Английская Википедия:Frequency of exceedance
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.
Definition
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.Шаблон:Sfn Counting exceedance of the critical value can be accomplished either by counting peaks of the process that exceed the critical valueШаблон:Sfn 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.Шаблон:SfnШаблон:Sfn 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.Шаблон:Sfn
Frequency of exceedance for a Gaussian process
Consider a scalar, zero-mean Gaussian process Шаблон:Math with variance Шаблон:Math and power spectral density Шаблон:Math, where Шаблон:Mvar is a frequency. Over time, this Gaussian process has peaks that exceed some critical value Шаблон:Math. Counting the number of upcrossings of Шаблон:Math, the frequency of exceedance of Шаблон:Math is given byШаблон:SfnШаблон:Sfn
- <math> N(y_{\max}) = N_0 e^{-\tfrac{1}{2}\left(\tfrac{y_{\max}}{\sigma_y}\right)^2}.</math>
Шаблон:Math is the frequency of upcrossings of 0 and is related to the power spectral density as
- <math> N_0 = \sqrt{\frac{\int_0^\infty{f^2 \Phi_y(f) \, df}}{\int_0^\infty{\Phi_y(f) \, df}}}.</math>
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 Шаблон:Math and for narrow band noise.Шаблон:Sfn
For power spectral densities that decay less steeply than Шаблон:Math as Шаблон:Math, the integral in the numerator of Шаблон:Math does not converge. Hoblit gives methods for approximating Шаблон:Math in such cases with applications aimed at continuous gusts.Шаблон:Sfn
Time and probability of exceedance
As the random process evolves over time, the number of peaks that exceeded the critical value Шаблон:Math grows and is itself a counting process. For many types of distributions of the underlying random process, including Gaussian processes, the number of peaks above the critical value Шаблон:Math converges to a Poisson process as the critical value becomes arbitrarily large. The interarrival times of this Poisson process are exponentially distributed with rate of decay equal to the frequency of exceedance Шаблон:Math.Шаблон:Sfn Thus, the mean time between peaks, including the residence time or mean time before the very first peak, is the inverse of the frequency of exceedance Шаблон:Math.
If the number of peaks exceeding Шаблон:Math grows as a Poisson process, then the probability that at time Шаблон:Mvar there has not yet been any peak exceeding Шаблон:Math is Шаблон:Math.Шаблон:Sfn Its complement,
- <math>p_{ex}(t) = 1 - e^{-N(y_{\max})t},</math>
is the probability of exceedance, the probability that Шаблон:Math has been exceeded at least once by time Шаблон:Mvar.Шаблон:SfnШаблон:Sfn This probability can be useful to estimate whether an extreme event will occur during a specified time period, such as the lifespan of a structure or the duration of an operation.
If Шаблон:Math is small, for example for the frequency of a rare event occurring in a short time period, then
- <math> p_{ex}(t) \approx N(y_{\max})t.</math>
Under this assumption, the frequency of exceedance is equal to the probability of exceedance per unit time, Шаблон:Math, and the probability of exceedance can be computed by simply multiplying the frequency of exceedance by the specified length of time.
Applications
- Probability of major earthquakes[1]
- Weather forecasting[2]
- Hydrology and loads on hydraulic structures[3]
- Gust loads on aircraftШаблон:Sfn
See also
Notes
References
