laa oe @ il 
a2nx 2n(1+a) 
S(l) = S(— 7) = #52) 
Its shape will then be as shown in Fig. 5.9. 
S(w )/S(0) 
Fig. 5.9. s(@) of AR(1). 
5.2.2.7 Estimation of a and oz. When we are given a set of data, X;- - - Xy, with 
numbers N, and the AR(1) model is to be fitted (the determination of order will be dis- 
cussed later in Section 5.5), we can easily estimate the values of a anda? by the 
minimum least squares method as 
N 
»S X; X14 
ga (5.53) 
OX 
t=2 
and the minimized residual sum is 
N 
Ue 7 wat (X,-4 Xia). (5.54) 
The same results can also be derived ae es 5.40 and Eq. 5.41 as 
pO) 
~ RO) (5.55) 
and 62=(1-—d7)R(0). (5.56) 
Equations 5.53 and 5.55 have the same content as if the sample correlation were replaced 
with R(0) and R(1). Obtaining a by the linear minimum least squares method from Eq. 
5.53 is actually done by Eq. 5.55. Then, the variance a? ofe, is estimated by Eq. 5.56. 
5.2.2.8 Example of AR(1). Figure 5.10 is an example of the AR(1) process simu- 
lated by an AR(1) model X,—0.5 X,_) =€,, where a = —0.5 in Eq. 5.7. In order to look 
at the pattern of variations, at the bottom of this figure, a part (t = 100 to 250) of this pro- 
cess is shown expanded on the time axis. Its readings are listed in Appendix Al as Table 
112 
