Nakagami distribution

Characterization

Its probability density function (pdf) is{{cite web | access-date = 2007-08-04 }}

: f(x;,m,\Omega) = \frac{2m^m}{\Gamma(m)\Omega^m}x^{2m-1}\exp\left(-\frac{m}{\Omega}x^2\right) \text{ for } x\geq 0. where m\geq 1/2 and \Omega0.

Its cumulative distribution function (CDF) is

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

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

Parameterization

The parameters m and \Omega are

: m = \frac{\left( \operatorname{E} [X^2] \right)^2 } {\operatorname{Var} [X^2]}, and : \Omega = \operatorname{E} [X^2].

No closed form solution exists for the median of this distribution, although special cases do exist, such as \sqrt{\Omega \ln(2)} when m = 1. For practical purposes the median would have to be calculated as the 50th-percentile of the observations.

Parameter estimation

An alternative way of fitting the distribution is to re-parametrize \Omega as σ = Ω/m.

Given independent observations X_1=x_1,\ldots,X_n=x_n from the Nakagami distribution, the likelihood function is

: 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

: \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

: \begin{align} \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{align} These derivatives vanish only when : \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, a maximum likelihood estimate for Ω is obtained as well. Here one could add something about the probability distribution of the MLE as a function of the n observed random variables. --

Random variate 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}. ,

Alternatively, 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 .

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 by Minoru Nakagami as a mathematical model for small-scale fading in long-distance high-frequency radio wave propagation. It has been used to model attenuation of wireless signals traversing multiple paths and to study the impact of fading channels on wireless communications.

Related distributions

• Restricting m to the unit interval (q = m; 0
• With 2m = k, the Nakagami distribution gives a scaled chi distribution.
• With m = \tfrac 1 2, the Nakagami distribution gives a scaled half-normal distribution.
• A Nakagami distribution is a particular form of generalized gamma distribution, with p = 2 and d = 2m.

References

References

1. 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
2. Mitra, Rangeet. (2012). "Maximum Likelihood Estimate of Parameters of Nakagami-m Distribution". International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012.
3. 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
4. Parsons, J. D. (1992) ''The Mobile Radio Propagation Channel''. New York: Wiley.
5. (2013). "2013 World Congress on Computer and Information Technology (WCCIT)".
6. (2009). "Nakagami-q (Hoyt) distribution function with applications". Electronics Letters.
7. "HoytDistribution".
8. "NakagamiDistribution".