Nakagami distribution

Nakagami

Probability density function
Cumulative distribution function
Parameters	${\displaystyle m\ or\ \mu >= 0.5}$ shape (real) ${\displaystyle \Omega\ or\ \omega > 0}$ spread (real)
Support	$x>0\!$
PDF	${\displaystyle \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1} \exp\left(-\frac{m}{\Omega}x^2 \right)}$
CDF	${\displaystyle \frac{\gamma \left(m,\frac{m}{\Omega} x^2\right)}{\Gamma(m)}}$
Mean	${\displaystyle \frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\left(\frac{\Omega}{m}\right)^{1/2}}$
Median	No simple closed form
Mode	${\displaystyle \frac{\sqrt{2}}{2} \left(\frac{(2m-1)\Omega}{m}\right)^{1/2}}$
Variance	${\displaystyle \Omega\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^2\right)}$

The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. It 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]

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

Its cumulative distribution function is^[1]

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

where P is the incomplete gamma function (regularized).

Differential equation

${\displaystyle \left\{x \Omega f'(x)+f(x) \left(2 m x^2-2 m \Omega +\Omega \right)=0,f(1)=\frac{2 m^m e^{-\frac{m}{\Omega }} \Omega ^{-m}}{\Gamma (m)}\right\} }$

Parameter estimation

The parameters $m$ and $\Omega$ are^[2]

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

and

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

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

${\displaystyle \Gamma(m)= \frac{x^{2m}}{\sigma^m}, }$

and

${\displaystyle \sigma= \frac{x^2}{m} }$

It is reported by authors 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.

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.

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

But it should be noted that 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.

Finally, there is also a more efficient generation method using efficient rejection-sampling.^[4]

History and applications

The Nakagami distribution is relatively new, being first proposed in 1960.^[5] It has been used to model attenuation of wireless signals traversing multiple paths.^[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."^[10]

References

1. Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved 2007-08-04.
2. R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography", Radioengineering, 13 (1), 8–12
3. 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.
4. Luengo, D.; Martino, L. "Almost rejectionless sampling from Nakagami-m distributions (m≥1)". Electronics Letters. 48 (24): 1559–1561. doi:10.1049/el.2012.3513.
5. 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.
6. Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.
7. "Nakagami-q (Hoyt) distribution function with applications". doi:10.1049/el:20093427.
8. "HoytDistribution".
9. "NakagamiDistribution".
10. Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2)".