\u03b3 0 E x 2 t q X k 1 \u03c3 2 k 45 which exhibits the overall variance as a sum of

Γ 0 e x 2 t q x k 1 σ 2 k 45 which exhibits the

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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
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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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