Biomedical Engineering Reference
In-Depth Information
Often instead of kurtosis parameter e -excessofcurtosis: e
3 is used. The sub-
traction of 3 at the end of this formula is often explained as a correction to make the
kurtosis of the normal distribution equal to zero. For the normally distributed vari-
ables (variables whose distribution is described by Gaussian), central odd moments
are equal to zero and central even moments take values:
=
β 2
m 2 n
2
m 2 n + 1
=
0
m 2 n
=(
2 n
1
)
(1.10)
Calculation of skewness and kurtosis can be used to assess if the distribution
is roughly normal. These measures can be computed using functions from the
MATLAB statistics toolbox: skewness and kurtosis .
The relation of two processesξ
(
t
)= {
x 1 (
t
) ,...,
x N (
t
) }
andη
(
t
)= {
y 1 (
t
) ,...,
y N (
t
) }
can be characterized by joint moments. Joint moment of the first order R xy (
t
)
and joined central moment C xy (
t
)
of process ξ
(
t
)
are called, respectively, cross-
correlation and cross-covariance:
R xy
(
t 1
,
τ
)=
E
[
ξ
(
t 1
)
η
(
t 1
+
τ
)]
(1.11)
C xy
(
t 1
,
τ
)=
E
[(
ξ
(
t 1
)
μ x
(
t 1
))(
η
(
t 1
+
τ
)
μ y
(
t 1
))]
(1.12)
where τ is the time shift between signals x and y .
A special case of the joint moments occurs when they are applied to the same
process, that is ξ
(
t
)=
η
(
t
)
. Then the first order joint moment R x
(
t
)
is called autocor-
relation and joined central moment C x
(
t
)
of process ξ
(
t
)
is called autocovariance.
Now we can define:
Stationarity: For the stochastic process ξ
the infinite number of moments and
joint moments can be calculated. If all moments and joint moments do not de-
pend on time, the process is called stationary in the strict sense. In a case when
mean value μ x and autocorrelation R x
(
t
)
(
)
do not depend on time the process
is called stationary in the broader sense, or weakly stationary. Usually weak
stationarity implies stationarity in the strict sense, and for testing stationarity
usually only mean value and autocorrelation are calculated.
τ
Ergodicity: The process is called ergodic when its mean value calculated in time
(for the infinite time) is equal to the mean value calculated by ensemble av-
eraging (according to equation 1.2). Ergodicity means that one realization is
representative for the whole process, namely that it contains the whole infor-
mation about the process. Stationarity of a process implies its ergodicity. For
ergodic processes we can describe the properties of the process by averaging
one realization over time, instead of ensemble averaging.
Under the assumption of ergodicity moment of order n is expressed by:
Z T
x n
m n
=
(
)
(
)
lim
T
t
p
x
dt
(1.13)
0
 
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