Typically the correlation function is an exponentially decaying function in time.
Having defined the mean value function and the cross-correlation function for the stochastic
process ( )X t the probability that the process remains within a certain safe domain D in the
time interval 0,t may be evaluated by:
Ptf()1-(()0 (0) )((0) ) PNt X DPX D (2.63)where N(t) is the number of out-crossings of the random process out of the domain D in the
time interval 0,t.
Statistical Assessment of Extreme Values
In risk and reliability assessments extreme values (small and large) of random processes in a
specified reference period are often of special interest. This is e.g. the case when considering
the maximum sea water level, maximum wave heights, minimum ground water reservoir level,
maximum wind pressures, strength of weakest link systems, maximum snow loads, etc.
For continuous time-varying loads, which can be described by a scalar, i.e. the water level or
the wind pressure one can define a number of related probability distributions. Often the
simplest, namely the “arbitrary point in time” distribution is considered.
If ( )xt^4 is a realisation of a single time-varying load at time t^4 then is the arbitrary
point in time cumulative distribution function of
FxX()
X()t defined by:
FPXtX(() 0)^4 (2.64)In Figure 2.15 a time series of daily measurements of the maximum water level are given
together with the histograms showing the sampling frequency distribution of the 5 days
maximal water level, i.e. the arbitrary point in time frequency distribution and the frequency
distribution of the 10 days maximal water levels.