Biomedical Engineering Reference
In-Depth Information
Notation
. There is no completely uniform notation for time series. Since it
will be convenient to refer to sequences of consecutive values. I will write all
the measurements starting at
s
and ending at
t
as
y
s
t
. Further, I will abbreviate the
set of all measurements up to time
t
,
y
d
, as
y
t
-
, and the future starting from
t
,
+
1
y
t
+1
, as
y
t
+
.
3.2. General Properties of Time Series
One of the most commonly assumed properties of a time series is
stationar-
ity
, which comes in two forms:
strong
or
strict
stationarity, and
weak
,
wide-
sense
or
second-order
stationarity. Strong stationarity is the property that the
probability distribution of sequences of observations does not change over time.
That is,
th
+
t
+ +
+
U
U
h
Pr(
Y
)
=
Pr(
Y
)
[12]
t
t
for all lengths of time
h
and all shifts forwards or backwards in time U. When a
series is described as "stationary" without qualification, it depends on context
whether strong or weak stationarity is meant.
Weak stationarity, on the other hand, is the property that the first and sec-
ond moments of the distribution do not change over time.
E
[
Y
t
] =
E
[
Y
t+
U
],
[13]
E
[
Y
t
Y
t+h
] =
E
[
Y
t
+
U
Y
t
+
U
+
h
].
[14]
If
Y
is a Gaussian process, then the two senses of stationarity are equivalent.
Note that both sorts of stationarity are statements about the true distribution, and
so cannot be simply read off from measurements.
Strong stationarity implies a property called
ergodicity
, which is much
more generally applicable. Roughly speaking, a series is ergodic if any suffi-
ciently long sample is representative of the entire process. More exactly, con-
sider the
time-average
of a well-behaved function
f
of
Y
,
tt
=
1
2
t
w
f
fY
( .
[15]
2
1
t
t
t
t
tt
=
2
1
This is generally a random quantity, since it depends on where the trajectory
started at
t
1
, and any random motion which may have taken place between then
and
t
2
. Its distribution generally depends on the precise values of
t
1
and
t
2
. The
series
Y
is ergodic if almost all time-averages converge eventually, i.e., if
