the method of order determination that will be mentioned later in Section 5.5. Now with 
the order as 1, a anda? were calculated from Eqs. 5.55 and 5.56, as @ = — 0.50933 and 
$2 = 1.04646, which are close to the a = — 0.5 anda? = 1.0 used to generate the process. 
We also estimated the correlation coefficient 6(r) = R(r) /R(0) from Eqs. 5.36 and 5.37, 
using these @ and Ge . These values are so close to the theoretical R(r), or just the same in 
o(r) = R(r)/R(O) in Fig. 5.11a that their drawing was omitted. Figure 5.12a is the theoret- 
ical spectrum from Eq. 5.50, using the design values of a = -0.5 anda? = 1. 
4.374 
0.06 012 019 O25 O31 40.37 0.44 ~°& 0.50 
7 @ 
Fig. 5.12a. Theoretical AR(1). 
Fig. 5.12. Spectrum for AR(1) process X;—0.5 X,,=€;, €+: N[0, 1]. 
The estimated spectrum s(w), with d = — 0.50933 and? = 1.04646 obtained by 
model fitting and also by Eq. 5.50, is given as b in Fig. 5.12. For comparison, as c in Fig. 
5.12, the spectrum 5(w) was also estimated by the nonparametric method as the Fourier 
transform of R(r) in Fig. 5.11b, using the Hanning window and maximum lag M = 60. It 
is interesting that the spectrum estimated by model fitting is very similar to the theoretical 
spectrum, Fig. 5.12a, of this generated process. On the other hand, the spectrum esti- 
mated by the nonparametric method Fig. 5.12c is more wavy than the theoretical one, 
although the fundamental shape is similar. The wavy fluctuations were found to come 
mostly from the wavy fluctuation of the input white noise {€,} that is shown in Fig. 5.5c, 
as the incomplete whiteness of the pure random process {€,} generated. In the figures, 
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