n

1
,
so there is a mirroring effect at the folding frequency of 1/2; consequently, the
periodogram is typically not plotted for frequencies higher than the folding
frequency. In addition, note that the heights of the scaled periodogram shown
in the figure are
P
(6
/
100) = 13
,
P
(10
/
100) = 41
,
P
(40
/
100) = 85
,
P
(
j/n
) =
P
(1

j/n
) and
P
(
j/n
) = 0 otherwise. These are exactly the values
of the squared amplitudes of the components generated in
Example 4.1
. This
outcome suggests that the periodogram may provide some insight into the
variance components, (
4.5
), of a real set of data.
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