Biomedical Engineering Reference
In-Depth Information
tT
t
+
lim
f
=
f
[16]
T
ld
for some constant
f
independent of the starting time
t
, the starting point
Y
t
, or
the trajectory
Y
t
.
Ergodic theorems
specify conditions under which ergodicity
holds; surprisingly, even completely deterministic dynamical systems can be
ergodic.
Ergodicity is such an important property because it means that statistical
methods are very directly applicable. Simply by waiting long enough one can
obtain an estimate of any desired property that will be closely representative of
the future of the process. Statistical inference
is
possible for non-ergodic proc-
esses, but it is considerably more difficult, and often requires multiple time se-
ries (51,52).
One of the most basic means of studying a time series is to compute the
autocorrelation function
(ACF), which measures the linear dependence be-
tween the values of the series at different points in time. This starts with
auto-
covariance function
:
C
(
s
,
t
)
E
[(
y
s
-
E
[
y
s
]) (
y
t
-
E
[
y
t
])].
[17]
(Statistical physicists, unlike everyone else, call
this
the "correlation function.")
The autocorrelation itself is the autocovariance, normalized by the variability of
the series:
Cst
(,)
S
w
(,)
st
,
(, ) (,)
[18]
CssCtt
S is 1 when
y
s
is a linear function of
y
t
. Note that the definition is symmetric, so
S(
s
,
t
) = S(
t
,
s
). For stationary or weakly stationary processes, one can show that S
depends only on the difference U between
t
and
s
. In this case one just writes
S
, with one argument. S(0) = 1, always. The time
t
c
such that S(
t
c
) = 1/
e
is
called the
(auto)correlation time
of the series.
The correlation function is a
time-domain
property, since it is basically
about the series considered as a sequence of values at distinct times. There are
also
frequency-domain
properties, which depend on reexpressing the series as a
sum of sines and cosines with definite frequencies. A function of time
y
has a
Fourier transform that is a function of frequency,
y
:
(
U
)
y
,
=
y
[19]
2
QO
t
T
i
=
y
e
y
,
[20]
T
O
t
t
=
1
