5.2. DISCRETE PARAMETRIC MODELS 
There are many statistical models for expressing time series. Among them the auto- 
regressive (AR) model, the moving average (MA) model, and the mixed autoregressive 
moving average (ARMA) model are the most representative linear models that we en- 
counter in the analysis of irregular records of observations. In Part I, only the linear 
models are introduced. Some types of nonlinear models will be referred to later in Part 
Ill. The order of the AR, MA, and ARMA models indicates the degree of simplicity or 
complexity of the models. 
Here in Part II, the discrete time process sampled from the continuous time process, 
with an interval Ar is expressed by {X;} and its realization by X;. Except as otherwise in- 
dicated, Ar is taken as 1. If Ar = 1, each function is easily transformed to proper form, as 
was already mentioned in Section 2.4.3. The explanation of the character of the elemen- 
tary models is based largely on the work of Priestly”? and Pandit and Wu.*° 
5.2.1 Pure Random Process 
{X;,} is called a pure random process if it is the sequence of uncorrelated random 
variables that are stationary up to order 2, and is written 
X,= €;. (5.1) 
The mean is 
E|X) =n, (5.2) 
the variance is 
E\&,-n)"| = 07 = 02 (5.3) 
and the covariance function is 
0 rz0 
= 2 (5.4) 
R(r) cov.{X,, Ge 32 cnc se 
R(r) is a function of r only and is normalized by a? as 
o(r) dk 0 r= 0 (5.5) 
ip 
Its spectrum is then 
fee) 
1 z 
s(w) Se > Rien =< = const. (for-z7 =w2z7). (5.6) 
r=0 
In this case 
1. When this process is also Gaussian, then X; is not only uncorrelated, but also 
X;, X;1,° + *X;-,, are independent of each other. 
2. The spectrum s(w) is flat for the interval of frequency w for —z toz, and is 
referred to as white noise. 
3. The process is also referred to as a memoryless process. 
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