Nakagami distribution

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

The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. It has two parameters: a shape parameter $m$ and a second parameter controlling spread, $\Omega$ .

Characterization

Its probability density function (pdf) is^[1]

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

Its cumulative distribution function is^[1]

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

where P is the incomplete gamma function (regularized).

Differential equation

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

Parameter estimation

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

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

and

\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:

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

and

\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 $Y \, \sim \textrm{Gamma}(k, \theta)$ , it is possible to obtain a random variable $X \, \sim \textrm{Nakagami} (m, \Omega)$ , by setting $k=m$ , $\theta=\Omega / m$ , and taking the square root of $Y$ :

X = \sqrt{Y} \,

The Nakagami distribution $f(y; \,m,\Omega)$ 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 $Y \sim \chi(2m)$ as below:

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

Currently, the more efficient generation method is provided in.^[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 2 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.
↑ 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.
↑ 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.
↑ Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.
↑ "Nakagami-q (Hoyt) distribution function with applications". doi:10.1049/el:20093427.
↑ "HoytDistribution".
↑ "NakagamiDistribution".
↑ Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package ‘shotGroups’ version 0.6.2)".

Probability distributions

List of probability distributions

Discrete univariate with finite support	Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory-Schulz Fréchet Gamma Gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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