Nakagami distribution

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Nakagami
Probability density function
Nakagami pdf.svg
Cumulative distribution function
Nakagami cdf.svg
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 \sqrt{\Omega}\!
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[edit]

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[edit]

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[who?] 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[edit]

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.

History and applications[edit]

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]

References[edit]

  1. ^ a b 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. ^ 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.
  5. ^ Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.