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| {{Probability distribution |
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| name =Nakagami|
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| type =density|
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| pdf_image =[[Image:Nakagami pdf.svg|325px]]|
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| cdf_image =[[Image:Nakagami cdf.svg|325px]]|
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| parameters =<math>m\ or\ \mu >= 0.5</math> [[shape parameter|shape]] ([[real number|real]])<br /><math>\Omega\ or\ \omega > 0</math> spread (real)|
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| support =<math>x > 0\!</math>|
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| pdf =<math>\frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1} \exp\left(-\frac{m}{\Omega}x^2 \right)</math>|
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| cdf =<math>\frac{\gamma \left(m,\frac{m}{\Omega} x^2\right)}{\Gamma(m)}</math>|
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| mean =<math>\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\left(\frac{\Omega}{m}\right)^{1/2}</math>|
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| median =<math>\sqrt{\Omega}\!</math>|
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| mode =<math>\frac{\sqrt{2}}{2} \left(\frac{(2m-1)\Omega}{m}\right)^{1/2}</math>|
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| variance =<math>\Omega\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^2\right)</math>|
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| skewness =|
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| kurtosis =|
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| entropy =|
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| mgf =|
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| char =|
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| }}
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| The '''Nakagami distribution''' or the '''Nakagami-''m'' distribution''' is a [[probability distribution]] related to the [[gamma distribution]]. It has two parameters: a [[shape parameter]] <math>m</math> and a second parameter controlling spread, <math>\Omega</math>.
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| == Characterization ==
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| Its [[probability density function]] (pdf) is<ref name='dl'>{{cite web
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| | last =Laurenson
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| | first =Dave
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| | authorlink =
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| | coauthors =
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| | title =Nakagami Distribution
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| | work =Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques
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| | publisher =
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| | date =1994
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| | url =http://www.see.ed.ac.uk/~dil/thesis_mosaic/section2_19.html
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| | format =
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| | doi =
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| | accessdate = 2007-08-04 }}</ref>
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| :<math> f(x;\,m,\Omega) = \frac{2m^m}{\Gamma(m)\Omega^m}x^{2m-1}\exp\left(-\frac{m}{\Omega}x^2\right).
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| </math>
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| Its [[cumulative distribution function]] is<ref name='dl'/>
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| :<math> F(x;\,m,\Omega) = P\left(m, \frac{m}{\Omega}x^2\right)</math>
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| where ''P'' is the [[incomplete gamma function]] (regularized).
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| == Parameter estimation ==
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| The parameters <math>m</math> and <math>\Omega</math> are<ref>R. Kolar, R. Jirik, J. Jan (2004) [http://www.radioeng.cz/fulltexts/2004/04_01_08_12.pdf "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography"], ''Radioengineering'', 13 (1), 8–12</ref>
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| :<math> m = \frac{\operatorname{E}^2 \left[X^2 \right]}
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| {\operatorname{Var} \left[X^2 \right]},
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| </math>
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| and
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| :<math> \Omega = \operatorname{E} \left[X^2 \right]. </math> | |
| An alternative way of fitting the distribution is to re-parametrize <math> \Omega </math> and m as σ = Ω/m and m.<ref name=paraest>{{cite journal|last=Mitra|first=Rangeet|coauthors=Mishra, Amit Kumar; Choubisa, Tarun|title=Maximum Likelihood Estimate of Parameters of Nakagami-m Distribution|journal=International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012 |date=2012|page=9-12}}</ref> Then, by taking the derivative of log likelihood with respect to each of the new parameters, the following equations are obtained and these can be solved using the [[Newton-Raphson]] method:
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| :<math> \Gamma(m)= \frac{x^{2m}}{\sigma^m},
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| </math>
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| and
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| :<math> \sigma= \frac{x^2}{m} </math>
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| It is reported by authors{{who|date=April 2013}} that modelling data with Nakagami distribution and estimating parameters by above mention method results in better performance for low data regime compared to moments based methods.
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| == Generation ==
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| The Nakagami distribution is related to the [[gamma distribution]].
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| In particular, given a random variable <math>Y \, \sim \textrm{Gamma}(k, \theta)</math>, it is possible to obtain a random variable <math>X \, \sim \textrm{Nakagami} (m, \Omega)</math>, by setting <math>k=m</math>, <math>\theta=\Omega / m </math>, and taking the square root of <math>Y</math>:
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| :<math> X = \sqrt{Y} \,</math>.
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| The Nakagami distribution <math>f(y; \,m,\Omega)</math> can be generated from the [[chi distribution]] with parameter <math>k</math> set to <math>2m</math> and then following it by a scaling transformation of random variables. That is, a Nakagami random variable <math>X</math> is generated by a simple scaling transformation on a Chi-distributed random variable <math>Y \sim \chi(2m) </math> as below:
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| :<math> X = \sqrt{(\Omega / 2 m)}\, Y.</math>
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| == History and applications ==
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| The Nakagami distribution is relatively new, being first proposed in 1960.<ref>Nakagami, M. (1960) "The m-Distribution, a general formula of intensity of rapid fading". In William C. Hoffman, editor, ''Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held June 18-20, 1958'', pp 3-36. Pergamon Press.</ref> It has been used to model attenuation of [[wireless]] signals traversing multiple paths.<ref>Parsons, J. D. (1992) ''The Mobile Radio Propagation Channel''. New York: Wiley.</ref>
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| {{refimprove|date=April 2013}}
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| {{inline|date=April 2013}}
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| == References ==
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| <references/>
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| {{ProbDistributions|continuous-semi-infinite}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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