Chain rule and Gumbel distribution: Difference between pages

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[[File:Gumbel-Density.svg|thumb|260px|Gumbel probability distribution function (PDF)]]
The '''chain rule''' is a way of finding the derivative of a function. It is used where the function is in another function. This is called a composite function. 
[[File:Gumbel-Cumulative.svg|thumb|260px|Gumbel cumulative distribution function (CDF)]]


If F(x) equals two functions that we can take a derivative of, such as:
The '''Gumbel distribution''' is a [[probability distribution]] of extreme values.
:<math>F(x)=f(g(x))</math>


then the derivative, F prime, is
In [[probability theory]] and [[statistics]], the Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. <ref>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.</ref>
:<math>F'(x)=f'(g(x))g'(x)</math>


==Steps==
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.  
'''1.''' Find the derivative of the outside function (all of it at once).


'''2.''' Find the derivative of the inside function (the bit between the brackets).
== Properties ==


'''3.''' Multiply the answer from the first step by the answer from the second step. This is basically the last step in solving for the derivative of a function.
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'' [[parameter]]s: the [[Mean (statistics)|mean]] ("[[average]]") of the distribution defines its location, and the [[standard deviation]] ("variability") defines the scale.


;Example;
One recognizes the Gumbel ''probability density function'' (PDF) and the Gumbel ''cumulative distribution function'' (CDF).


:<math>F(x)=(x^2+5)^3</math>
=== PDF ===
:<math>F(x)=3(x^2+5)^2</math>
:<math>F'(x)=3(x^2+5)^2(2x)</math>
:<math>F'(x)=6x(x^2+5)^2</math>


[[File:Standard deviation diagram.svg|thumb|260px|The [[Normal distribution|normal probability density function]] (PDF) is symmetric.]]


In this example, the cubed sign (<sup>3</sup>) is the outside function and <math>x^2+5</math> is the inside function. The derivative of the outside function would be <math>3x^2</math>, where the inside function is plugged in for x. The derivative of the inside function would be 2x, which is multiplied by <math>3(x^2+5)^2</math> to get <math>6x(x^2+5)^2</math>.  
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.


{{stub}}
:{| class="wikitable"
| bgcolor=#faf0e6 | Example of probability in the PDF
|-
| In the figure of the [[Normal distribution|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.<br>
μ = [[Mean (statistics)|mean]], σ = [[standard deviation]].<br>
The areas under the curve in the intervals, each with a width of one standard deviation, give the probability of occurrence in those intervals.<br>
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
|}


[[Category:Mathematics]]
Contrary to the normal distribution, the Gumbel PDF is a-symmetrical and skew to the right.
 
=== CDF ===
 
In the CDF, the probability that a value V is less than A is found directly as the CDF value at A:
 
:<math>P(V \leq A) = CDF(A) </math>.
 
:{| class="wikitable"
| bgcolor=#faf0e6 | 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 ==
 
=== The CDF ===
[[File:Comparison standard deviations.svg|thumb|250px|There are two data series: red and blue. Both have the same ''mean (average)'' : 100, but the blue group has a larger ''standard deviation'' (SD=σ=50) than the red group (SD=σ=10).]]
 
The mathematical expression of the CDF is:
 
:<math>CDF(A) = e^{-e^{-(A-\mu)/\beta}} ,</math>
 
where μ is the ''mode'' (the value where the probability density function reaches its peak), ''e'' is a [[e (mathematical constant)|mathematical constant]], about 2.718, and β is a value related to the ''[[standard deviation]]'' (σ) :
 
:<math> \beta = \sigma \sqrt{6}/ \pi , </math>
 
where π is the Greek symbol for [[Pi (math)|Pi]] whose value is close to 22/7 or 3.142, and the symbol <math>\sqrt{\,\,}</math> stands for the [[square root]].
 
=== Mode and median ===
 
The ''mode'' μ can be found from the ''[[median]]'' M, being the value of A where CDF(A)=0.5,  and β:
 
:<math>\mu = M+\beta \ln\left(\ln 2\right) ,</math>
 
where ''ln'' is the [[Logarithm#Natural logarithms|natural logarithm]].
 
=== Mean ===
 
The ''[[Mean (statistics)|mean]]'', E(x), given by:
 
:<math>\operatorname{E}(x)=\mu+c\beta ,</math>
 
where <math>c</math> = ''Euler constant'' <math>\approx</math> 0.5772.
 
== Estimation ==
 
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.
 
[[File:FitGumbelDistr.tif|thumb|260px|Fitted cumulative Gumbel distribution to maximum one-day
October rainfalls using CumFreq <ref> [http://www.waterlog.info/cumfreq.htm CumFreq software] for distribution fitting</ref> ]]
 
== Application ==
 
In [[hydrology]], the Gumbel distribution is used to analyze such variables as monthly and annual maximum values of daily rainfall and river discharge volumes,<ref>{{cite book|last=Ritzema (ed.)|first=H.P.|title=Frequency and Regression Analysis|year=1994|publisher=Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands|pages=175–224|url=http://www.waterlog.info/pdf/freqtxt.pdf|isbn=90-70754-33-9}}</ref> and also to describe droughts.<ref>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'' </ref>
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 ==
{{reflist}}
 
[[Category:Probability distributions]]

Latest revision as of 11:55, 17 April 2013

File:Gumbel-Density.svg
Gumbel probability distribution function (PDF)
File:Gumbel-Cumulative.svg
Gumbel cumulative distribution function (CDF)

The Gumbel distribution is a probability distribution of extreme values.

In probability theory and statistics, the Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. [1]

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

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

File:Standard deviation diagram.svg
The normal probability density function (PDF) is symmetric.

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

In the CDF, the probability that a value V is less than A is found directly as the CDF value at A:

[math]\displaystyle{ P(V \leq A) = CDF(A) }[/math].
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

The CDF

File:Comparison standard deviations.svg
There are two data series: red and blue. Both have the same mean (average) : 100, but the blue group has a larger standard deviation (SD=σ=50) than the red group (SD=σ=10).

The mathematical expression of the CDF is:

[math]\displaystyle{ CDF(A) = e^{-e^{-(A-\mu)/\beta}} , }[/math]

where μ is the mode (the value where the probability density function reaches its peak), e is a mathematical constant, about 2.718, and β is a value related to the standard deviation (σ) :

[math]\displaystyle{ \beta = \sigma \sqrt{6}/ \pi , }[/math]

where π is the Greek symbol for Pi whose value is close to 22/7 or 3.142, and the symbol [math]\displaystyle{ \sqrt{\,\,} }[/math] stands for the square root.

Mode and median

The mode μ can be found from the median M, being the value of A where CDF(A)=0.5, and β:

[math]\displaystyle{ \mu = M+\beta \ln\left(\ln 2\right) , }[/math]

where ln is the natural logarithm.

Mean

The mean, E(x), given by:

[math]\displaystyle{ \operatorname{E}(x)=\mu+c\beta , }[/math]

where [math]\displaystyle{ c }[/math] = Euler constant [math]\displaystyle{ \approx }[/math] 0.5772.

Estimation

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.

File:FitGumbelDistr.tif
Fitted cumulative Gumbel distribution to maximum one-day October rainfalls using CumFreq [2]

Application

In hydrology, the Gumbel distribution is used to analyze such variables as monthly and annual maximum values of daily rainfall and river discharge volumes,[3] and also to describe droughts.[4]

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

Template:Reflist

  1. ↑ 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.
  2. ↑ CumFreq software for distribution fitting
  3. ↑ Template:Cite book
  4. ↑ 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
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