Wakeby distribution

Wakeby distribution
Parameters	${\displaystyle \alpha ,\beta ,\gamma ,\delta ,\xi }$
Support	${\displaystyle \xi }$ to ${\displaystyle \infty }$, if ${\displaystyle \delta \geq 0,\gamma >0}$ ${\displaystyle \xi }$ to ${\displaystyle \xi +(\alpha /\beta )-(\gamma /\delta )}$, otherwise
Quantile	${\displaystyle \xi +{\frac {\alpha }{\beta }}(1-(1-p)^{\beta })-{\frac {\gamma }{\delta }}(1-(1-p)^{-\delta })}$

The Wakeby distribution ^[1] is a five-parameter probability distribution defined by its quantile function,

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

and by its quantile density function,

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

where ${\displaystyle 0\leq p\leq 1}$, ξ is a location parameter, α and γ are scale parameters and β and δ are shape parameters. ^[1]

This distribution was first proposed by Harold A. Thomas Jr., who named it after Wakeby Pond in Cape Cod. ^{[2] [3]}

Applications

The Wakeby distribution has been used for modeling distributions of

• flood flows, ^{[4] [5]}
• citation counts, ^[6]
• extreme rainfall, ^{[7] [8]}
• tidal current speeds, ^[9]
• and peak flows of rivers. ^[10]

Parameters and domain

The following restrictions apply to the parameters of this distribution:

• ${\displaystyle \beta +\delta \geq 0}$
• Either ${\displaystyle \beta +\delta >0}$ or ${\displaystyle \beta =\gamma =\delta =0}$
• If ${\displaystyle \gamma >0}$, then ${\displaystyle \delta >0}$
• ${\displaystyle \gamma \geq 0}$
• ${\displaystyle \alpha +\gamma \geq 0}$

The domain of the Wakeby distribution is

• ${\displaystyle \xi }$ to ${\displaystyle \infty }$, if ${\displaystyle \delta \geq 0}$ and ${\displaystyle \gamma >0}$
• ${\displaystyle \xi }$ to ${\displaystyle \xi +(\alpha /\beta )-(\gamma /\delta )}$, if ${\displaystyle \delta <0}$ or ${\displaystyle \gamma =0}$

With two shape parameters, the Wakeby distribution can model a wide variety of shapes. ^[1]

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 ^[11] ):

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

where F is the cumulative distribution function and

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

An implementation that computes the probability density function of the Wakeby distribution is included in the Dataplot scientific computation library, as routine WAKPDF. ^[1]

An alternative to the above method is to define the PDF parametrically as ${\displaystyle (W(p),1/w(p)),\ 0\leq p\leq 1}$. This can be set up as a probability density function, ${\displaystyle f(x)}$, by solving for the unique ${\displaystyle p}$ in the equation ${\displaystyle W(p)=x}$ and returning ${\displaystyle 1/w(p)}$.^{[ citation needed ]}

See also

• Generalized Pareto distribution

References

1. "Dataplot reference manual: WAKPDF". NIST. Retrieved 20 August 2015.
2. Rodda, John C.; Robinson, Mark (2015-08-26). Progress in Modern Hydrology: Past, Present and Future. John Wiley & Sons. p. 75. ISBN   978-1-119-07429-8.
3. Katchanov, Yurij L.; Markova, Yulia V. (2015-02-26). "On a heuristic point of view concerning the citation distribution: introducing the Wakeby distribution". SpringerPlus. 4 (1): 94. doi: 10.1186/s40064-015-0821-1 . ISSN   2193-1801. PMC   4352413 . PMID   25763305.
4. John C. Houghton (October 14, 1977). "Birth of a Parent: The Wakeby Distribution for Modeling Flood Flows; Working Paper No. MIT-EL77-033WP" (PDF). MIT.
5. GRIFFITHS, GEORGE A. (1989-06-01). "A theoretically based Wakeby distribution for annual flood series". Hydrological Sciences Journal. 34 (3): 231–248. Bibcode:1989HydSJ..34..231G. CiteSeerX   10.1.1.399.6501 . doi:10.1080/02626668909491332. ISSN   0262-6667.
6. Katchanov, Yurij L.; Markova, Yulia V. (2015-02-26). "On a heuristic point of view concerning the citation distribution: introducing the Wakeby distribution". SpringerPlus. 4 (1): 94. doi: 10.1186/s40064-015-0821-1 . ISSN   2193-1801. PMC   4352413 . PMID   25763305.
7. Park, Jeong-Soo; Jung, Hyun-Sook; Kim, Rae-Seon; Oh, Jai-Ho (2001). "Modelling summer extreme rainfall over the Korean peninsula using Wakeby distribution". International Journal of Climatology. 21 (11): 1371–1384. Bibcode:2001IJCli..21.1371P. doi: 10.1002/joc.701 . ISSN   1097-0088. S2CID   130799481.
8. Su, Buda; Kundzewicz, Zbigniew W.; Jiang, Tong (2009-05-01). "Simulation of extreme precipitation over the Yangtze River Basin using Wakeby distribution" . Theoretical and Applied Climatology. 96 (3): 209–219. Bibcode:2009ThApC..96..209S. doi:10.1007/s00704-008-0025-5. ISSN   1434-4483. S2CID   122488492.
9. Liu, Mingjun; Li, Wenyuan; Billinton, Roy; Wang, Caisheng; Yu, Juan (2015-10-01). "Modeling tidal current speed using a Wakeby distribution". Electric Power Systems Research. 127: 240–248. doi: 10.1016/j.epsr.2015.06.014 . ISSN   0378-7796.
10. Öztekin, Tekin (2011-03-01). "Estimation of the Parameters of Wakeby Distribution by a Numerical Least Squares Method and Applying it to the Annual Peak Flows of Turkish Rivers" . Water Resources Management. 25 (5): 1299–1313. doi:10.1007/s11269-010-9745-2. ISSN   1573-1650. S2CID   154960776.
11. Johnson, Norman Lloyd; Kotz, Samuel; Balakrishnan, Narayanaswamy (1994). Continuous univariate distributions. Vol1 (2 ed.). New York: Wiley. p. 46. ISBN   0-471-58495-9. OCLC   29428092.

External links

• 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