Wakeby distribution

{{expert needed|statistics|reason=New article adapted from NIST document, needs checking and expansion by someone familiar with the topic|date=August 2015}}

{{Infobox probability distribution

|name=Wakeby distribution

|parameters= $\alpha, \beta, \gamma, \delta, \xi$

|quantile= $\xi + \frac{\alpha}{\beta}(1 - (1-p)^{\beta}) - \frac{\gamma}{\delta}(1 - (1-p)^{-\delta})$

|support= $\xi$ to $\infty$ , if $\delta \ge 0, \gamma > 0$

$\xi$ to $\xi + (\alpha/ \beta) - (\gamma/ \delta)$ , otherwise

|type=density

}}

The Wakeby distribution{{cite web|url= http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/wakpdf.htm|title=Dataplot reference manual: WAKPDF |publisher=NIST|accessdate=20 August 2015}} is a five-parameter probability distribution defined by its quantile function,

: $W(p) =\xi + \frac{\alpha}{\beta}(1 - (1-p)^{\beta}) - \frac{\gamma}{\delta}(1 - (1-p)^{-\delta})$ ,

and by its quantile density function,

: $W'(p) = w(p) = \alpha (1-p)^{\beta - 1} + \gamma (1-p)^{-\delta - 1}$ ,

where $0 \le p \le 1$ , ξ is a location parameter, α and γ are scale parameters and

β and δ are shape parameters.

This distribution was first proposed by Harold A. Thomas Jr., who named it after Wakeby Pond in Cape Cod.{{Cite book|last1=Rodda|first1=John C.|url=https://books.google.com/books?id=DzV0CgAAQBAJ&q=%22wakeby+pond%22|title=Progress in Modern Hydrology: Past, Present and Future|last2=Robinson|first2=Mark|date=2015-08-26|publisher=John Wiley & Sons|isbn=978-1-119-07429-8|pages=75|language=en}}{{Cite journal|last1=Katchanov|first1=Yurij L.|last2=Markova|first2=Yulia V.|date=2015-02-26|title=On a heuristic point of view concerning the citation distribution: introducing the Wakeby distribution|journal=SpringerPlus|language=en|volume=4|issue=1|pages=94|doi=10.1186/s40064-015-0821-1|issn=2193-1801|pmc=4352413|pmid=25763305 |doi-access=free }}

Applications

The Wakeby distribution has been used for modeling distributions of

flood flows,{{cite web|url=http://dspace.mit.edu/bitstream/handle/1721.1/31278/mit-el-77-033wp-04146753.pdf?sequence=1|title=Birth of a Parent: The Wakeby Distribution for Modeling Flood Flows; Working Paper No. MIT-EL77-033WP|author=John C. Houghton|date=October 14, 1977|publisher=MIT}}{{Cite journal|last=GRIFFITHS|first=GEORGE A.|date=1989-06-01|title=A theoretically based Wakeby distribution for annual flood series|journal=Hydrological Sciences Journal|volume=34|issue=3|pages=231–248|doi=10.1080/02626668909491332|bibcode=1989HydSJ..34..231G |issn=0262-6667|citeseerx=10.1.1.399.6501}}
citation counts,{{Cite journal|last1=Katchanov|first1=Yurij L.|last2=Markova|first2=Yulia V.|date=2015-02-26|title=On a heuristic point of view concerning the citation distribution: introducing the Wakeby distribution|journal=SpringerPlus|language=En|volume=4|issue=1|pages=94|doi=10.1186/s40064-015-0821-1|pmid=25763305|issn=2193-1801|pmc=4352413 |doi-access=free }}
extreme rainfall,{{Cite journal|last1=Park|first1=Jeong-Soo|last2=Jung|first2=Hyun-Sook|last3=Kim|first3=Rae-Seon|last4=Oh|first4=Jai-Ho|date=2001|title=Modelling summer extreme rainfall over the Korean peninsula using Wakeby distribution|journal=International Journal of Climatology|language=en|volume=21|issue=11|pages=1371–1384|doi=10.1002/joc.701|bibcode=2001IJCli..21.1371P |s2cid=130799481 |issn=1097-0088|doi-access=free}}{{Cite journal|last1=Su|first1=Buda|last2=Kundzewicz|first2=Zbigniew W.|last3=Jiang|first3=Tong|date=2009-05-01|title=Simulation of extreme precipitation over the Yangtze River Basin using Wakeby distribution|url=https://doi.org/10.1007/s00704-008-0025-5|journal=Theoretical and Applied Climatology|language=en|volume=96|issue=3|pages=209–219|doi=10.1007/s00704-008-0025-5|bibcode=2009ThApC..96..209S |s2cid=122488492|issn=1434-4483}}
tidal current speeds,{{Cite journal|date=2015-10-01|title=Modeling tidal current speed using a Wakeby distribution|journal=Electric Power Systems Research|language=en|volume=127|pages=240–248|doi=10.1016/j.epsr.2015.06.014|issn=0378-7796|last1=Liu|first1=Mingjun|last2=Li|first2=Wenyuan|last3=Billinton|first3=Roy|last4=Wang|first4=Caisheng|last5=Yu|first5=Juan|doi-access=free}}
and peak flows of rivers.{{Cite journal|last=Öztekin|first=Tekin|date=2011-03-01|title=Estimation of the Parameters of Wakeby Distribution by a Numerical Least Squares Method and Applying it to the Annual Peak Flows of Turkish Rivers|url=https://doi.org/10.1007/s11269-010-9745-2|journal=Water Resources Management|language=en|volume=25|issue=5|pages=1299–1313|doi=10.1007/s11269-010-9745-2|s2cid=154960776|issn=1573-1650}}

Parameters and domain

The following restrictions apply to the parameters of this distribution:

$\beta + \delta \ge 0$
Either $\beta + \delta > 0$ or $\beta = \gamma = \delta = 0$
If $\gamma > 0$ , then $\delta > 0$
$\gamma \ge 0$
$\alpha + \gamma \ge 0$

The domain of the Wakeby distribution is

$\xi$ to $\infty$ , if $\delta \ge 0$ and $\gamma > 0$
$\xi$ to $\xi + (\alpha/ \beta) - (\gamma/ \delta)$ , if $\delta < 0$ or $\gamma = 0$

With two shape parameters, the Wakeby distribution can model a wide variety of shapes.

CDF and PDF

The cumulative distribution function is computed by numerically inverting the quantile function given above. The probability density function is then found by using the following relation (given on page 46 of Johnson, Kotz, and Balakrishnan{{Cite book|last1=Johnson|first1=Norman Lloyd|url=https://www.worldcat.org/oclc/29428092|title=Continuous univariate distributions. Vol1|last2=Kotz|first2=Samuel|last3=Balakrishnan|first3=Narayanaswamy|publisher=Wiley|year=1994|isbn=0-471-58495-9|edition=2|location=New York|pages=46|oclc=29428092}}):

: $f(x) = \frac{(1 - F(x))^{(\delta+1)}}{\alpha t + \gamma}$

where F is the cumulative distribution function and

: $t = (1 - F(x))^{(\beta + \delta)}$

An implementation that computes the probability density function of the Wakeby distribution is included in the Dataplot scientific computation library, as routine WAKPDF.

An alternative to the above method is to define the PDF parametrically as $(W(p),1/w(p)), \ 0\le p \le 1$ . This can be set up as a probability density function, $f(x)$ , by solving for the unique $p$ in the equation $W(p)=x$ and returning $1/w(p)$ .{{Citation needed|date=June 2021}}

References

External links

[https://stats.stackexchange.com/q/149228 Discussion of the naming of the distribution on Stack Exchange]

:Note: this work is based on a NIST document that is in the public domain as a work of the U.S. federal government

Category:Continuous distributions