CHAPTER 6 
MODEL FITTING TO THE RESPONSE OF THE LINEAR DYNAMIC 
SYSTEM TO IRREGULAR INPUT 
6.1 INTRODUCTION 
In Chapter 5, the characteristics, stability, and invertibility of a single linear stochas- 
tic process X; were discussed and it was concluded that AR, MA, or ARMA models of 
appropriate order can usually be fitted to most of the linear stationary time series to 
represent their statistical properties. 
The characteristics of the model fitting techniques is, the author believes, that all of 
these AR, MA, and ARMA models relate the process X; to the pure random process 
€,, although the relations that connect €; to X; are different for the different models. 
For 
AR: X,+@)X;-1 + a2Xp2 +... tQnXpn = €1; 
MA: X,=€; + by€+-1+ b2€-2 . . . tbm€rm; and (6.1) 
ARMA : X, + A1Xj_1 + a2Xp-2 6 0-0 + AnXp-n = Ey+ by€;-1 
+ b2€ 12 GP 60 6 FF bmE tm . 
In all of these models, €, was supposed to be the input to an imaginary system, and 
X, was treated as the output of the same system. All the characteristics of X, are expressed 
in relation to those of the pure random (white) process. 
6.2 RESPONSE SYSTEM WITH FEEDBACK 
6.2.1 One Input / One Output System 
Here we consider the output Y of a real linear system with input X;, usually ex- 
pressed as 
Y; = yy SuXtu + €;. (6.2) 
u=0 
Here g,, is the so—called impulse response function of Y; to X; or the weighting function of 
this linear system to X;. Now suppose we have N observations of the input/output pair 
{X,, Yi}, t=1 to N. 
If €; is a pure random (white) noise, then we can estimate the coefficient g,, to 
minimize 
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