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 An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...

and

flood A flood is an overflow of water ( or rarely other fluids) that submerges land that is usually dry. In the sense of "flowing water", the word may also be applied to the inflow of the tide. Floods are an area of study of the discipline hydrol ...

Definition

The frequency of exceedance is the number of times a

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

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.

Frequency of exceedance for a Gaussian process

Consider a scalar, zero-mean

Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. e ...

with

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...

and

power spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, o ...

, where is a frequency. Over time, this Gaussian process has peaks that exceed some critical value . Counting the number of upcrossings of , the frequency of exceedance of is given by :

N(y_) = N_0 e^.

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

N_0 = \sqrt.

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 and for narrow band noise. For power spectral densities that decay less steeply than as , the integral in the numerator of does not converge. Hoblit gives methods for approximating in such cases with applications aimed at

continuous gusts Continuous gusts or stochastic gusts are winds that vary randomly in space and time. Models of continuous gusts are used to represent atmospheric turbulence, especially clear air turbulence and turbulent winds in storms. The Federal Aviation Admini ...

Time and probability of exceedance

As the random process evolves over time, the number of peaks that exceeded the critical value grows and is itself a

counting process A counting process is a stochastic process with values that are non-negative, integer, and non-decreasing: # ''N''(''t'') ≥ 0. # ''N''(''t'') is an integer. # If ''s'' ≤ ''t'' then ''N''(''s'') ≤ ''N''(''t''). If ''s'' < ''t'', then ''N''(' ...

. For many types of distributions of the underlying random process, including Gaussian processes, the number of peaks above the critical value converges to a

Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...

as the critical value becomes arbitrarily large. The interarrival times of this Poisson process are

exponentially distributed In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...

with rate of decay equal to the frequency of exceedance . Thus, the mean time between peaks, including the

residence time The residence time of a fluid parcel is the total time that the parcel has spent inside a control volume (e.g.: a chemical reactor, a lake, a human body). The residence time of a set of parcels is quantified in terms of the frequency distribution ...

or mean time before the very first peak, is the inverse of the frequency of exceedance . If the number of peaks exceeding grows as a Poisson process, then the probability that at time there has not yet been any peak exceeding is . Its complement, :

p_(t) = 1 - e^,

is the probability of exceedance, the probability that has been exceeded at least once by time . 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 is small, for example for the frequency of a rare event occurring in a short time period, then :

p_(t) \approx N(y_)t.

Under this assumption, the frequency of exceedance is equal to the probability of exceedance per unit time, , 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 * Weather forecasting * Hydrology and loads on hydraulic structures * Gust loads on aircraft

Notes

References