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Concept
Frequency of exceedance
Summary
The frequency of exceedance, sometimes called the annual rate of exceedance, is the frequency with which a random process exceeds some critical value. Typically, the critical value is far from the mean. It is usually defined in terms of the number of peaks of the random process that are outside the boundary. It has applications related to predicting extreme events, such as major earthquakes and floods.
The frequency of exceedance is the number of times a stochastic process exceeds some critical value, usually a critical value far from the process' mean, per unit time. Counting exceedance of the critical value can be accomplished either by counting peaks of the process that exceed the critical value or by counting upcrossings of the critical value, where an upcrossing is an event where the instantaneous value of the process crosses the critical value with positive slope. This article assumes the two methods of counting exceedance are equivalent and that the process has one upcrossing and one peak per exceedance. However, processes, especially continuous processes with high frequency components to their power spectral densities, may have multiple upcrossings or multiple peaks in rapid succession before the process reverts to its mean.
Consider a scalar, zero-mean Gaussian process y(t) with variance σy2 and power spectral density Φy(f), where f is a frequency. Over time, this Gaussian process has peaks that exceed some critical value ymax > 0. Counting the number of upcrossings of ymax, the frequency of exceedance of ymax is given by
N0 is the frequency of upcrossings of 0 and is related to the power spectral density as
For a Gaussian process, the approximation that the number of peaks above the critical value and the number of upcrossings of the critical value are the same is good for ymax/σy > 2 and for narrow band noise.
For power spectral densities that decay less steeply than f−3 as f→∞, the integral in the numerator of N0 does not converge. Hoblit gives methods for approximating N0 in such cases with applications aimed at continuous gusts.
Official source
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