Figure 5.27 shows the spectra, the theoretical one from Eq. 5.182” in Fig. 5.27a, the one 
estimated from the fitted model ARMA(2.1), using the estimated parameters in Fig. 
5.27b, and the one estimated by the nonparametric method from the generated process of 
Fig. 5.25 in Fig. 5.27c, with a maximum lag number of 60 and Hanning window. Spectra 
a and b are similar and close to each other, and c fluctuated as expected. In another trial, 
instead of the ARMA (n,m) model, an AR model was fitted to this generated ARMA(2.1) 
process. The optimum order n was searched by AIC criteria as AR(n), and was found to 
be n = 6. Its parameters 4) to dg and G2 were estimated by the method described in 
Sections 5.4.3 and 5.4.4 as 
@, =—0.35801, dp =—0.64448, a3 =—0.45424, 
G4 =—0.34915, ds =—0.10554, dg =— 0.11502, 
o¢ = 1.03499. 
These values are listed in Table 5.1. 
The value of G2 is very close to the theoretical value of ¢2 = 1.0. The spectrum 
of this fitted AR(6) is shown as Fig. 5.274. It is interesting to find that this spectrum 
Fig. 5.27d is very smooth, but its shape looks like the smoother shape of spectrum 
Fig. 5.27c, the one estimated by the nonparametric method. In Fig. 5.27c, by vertical 
lines, the 90% level of confidence interval of this estimate, based on the y7—distribution 
of the equivalent degree of freedom (here approximately equal to 27) as is explained in 
Section 2.5.5 is shown for reference. 
As another example of the ARMA(2.1) process, the sign of the parameter bj in the 
previous example was changed, and another ARMA(2.1) process was generated, keeping 
a}, 22, anda? the same, 
a,=-0.3, a=+04, bj=+0.7, o2=1.0. 
As for the previous example, its time history (readings are listed in Appendix A1 as Table 
A1.5; pp. 251, 256, and 257), autocorrelation, and spectra are shown as Figs. 5.28, 5.29, 
and 5.30. 
In this example, the AIC criteria gave ARMA(3.1) as the optimum model to be 
fitted, instead of ARMA(2.1), though the difference in AIC is not so large. The estimated 
spectrum of this optimum ARMA(3.1) is shown as Fig. 5.30b, and for reference the 
spectrum of the fitted ARMA(2.1) model is shown as Fig. 5.30e. ARMA(3.1) gave the 
minimum value of AIC and ARMA(2.1) did not. However, the ARMA(2.1) spectrum Fig. 
5.30e looks more like the theoretical spectrum Fig. 5.30a than does that of ARMA(3.1), 
spectrum Fig. 5.30b. Spectrum Fig. 5.30d shows the spectrum of the fitted optimum 
AR(n) model, where n appeared to be 8 by AIC criteria. It is interesting to find that this 
spectrum Fig. 5.30d shows again the same pattern of variation as spectrum Fig. 5.30c 
obtained by the nonparametric method. The estimated values of parameters 
G},4>,- - -dgandG2 are listed in Table 5.1, and G2 = 1.04911 is very close to the 
theoretical value d2= 1.0. 
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