Probability of exceedance

The frequency of exceedance, sometimes called the annual rate of exceedance, is the number of times per unit time that a random process exceeds some critical value, normally a critical value far from the mean. It is usually defined in terms of the number of peaks of the random process that are outside the boundary or in terms of upcrossings of the boundary itself. It has applications related to predicting extreme events, such as major earthquakes and floods.

The frequency of exceedance is the number of times a 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 $σ y 2$ and power spectral density $Φ y (f)$ , where $f$ is a frequency. Over time, this Gaussian process has peaks that exceed some critical value $y max > 0$ . Counting the number of upcrossings of $y max$ , the frequency of exceedance of $y max$ is given by

$N 0$ 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 $y max /σ y 2 > 2$ and for narrow band noise.

For power spectral densities that decay less steeply than $f -3$ as $f \to\infty$ , the integral in the numerator of $N 0$ does not converge. Hoblit gives methods for approximating $N 0$ in such cases with applications aimed at continuous gusts.

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