Biomedical Engineering Reference
In-Depth Information
well perform the same kind of decomposition in any other basis. The trigono-
metric basis is particularly useful for stationary time series because the basis
functions are themselves evenly spread over all times (56, ch. 2). Other bases,
localized in time, are more convenient for nonstationary situations. The most
well-known of these alternate bases, currently, are wavelets (57), but there is,
literally, no counting the other possibilities.
3.3. The Traditional Statistical Approach
The traditional statistical approach to time series is to represent them
through linear models of the kind familiar from applied statistics.
The most basic kind of model is that of a
moving average
, which is espe-
cially appropriate if
x
is highly correlated up to some lag, say
q
, after which the
ACF decays rapidly. The moving average model represents
x
as the result of
smoothing
q
+ 1 independent random variables. Specifically, the MA(
q
) model
of a weakly stationary series is
q
=++
y
N
w
R
w
,
[23]
t
t
k
t
k
k
=
1
where N is the mean of
y
, the R
i
are constants and the
w
t
are white noise variables.
q
is called the
order
of the model. Note that there is no direct dependence be-
tween successive values of
y
; they are all functions of the white noise series
w
.
Note also that
y
t
and y
t+q+
1
are completely independent; after
q
time-steps, the
effects of what happened at time
t
disappear.
Another basic model is that of an
autoregressive process
, where the next
value of
y
is a linear combination of the preceding values of
y
. Specifically, an
AR(
p
) model is
p
y
=+
B
G
y
+
w
,
[24]
t
k
t
k
t
k
=
1
=+
. The order of the model, again is
p
.
This is the multiple regression of applied statistics transposed directly on to time
series, and is surprisingly effective. Here, unlike the moving average case, ef-
fects propagate indefinitely—changing
y
t
can affect all subsequent values of
y
.
The remote past only becomes irrelevant if one controls for the last
p
values of
the series. If the noise term
w
t
were absent, an AR(
p
) model would be a
p
th or-
der linear difference equation, the solution to which would be some combination
of exponential growth, exponential decay and harmonic oscillation. With noise,
they become oscillators under stochastic forcing (58).
where G
i
are constants and
BN G
=
p
k
k
1
