CHAPTER 13 
EXTENSION OF MODEL FITTING TECHNIQUES TO 
NONLINEAR PROCESS 
13.1 INTRODUCTION 
Suppose we have a discrete random process X; and a purely random process €,, and 
X; is expressed by 
X, = Qa(Xi-1, X12," + - Xin) + €2- (13.1) 
When Q, is a linear function, 
X,= a, Xp) + a2 Xp2+- > ++ Qn Xen t€:, (13.2) 
X, is called as an autoregressive process (AR) of nth order, as discussed in Part II, 
Chapter 5. 
If X, is expressed as 
Xr = Om(Er-1,€1-2€1-3,€-m) + Er, (13.3) 
and Q,, is a linear function, 
X, = by€,-1 + bz €-2+- + -+ Dn Epmt Er, (13.4) 
then X; is a moving average model (MA) of mth order, also discussed in detail in Part I, 
Chapter 5. 
When these Q, or Qj functions are not linear functions but, for example, the poly- 
nomials of X;_, Or €;» , then X; is no longer a linear process and is called an expanded 
AR model or an expanded MA model. 
A Voltera type process, as we saw in Section 11.1 in Eq. 11.1, is this expanded 
moving average process. More generally, 
X,= QaXi1 Xr-2, Xin) + Om(Er-1,€+-2€r-m) + € (13.5) 
is an expanded ARMA model when Q,, Q,, are polynomials of variables. 
For expanded AR, MA, and ARMA models, there is no general way to solve the 
process. When the process is expressed as, 
X,+4, Xp1+ 42 Xpoat+- > > +ay Xx 
m v 
=€,+ dy €3+: °° +b; €pj+ Sy Gi; Ei Xj, (13.6) 
i=1j=l 
Eq. 13.6 is called a bilinear model. The scope of nonlinear models that have been analyti- 
cally developed is rather limited. A few efforts have been made along the following 
models: i) a simple bilinear model, ii) a threshold autoregressive model, and iii) an 
exponential autoregressive model. 
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