The Gumbel distribution is a probability distribution of extreme values.
Such a distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years. It is also useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur.
Properties[change | edit source]
The Gumbel distribution is a continuous probability distribution. Gumbel distributions are a family of distributions of the same general form. These distributions differ in their location and scale parameters: the mean ("average") of the distribution defines its location, and the standard deviation ("variability") defines the scale.
One recognizes the Gumbel probability density function (PDF) and the Gumbel cumulative distribution function (CDF).
PDF[change | edit source]
In the PDF, the probability P of a value V to occur between limits A and B, briefly written as P(A<V<B), is found by the area under the PDF curve between A and B.
Example of probability in the PDF In the figure of the normal probability density function, the values on the horizontal axis should read: μ-3σ, μ-2σ, μ-1σ, μ+1σ, μ+2σ, and μ+3σ respectively.
μ = mean, σ = standard deviation.
The areas under the curve in the intervals, each with a width of one standard deviation, give the probability of occurrence in those intervals.
Example: the probability of a value V to occur in the interval between A=μ+1σ and B=μ+2σ is P(μ+1σ<V<μ+2σ)=13.6% or 0.136
Contrary to the normal distribution, the Gumbel PDF is a-symmetrical and skew to the right.
CDF[change | edit source]
In the CDF, the probability that a value V is less than A is found directly as the CDF value at A:
Example of probability in the CDF In the Gumbel CDF figure, the red curve indicates that the probability of V to be less than 5 is 0.9 (or 90%), whereas for the dark blue line this probability is 0.7 or 70%
Mathematics[change | edit source]
The CDF[change | edit source]
The mathematical expression of the CDF is:
Mode and median[change | edit source]
The mode μ can be found from the median M, being the value of A where CDF(A)=0.5, and β:
where ln is the natural logarithm.
Mean[change | edit source]
The mean, E(x), given by:
where = Euler constant 0.5772.
Estimation[change | edit source]
In a data series, the parameters mode (μ) and β can be estimated from the average, median and standard deviation. The calculation of the last three quantities is explained in the respective Wiki pages. Then, with the help of formulas given in the previous section, the factors μ and β can be calculated. In this way, the CDF of the Gumbel distribution belonging to the data can be determined and the probability of interesting data values can be found.
Application[change | edit source]
The blue picture illustrates an example of fitting the Gumbel distribution to ranked maximum one-day October rainfalls showing also the 90% confidence belt based on the binomial distribution.
References[change | edit source]
- Gumbel, E.J. 1954. "Statistical theory of extreme values and some practical applications". Applied Mathematics Series, 33. U.S. Department of Commerce, National Bureau of Standards.
- CumFreq software for distribution fitting
- Ritzema (ed.), H.P. (1994). Frequency and Regression Analysis. Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. pp. 175–224. ISBN 90-70754-33-9. http://www.waterlog.info/pdf/freqtxt.pdf.
- Burke, E.J.; Perry R.H.J.; Brown, S.J. (2010) "An extreme value analysis of UK drought and projections of change in the future", Journal of Hydrology