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1.2 Time Series Statistical Models 9
It is conventional to display a sample time series graphically by plotting the values
of the random variables on the vertical axis, or ordinate, with the time scale as the
abscissa. It is usually convenient to connect the values at adjacent time periods to
reconstruct visually some original hypothetical continuous time series that might
have produced these values as a discrete sample. Many of the series discussed in
the previous section, for example, could have been observed at any continuous point
in time and are conceptually more properly treated as continuous time series.The
approximation of these series by discrete time parameter series sampled at equally
spaced points in time is simply an acknowledgment that sampled data will, for the
most part, be discrete because of restrictions inherent in the method of collection.
Furthermore, the analysis techniques are then feasible using computers, which are
limited to digital computations. Theoretical developments also rest on the idea that a
continuous parameter time series should be specified in terms of finite-dimensional
distribution functions defined over a finite number of points in time. This is not to
say that the selection of the sampling interval or rate is not an extremely important
consideration. The appearance of data can be changed completely by adopting an
insufficient sampling rate. We have all seen wheels in movies appear to be turning
backwards because of the insufficient number of frames sampled by the camera. This
phenomenon leads to a distortion called aliasing (see Sect. 4.1).
The fundamental visual characteristic distinguishing the different series shown
in Example 1.1–Example 1.7 is their differing degrees of smoothness. One possible
explanation for this smoothness is that it is being induced by the supposition that
adjacent points in time are correlated, so the value of the series at time t,say,
x
t
, depends in some way on the past values x
t−1
, x
t−2
,.... This model expresses a
fundamental way in which we might think about generating realistic-looking time
series. To begin to develop an approach to using collections of random variables to
model time series, consider Example 1.8.
Example 1.8 White Noise (3 Flavors)
A simple kind of generated series might be a collection of uncorrelated random
variables, w
t
, with mean 0 and finite variance σ
2
w
. The time series generated from
uncorrelated variables is used as a model for noise in engineering applications,
where it is called white noise; we shall denote this process as w
t
∼ wn(0,σ
2
w
).The
designation white originates from the analogy with white light and indicates that
all possible periodic oscillations are present with equal strength.
We will sometimes require the noise to be independent and identically dis-
tributed (iid) random variables with mean 0 and variance σ
2
w
. We distinguish this
by writing w
t
∼ iid(0,σ
2
w
) or by saying white independent noise or iid noise.
A particularly useful white noise series is Gaussian white noise, wherein the w
t
are independent normal random variables, with mean 0 and variance σ
2
w
;ormore
succinctly, w
t
∼ iid N(0,σ
2
w
). Figure 1.8 shows in the upper panel a collection of
500 such random variables, with σ
2
w
= 1, plotted in the order in which they were
drawn. The resulting series bears a slight resemblance to the explosion in Fig. 1.7
but is not smooth enough to serve as a plausible model for any of the other experi-