Thefrequency of exceedance, sometimes called theannual 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 majorearthquakes andfloods.

Thefrequency of exceedance is the number of times astochastic process exceeds some critical value, usually a critical value far from the process' mean, per unit time.^[1] Counting exceedance of the critical value can be accomplished either by counting peaks of the process that exceed the critical value^[1] or by counting upcrossings of the critical value, where anupcrossing is an event where the instantaneous value of the process crosses the critical value with positive slope.^[1][2] 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.^[3]

Consider a scalar, zero-meanGaussian processy(t) withvarianceσ_y² andpower spectral densityΦ_y(f), wheref is a frequency. Over time, this Gaussian process has peaks that exceed some critical valuey_max > 0. Counting the number of upcrossings ofy_max, thefrequency of exceedance ofy_max is given by^[1][2]

${\displaystyle N(y_{max})=N_0{e}^{-\frac{1}{2}{\left(\frac{y_{max}}{{\sigma }_y}\right)}^2}.}$

N₀ is the frequency of upcrossings of 0 and is related to the power spectral density as

${\displaystyle N_0=\sqrt{\frac{{\int }_0^{\mathrm{\infty }}{f}^2{\mathrm{\Phi }}_y(f)df}{{\int }_0^{\mathrm{\infty }}{\mathrm{\Phi }}_y(f)df}}.}$

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 fory_max/σ_y > 2 and fornarrow band noise.^[1]

For power spectral densities that decay less steeply thanf⁻³ asf→∞, the integral in the numerator ofN₀ does not converge. Hoblit gives methods for approximatingN₀ in such cases with applications aimed atcontinuous gusts.^[4]

As the random process evolves over time, the number of peaks that exceeded the critical valuey_max grows and is itself acounting process. For many types of distributions of the underlying random process, including Gaussian processes, the number of peaks above the critical valuey_max converges to aPoisson process as the critical value becomes arbitrarily large. The interarrival times of this Poisson process areexponentially distributed with rate of decay equal to the frequency of exceedanceN(y_max).^[5] Thus, the mean time between peaks, including theresidence time or mean time before the very first peak, is the inverse of the frequency of exceedanceN⁻¹(y_max).

If the number of peaks exceedingy_max grows as a Poisson process, then the probability that at timet there has not yet been any peak exceedingy_max ise^−N(y_max)t.^[6] Its complement,

${\displaystyle p_{ex}(t)=1-e^{-N(y_{max})t},}$

is theprobability of exceedance, the probability thaty_max has been exceeded at least once by timet.^[7][8] 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.

IfN(y_max)t is small, for example for the frequency of a rare event occurring in a short time period, then

${\displaystyle p_{ex}(t)\approx N(y_{max})t.}$

Under this assumption, the frequency of exceedance is equal to theprobability of exceedance per unit time,p_ex/t, and the probability of exceedance can be computed by simply multiplying the frequency of exceedance by the specified length of time.

• Probability of major earthquakes^[9]
• Weather forecasting^[10]
• Hydrology and loads on hydraulic structures^[11]
• Gust loads on aircraft^[12]

• 100-year flood
• Cumulative frequency analysis
• Extreme value theory
• Rice's formula

• Hoblit, Frederic M. (1988).Gust Loads on Aircraft: Concepts and Applications. Washington, DC: American institute of Aeronautics and Astronautics, Inc.ISBN 0930403452.
• Feller, William (1968).An Introduction to Probability Theory and Its Applications. Vol. 1 (3rd ed.). New York: John Wiley and Sons.ISBN 9780471257080.
• Leadbetter, M. R.; Lindgren, Georg; Rootzén, Holger (1983).Extremes and Related Properties of Random Sequences and Processes. New York: Springer–Verlag.ISBN 9781461254515.
• Rice, S. O. (1945). "Mathematical Analysis of Random Noise: Part III Statistical Properties of Random Noise Currents".Bell System Technical Journal.24 (1):46–156.doi:10.1002/(ISSN)1538-7305c.
• Richardson, Johnhenri R.;Atkins, Ella M.; Kabamba, Pierre T.; Girard, Anouck R. (2014). "Safety Margins for Flight Through Stochastic Gusts".Journal of Guidance, Control, and Dynamics.37 (6). AIAA:2026–2030.doi:10.2514/1.G000299.hdl:2027.42/140648.

• Extreme value data
• Reliability analysis
• Stochastic processes
• Survival analysis