Xin] = AX, + CX, €;+€ p41. (13.14) 
We can modify this equation to 
{1 +X, Ble xX aXe (13.15) 
Here B is the backward shift operator. For small c, Eq. 13.15 can be inverted to 
emi ={1—c X, B}Xi41—aXi] 
SG SO SOG lO GF Xs (13.16) 
This is a generalized autoregressive model. If the product term X; X,_1, is neglected, initial 
estimates of a and c can be obtained by a least squares approach. If the product term is 
not neglected, a nonlinear least squares approach can be used to get a andc. After initial 
estimates dp and Cp are obtained, €, can be estimated by 
Enya = X41 — AX,;— CX, €; for t= ee -- + N. (13.17) 
Recursively starting from€, = 0, and using {X,.4] fort=1, °° ‘N, new estimates for a 
and c can be found to minimize 
N 2 
» {Kee —ax,-— X,e . (13.18) 
t=1 
Until the estimates converge, this procedure is repeated. Subba Rao®? studied this 
bilinear model and gave several examples. 
13.3 THRESHOLD AUTOREGRESSIVE MODEL 
The threshold autoregressive model introduced by Tong™ is generally expressed as 
GeO ante o oO San (13.19) 
() 
1 
where a”. - -a{ are constants in Region R® for 
OG SON. Gaal. Grea iige o 3/2 (13.20) 
Region R® is in the dimensional Euclidean space R’. For example, the first order 
threshold autoregressive model TAR(1) is 
- f DO GN Aan < a 
bce 9 
ae (13.21) 
aX, +e, #X,.2d 
where the coefficient a differs by the size of X,_). Priestley’ indicates that, if we consider 
a bilinear system in which the physical input U;, output X,, and noise €; are related by 
X41 = aX,+cU, Xp+ Epa 9 (13.22) 
and if U, is determined by a feedback mechanism of the form 
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