0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 
8.0 
VAAN IAAAAAAANA 
aoe cee | | } 
\ Vi yi / | \ 
f 
| \ V V 
800.0 900.0 1000.0 
in Vd 
V \ / \ i | i \ | \ / ; / 
YW YOY WY 
0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 
Fig. 13.7. X;= (1.95 + 0.23671) X,, — (0.96 + 0.246)X, 2, with different 
initial values (without white noise). 
(From Ozaki.98) 
More generally, Ozaki® proposed the higher type Duffing and Van del Pol nonlinear 
model as 
X,= 91(Xr-1) Xr1 + B2(X-1) Xr2 + €r, (13.54) 
where 
(X11) = git (xg? Xt Xe ni))x",) exh 
22(X1-1) = 2+ (af? + TOaxXen + Wx? | aro © OCs 7 Xs) exe, (13.56) 
The term g(-) is expressed by a constant plus a Hermite type polynomial. When 
r,; = 0,72 = 0 in these equations, an equation of the type of Eq. 13.50 is obtained. 
To determine the order, Akaike’s criterion (AIC) was shown to be applicable for 
these nonlinear models, too. 
(13.55) 
13.5 NONLINEAR THRESHOLD AUTOREGRESSIVE MODEL 
Ozaki®? extended and generalized his model further and proposed his nonlinear 
threshold autoregressive model and a unified explanation of the nonlinear models. 
For example, the exponential autoregressive model that he used as an example of a 
Van del Pol type nonlinear process, Eq. 13.53, 
X, = (1.95 + 0.236%) X,1— (0.96 + 0.24e%") Xp +e; 
was shown to be approximated by Tong’s linear threshold autoregressive model, as 
x = 1.95 Xi-1 4 0.96 X12 + €; if X,_1! 20.5 (13.57) 
2.18 X;1—1.20 X2+¢€, if X11 < 0.5. 
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