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The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter \( {\displaystyle m\geq 1/2} \) and a second parameter controlling spread \( {\displaystyle \Omega >0}. \)

Characterization

Its probability density function (pdf) is[1]

\( {\displaystyle f(x;\,m,\Omega )={\frac {2m^{m}}{\Gamma (m)\Omega ^{m}}}x^{2m-1}\exp \left(-{\frac {m}{\Omega }}x^{2}\right),\forall x\geq 0.} \)

where \( {\displaystyle (m\geq 1/2,{\text{ and }}\Omega >0)} \)

Its cumulative distribution function is[1]

\( {\displaystyle F(x;\,m,\Omega )=P\left(m,{\frac {m}{\Omega }}x^{2}\right){\Gamma }^{-1}(m)} \)

where P is the incomplete gamma function (regularized).
Parametrization

The parameters m {\displaystyle m} m and Ω {\displaystyle \Omega } \Omega are[2]

\( {\displaystyle m={\frac {\left(\operatorname {E} \left[X^{2}\right]\right)^{2}}{\operatorname {Var} \left[X^{2}\right]}},} \)

and

\( {\displaystyle \Omega = \operatorname{E} \left[X^2 \right]. } \)

Parameter estimation

An alternative way of fitting the distribution is to re-parametrize \( \Omega\) and m as σ = Ω/m and m.[3]

Given independent observations \( {\textstyle X_{1}=x_{1},\ldots ,X_{n}=x_{n}} \) from the Nakagami distribution, the likelihood function is

\( {\displaystyle L(\sigma ,m)=\left({\frac {2}{\Gamma (m)\sigma ^{m}}}\right)^{n}\left(\prod _{i=1}^{n}x_{i}\right)^{2m-1}\exp \left(-{\frac {\sum _{i=1}^{n}x_{i}^{2}}{\sigma }}\right).} \)

Its logarithm is

\( {\displaystyle \ell (\sigma ,m)=\log L(\sigma ,m)=-n\log \Gamma (m)-nm\log \sigma +(2m-1)\sum _{i=1}^{n}\log x_{i}-{\frac {\sum _{i=1}^{n}x_{i}^{2}}{\sigma }}.} \)

Therefore

\( {\displaystyle {\begin{aligned}{\frac {\partial \ell }{\partial \sigma }}={\frac {-nm\sigma +\sum _{i=1}^{n}x_{i}^{2}}{\sigma ^{2}}}\quad {\text{and}}\quad {\frac {\partial \ell }{\partial m}}=-n{\frac {\Gamma '(m)}{\Gamma (m)}}-n\log \sigma +2\sum _{i=1}^{n}\log x_{i}.\end{aligned}}} \)

These derivatives vanish only when

\( {\displaystyle \sigma ={\frac {\sum _{i=1}^{n}x_{i}^{2}}{nm}}} \)

and the value of m for which the derivative with respect to m vanishes is found by numerical methods including the Newton–Raphson method.

It can be shown that at the critical point a global maximum is attained, so the critical point is the maximum-likelihood estimate of (m,σ). Because of the equivariance of maximum-likelihood estimation, one then obtains the MLE for Ω as well.
Generation

The Nakagami distribution is related to the gamma distribution. In particular, given a random variable \) {\displaystyle Y \, \sim \textrm{Gamma}(k, \theta)} \) , it is possible to obtain a random variable\) {\displaystyle X \, \sim \textrm{Nakagami} (m, \Omega)} \) , by setting k=m, \( {\displaystyle \theta=\Omega / m } \), and taking the square root of Y:

\( {\displaystyle X={\sqrt {Y}}.\,} \)

Alternatively, the Nakagami distribution \( {\displaystyle f(y; \,m,\Omega)} \) can be generated from the chi distribution with parameter k set to 2m and then following it by a scaling transformation of random variables. That is, a Nakagami random variable X is generated by a simple scaling transformation on a Chi-distributed random variable \( {\displaystyle Y \sim \chi(2m) } \) as below.

\( {\displaystyle X={\sqrt {(\Omega /2m)Y}}.} \)

For a Chi-distribution, the degrees of freedom 2m must be an integer, but for Nakagami the m can be any real number greater than 1/2. This is the critical difference and accordingly, Nakagami-m is viewed as a generalization of Chi-distribution, similar to a gamma distribution being considered as a generalization of Chi-squared distributions.

History and applications

The Nakagami distribution is relatively new, being first proposed in 1960.[4] It has been used to model attenuation of wireless signals traversing multiple paths [5] and to study the impact of fading channels on wireless communications.[6]
Related distributions

Restricting m to the unit interval (q = m; 0 < q < 1) defines the Nakagami-q distribution, also known as Hoyt distribution.[7][8][9]

"The radius around the true mean in a bivariate normal random variable, re-written in polar coordinates (radius and angle), follows a Hoyt distribution. Equivalently, the modulus of a complex normal random variable does."

References

Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved 2007-08-04.
R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography", Radioengineering, 13 (1), 8–12
Mitra, Rangeet; Mishra, Amit Kumar; Choubisa, Tarun (2012). "Maximum Likelihood Estimate of Parameters of Nakagami-m Distribution". International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012: 9–12.
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., doi:10.1016/B978-0-08-009306-2.50005-4
Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.
Ramon Sanchez-Iborra; Maria-Dolores Cano; Joan Garcia-Haro (2013). Performance evaluation of QoE in VoIP traffic under fading channels. World Congress on Computer and Information Technology (WCCIT). pp. 1–6. doi:10.1109/WCCIT.2013.6618721. ISBN 978-1-4799-0462-4.
Paris, J.F. (2009). "Nakagami-q (Hoyt) distribution function with applications". Electronics Letters. 45 (4): 210. doi:10.1049/el:20093427.
"HoytDistribution".
"NakagamiDistribution".

Mathematics Encyclopedia

Hellenica World - Scientific Library

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