Download a complete copy of this risk
analysis resource for free here.
|
Format: VoseRayleigh(b, U)
Originally derived by Lord Rayleigh (or by his less glamorous name J.W. Strutt) in the field of acoustics.
The graph below shows various Rayleigh distributions. The distribution in black is a Rayleigh(1), sometimes referred to as the standard Rayleigh distribution.
The Rayleigh1 distribution is frequently used to model wave heights in oceanography, and in communication theory to describe hourly median and instantaneous peak power of received radio signals. It has been used to model the frequency of different wind speeds over a year at wind turbine sites.
The distance from one individual to its nearest neighbour when the spatial pattern is generated by a Poisson distribution follows a Rayleigh distribution. This example shows how that turns out to be very useful.
Consider the location of an object in two dimensions {x,y} relative to some point at location {0,0}. Imagine that x = Normal(0,s) and y = x = Normal(0,s), where the two distributions are independent. Then the distance of the object from point {0,0} is given by a Rayleigh(s) distribution. In other words, SQRT( Normal(0,s)^2 + Normal(0,s)^2 ) = Rayleigh(s)
The Rayleigh distribution is a special case of the Weibull distribution since Rayleigh(b) = Weibull(2, b√2), and as such is a suitable distribution for modeling the lifetime of a device that has a linearly increasing instantaneous failure rate: z(x) = x/b2.
1John William Strutt, 3rd Baron Rayleigh (1842 - 1919) was a British physicist who co-discovered the element argon, which earned him the Nobel Prize for Physics. He also discovered 'Rayleigh scattering' and predicted the existence of the surface waves now known as Rayleigh waves. Smart guy.
Other identities: [Rayleigh (1)]2 = ChiSq (2) and [Rayleigh(β)]2 = Expon(1/(2β2)).
VoseRayleigh generates values from this distribution or calculates a percentile
VoseRayleighObject constructs a distribution object for this distribution
VoseRayleighProb returns the probability density or cumulative distribution function for this distribution
VoseRayleighProb10 returns the log10 of the probability density or cumulative distribution function
VoseRayleighFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution
VoseRayleighFitObject constructs a distribution object of this distribution fitted to data
VoseRayleighFitP returns the parameters of this distribution fitted to data
