Civil Engineering Reference
In-Depth Information
Fig. 2.3
Ensemble statistics of simultaneous events
The statistical properties of the data set of extracted mean values will then represent
an example of long term ensemble statistics. Typically, the probability density
distribution of a data set of mean values may attain a shape that may be fitted to a
Weibull or a Rayleigh distribution as illustrated in Fig. 2.4.b.
Apart from fitting the data from a random variable to a suitable parent probability
distribution and estimating its mean value and variance (see chapter 2.1 above), it is the
properties of correlation and covariance that are of particular interest. These are both
providing information about possible relationships in the time domain or ensemble data
that have been extracted from the process. Correlation estimates are taken on the full
value of the process variable, i.e. on
()
()
, while covariance is estimated from
X t
=+
xxt
()
zero mean variables
i
xt
.
Given two realisations
()
()
()
()
=+
, either two of the
same process at different time or location, or of two entirely different processes. Then
the correlation and covariance between these two process variables are defined by
X
t
xxt
and
X
t
xxt
=+
1
1
1
2
2
2
T
1
()
()
()
()
R
EX t X t
lim
∫
X t X tdt
=
⎡
⋅
⎤
=
⋅
(2.10)
⎣
⎦
xx
1
2
1
2
12
T
T
→∞
0
