Extreme Value Principles |
 
|
Extreme Value Principles |
|
An estimate of the extreme maximum/minimum value is determined from the probability distribution of the largest/smallest value in a timetrace that contains N individual maxima/minima. The theory recognises that several simulations of a stochastic timetrace (with different seeds) will produce a sequence of largest/smallest values. The extreme value is derived from the probability distribution of these largest/smallest values.
Individual maxima with a Rayleigh or Weibull probability distribution of produce an extreme value that has a Gumbel distribution. Note, however, the extreme value can be determined directly from the parameters of the Rayleigh or Weibull distribution. It is not necessary to explicitly compute parameters of the Gumbel distribution to determine the extreme values. The following describes the extreme value prediction of the maximum value. The same approach is used for the extreme value prediction of the minimum value, although in this case, the timetrace is negated beforehand so the individual minima are converted to maxima. This conversion is only explicitly required for the Weibull option. The Rayleigh distribution assumes complete symmetry between the maximum and minimum values.
The exceedance distribution of the extreme value is determined from the relationship:
where xextreme denotes the extreme value, F the cumulative distribution of the individual maxima in the timetrace and N the number of maxima (e.g. 1000). The extreme value is determined from the above equation in a two-stage process. Let p denote a specified exceedance probability (e.g. 0.01); then the relationship for F, p and N simplifies to:
(2)
The approximation on the right holds for small p and large N. Solving for x in Equation (2) depends on the cumulative distribution F of the individual maxima in a timetrace.
•*STATISTICS is used to request the creation of statistics plots.
•*EXTREMA is used to specify the procedure and input parameters to be used in the computation of extreme values.
