Biomedical Engineering Reference
In-Depth Information
CHAPTER
8
Nonparametric Statistic
and the Runs Test for
Stationarity
8.1 INTRODUCTION
8.1.1 Conditions for a Random Process to Be Considered as Stationary
Recall that Chapter 7 contained a discussion on how the properties of a random process
can be determined to be stationary by computing the ensemble averages at specific
instants of time. A stationary random process was considered to be weakly stationary or
stationary in the wide sense, when the following conditions were met:
(a)
the mean,
μ
x
(
t
1
), termed the first moment of the random processes at all times
t
i
and
(b)
the autocorrelation function,
R
x
(
t
1
,
t
1
), or joint moment between the values
of the random process at two different times does not vary as time
t
varies.
+
τ
N
k
=1
x
k
(
t
1) and the autocorrelation
1
That is, both the ensemble mean
μ
x
(
t
1)
=
Lim
N
→∞
N
k
=1
x
k
(
t
1
)
x
(
t
1
1
function
R
x
(
t
1
,
+
τ
)
=
R
x
(
λ
)
=
Lim
N
=
τ
) are time invariant.
t
1
→∞
If all the higher moments and all the joint moments could be collected, one would
have a complete family of probability distribution functions describing the processes. For
the case where all possible moments and joint moments are time invariant, the random
process is strongly stationary or stationary in the strict sense.
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