Here Ar is assumed to be 1, as was mentioned at the beginning of Part II. As a 
result, S,(@) are obtained in the range —-2 <w <2. However, if Ar =~ 1 and the 
spectrum is calculated for —2/Ar < w </At, then the spectrum is easily inverted into 
oe Be)’ 
ae ae) | a (5.183) 
SA@) = At- 
52.4.7 Example of ARMA(2.1) Simulated. Figure 5.25 is an example of a gener- 
ated ARMA(2.1) process over t= 1 to 600, when a = -0.3, a7=+ 0.4, b} =— 0.7, and 
o2 = 1.0, namely by 
X,—0.3 Xie oF 0.4 X12 = €,—0.7 € 7-1 : 
Here €, is the pure random process we generated as AR(0) in Fig. 5.3. Its readings 
are listed in Appendix Al as Table A1.4; pp. 251, 254, and 255. The correlations and 
spectra of this example are shown in Figs. 5.26 and 5.27. Figure 5.26a shows the theoreti- 
cal autocorrelation coefficient o(r) = R(r)/R(O) of this generated ARMA(2.1), calcu— 
lated by Eqs. 5.160, 5.161, 5.162, or 5.163, 5.164, 5.165, using the design values 
of a; =— 0.3, a2 = 0.4, b} =—0.7 anda? = 1.0. Figure 5.26b shows the estimated 
o(r) = R(r) /R(O), calculated from the generated process. 
We can see in Fig. 5.26b that the autocorrelation coefficient at higher lag r continues 
to fluctuate, even after the theoretical coefficient, shown in Fig. 5.26a, has died down and 
converged to zero. On the other hand, the values at low lag numbers look very similar to 
the theoretical values. An ARMA(n,m) process was fitted and its optimum orders n and 
m were searched by AIC criteria as explained in Section 5.5.4, using these autocorrela- 
tions b, and actually were found to be n = 2, m= 1. Then the parameters were estimated 
by the method described in this section as Eqs. 5.174, 5.175, 5.155, and 5.156. The values 
thus obtained are 
Gd, = — 0.37842, dz = 0.40130, b, =— 0.73968, G2= 1.0400 
which are not very close, but pretty close to the values actually used to generate this 
process. The autocorrelation coefficients 6(r) = R(r) /R(0) were then estimated using 
these estimated parameters by Eqs. 5.160, 5.161, and 5.162, just as for the theoretical 
values. The results are similar and very close to the theoretical values shown in 
Fig. 5.26a, so that drawings were again omitted. 
146 
