rice distribution

File:Rice distribution motivation.svg with standard deviation σ (blue region). If R is the distance from these points to the origin, then R has a Rice distribution.]]

{{Probability distribution

| type = continuous

| pdf_image = Image:Rice distributiona PDF.png

| cdf_image = Image:Rice distributiona CDF.png

| parameters = $\nu\ge 0$ , distance between the reference point and the center of the bivariate distribution,
$\sigma\ge 0$ , scale

| support = $x \in [0,\infty)$

| pdf = $\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)}
{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)$

| cdf = $1-Q_1\left(\frac{\nu}{\sigma },\frac{x}{\sigma }\right)$

where Q₁ is the Marcum Q-function

| mean = $\sigma \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)$

| median =

| mode =

| variance = $2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)$

| skewness = (complicated)

| kurtosis = (complicated)

| entropy =

| mgf =

| cf =

}}

In probability theory, the Rice distribution or Rician distribution (or, less commonly, Ricean distribution) is the probability distribution of the magnitude of a circularly-symmetric bivariate normal random variable, possibly with non-zero mean (noncentral). It was named after Stephen O. Rice (1907–1986).

Characterization

The probability density function is

: $f(x\mid\nu,\sigma) = \frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)}
{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right),$

where I₀(z) is the modified Bessel function of the first kind with order zero.

In the context of Rician fading, the distribution is often also rewritten using the Shape Parameter $K = \frac{\nu^2}{2\sigma^2}$ , defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the Scale parameter $\Omega = \nu^2+2\sigma^2$ , defined as the total power received in all paths.Abdi, A. and Tepedelenlioglu, C. and Kaveh, M. and Giannakis, G., [https://dx.doi.org/10.1109/4234.913150 "On the estimation of the K parameter for the Rice fading distribution]", IEEE Communications Letters, March 2001, p. 92–94

The characteristic function of the Rice distribution is given as:Liu 2007 (in one of Horn's confluent hypergeometric functions with two variables).Annamalai 2000 (in a sum of infinite series).

: $\begin{align}
\chi_X(t\mid\nu,\sigma)
= \exp \left( -\frac{\nu^2}{2\sigma^2} \right) & \left[ \Psi_2 \left( 1; 1, \frac{1}{2}; \frac{\nu^2}{2\sigma^2}, -\frac{1}{2} \sigma^2 t^2 \right) \right. \\[8pt]
& \left. {} + i \sqrt{2} \sigma t
\Psi_2 \left( \frac{3}{2}; 1, \frac{3}{2}; \frac{\nu^2}{2\sigma^2}, -\frac{1}{2} \sigma^2 t^2 \right) \right],
\end{align}$

where $\Psi_2 \left( \alpha; \gamma, \gamma'; x, y \right)$ is one of Horn's confluent hypergeometric functions with two variables and convergent for all finite values of $x$ and $y$ . It is given by:Erdelyi 1953.Srivastava 1985.

: $\Psi_2 \left( \alpha; \gamma, \gamma'; x, y \right) = \sum_{n=0}^{\infty}\sum_{m=0}^\infty \frac{(\alpha)_{m+n}}{(\gamma)_m(\gamma')_n} \frac{x^m y^n}{m!n!},$

where

: $(x)_n = x(x+1)\cdots(x+n-1) = \frac{\Gamma(x+n)}{\Gamma(x)}$

is the rising factorial.

Properties

=Moments=

The first few raw moments are:

: $\begin{align}
\mu_1^{'}&= \sigma \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2) \\
\mu_2^{'}&= 2\sigma^2+\nu^2\, \\
\mu_3^{'}&= 3\sigma^3\sqrt{\pi/2}\,\,L_{3/2}(-\nu^2/2\sigma^2) \\
\mu_4^{'}&= 8\sigma^4+8\sigma^2\nu^2+\nu^4\, \\
\mu_5^{'}&=15\sigma^5\sqrt{\pi/2}\,\,L_{5/2}(-\nu^2/2\sigma^2) \\
\mu_6^{'}&=48\sigma^6+72\sigma^4\nu^2+18\sigma^2\nu^4+\nu^6
\end{align}$

and, in general, the raw moments are given by

: $\mu_k^{'}=\sigma^k2^{k/2}\,\Gamma(1\!+\!k/2)\,L_{k/2}(-\nu^2/2\sigma^2). \,$

Here L_q(x) denotes a Laguerre polynomial:

: $L_q(x)=L_q^{(0)}(x)=M(-q,1,x)=\,_1F_1(-q;1;x)$

where $M(a,b,z) = _1F_1(a;b;z)$ is the confluent hypergeometric function of the first kind. When k is even, the raw moments become simple polynomials in σ and ν, as in the examples above.

For the case q = 1/2:

: $\begin{align}
L_{1/2}(x) &=\,_1F_1\left( -\frac{1}{2};1;x\right) \\
&= e^{x/2} \left[\left(1-x\right)I_0\left(-\frac{x}{2}\right) -xI_1\left(-\frac{x}{2}\right) \right].
\end{align}$

The second central moment, the variance, is

: $\mu_2= 2\sigma^2+\nu^2-(\pi\sigma^2/2)\,L^2_{1/2}(-\nu^2/2\sigma^2) .$

Note that $L^2_{1/2}(\cdot)$ indicates the square of the Laguerre polynomial $L_{1/2}(\cdot)$ , not the generalized Laguerre polynomial $L^{(2)}_{1/2}(\cdot).$

Related distributions

$R \sim \mathrm{Rice}\left(|\nu|,\sigma\right)$ if $R = \sqrt{X^2 + Y^2}$ where $X \sim N\left(\nu\cos\theta,\sigma^2\right)$ and $Y \sim N\left(\nu \sin\theta,\sigma^2\right)$ are statistically independent normal random variables and $\theta$ is any real number.
Another case where $R \sim \mathrm{Rice}\left(\nu,\sigma\right)$ comes from the following steps:
# Generate $P$ having a Poisson distribution with parameter (also mean, for a Poisson) $\lambda = \frac{\nu^2}{2\sigma^2}.$
# Generate $X$ having a chi-squared distribution with {{nowrap|2P + 2}} degrees of freedom.
# Set $R = \sigma\sqrt{X}.$
If $R \sim \operatorname{Rice}(\nu,1)$ then $R^2$ has a noncentral chi-squared distribution with two degrees of freedom and noncentrality parameter $\nu^2$ .
If $R \sim \operatorname{Rice}(\nu,1)$ then $R$ has a noncentral chi distribution with two degrees of freedom and noncentrality parameter $\nu$ .
If $R \sim \operatorname{Rice}(0,\sigma)$ then $R \sim \operatorname{Rayleigh}(\sigma)$ , i.e., for the special case of the Rice distribution given by $\nu = 0$ , the distribution becomes the Rayleigh distribution, for which the variance is $\mu_2= \frac{4-\pi}{2}\sigma^2$ .
If $R \sim \operatorname{Rice}(0,\sigma)$ then $R^2$ has an exponential distribution.Richards, M.A., [http://users.ece.gatech.edu/mrichard/Rice%20power%20pdf.pdf Rice Distribution for RCS], Georgia Institute of Technology (Sep 2006)
If $R \sim \operatorname{Rice}\left(\nu,\sigma\right)$ then $1/R$ has an Inverse Rician distribution.Jones, Jessica L., Joyce McLaughlin, and Daniel Renzi. [https://iopscience.iop.org/article/10.1088/1361-6420/aa6163/ampdf "The noise distribution in a shear wave speed image computed using arrival times at fixed spatial positions."], Inverse Problems 33.5 (2017): 055012.
The folded normal distribution is the univariate special case of the Rice distribution.

Limiting cases

For large values of the argument, the Laguerre polynomial becomesAbramowitz and Stegun (1968) [http://www.math.sfu.ca/~cbm/aands/page_508.htm §13.5.1]

: $\lim_{x \to -\infty}L_\nu(x)=\frac{|x|^\nu}{\Gamma(1+\nu)}.$

It is seen that as ν becomes large or σ becomes small the mean becomes ν and the variance becomes σ².

The transition to a Gaussian approximation proceeds as follows. From Bessel function theory we have

: $I_\alpha(z) \to \frac{e^z}{\sqrt{2\pi z}} \left(1 - \frac{4 \alpha^2 - 1}{8z} + \cdots \right) \text { as } z \rightarrow \infty$

so, in the large $x\nu/\sigma^2$ region, an asymptotic expansion of the Rician distribution:

: $\begin{align} f(x,\nu,\sigma) = {} & \frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)}
{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right) \\
\text{ is } \\
& \frac{x}{\sigma^2}\exp\left(\frac{-(x^2 + \nu^2)}
{2\sigma^2}\right) \sqrt{\frac{\sigma^2}{2\pi x \nu}} \exp \left( \frac {2x \nu}{2\sigma^2} \right) \left(1 + \frac{\sigma^2}{8x\nu} + \cdots \right)\\
\rightarrow {} & \frac{1}{\sigma \sqrt{2 \pi}}\exp\left(-\frac{(x - \nu)^2}
{2\sigma^2}\right) \sqrt{ \frac{x}{\nu} } , \;\;\;
\text{ as } \frac{x\nu}{\sigma^2} \rightarrow \infty
\end{align}$

Moreover, when the density is concentrated around $\nu$ and $|x - \nu| \ll \sigma$ because of the Gaussian exponent, we can also write $\sqrt{ {x}/{\nu} } \approx 1$ and finally get the Normal approximation

: $f(x,\nu,\sigma) \approx \frac{1}{\sigma \sqrt{2\pi}} \exp\left(- \frac{(x - \nu)^2}{2\sigma^2}\right) , \;\;\; \frac{\nu}{\sigma} \gg 1$

The approximation becomes usable for $\frac{\nu}{\sigma} > 3$

Parameter estimation (the Koay inversion technique)

There are three different methods for estimating the parameters of the Rice distribution, (1) method of moments,Talukdar et al. 1991Bonny et al. 1996Sijbers et al. 1998den Dekker and Sijbers 2014 (2) method of maximum likelihood,Varadarajan and Haldar 2015 and (3) method of least squares.{{citation needed|date=June 2012}} In the first two methods the interest is in estimating the parameters of the distribution, ν and σ, from a sample of data. This can be done using the method of moments, e.g., the sample mean and the sample standard deviation. The sample mean is an estimate of μ₁^' and the sample standard deviation is an estimate of μ₂^1/2.

The following is an efficient method, known as the "Koay inversion technique".Koay et al. 2006 (known as the SNR fixed point formula). for solving the estimating equations, based on the sample mean and the sample standard deviation, simultaneously . This inversion technique is also known as the fixed point formula of SNR. Earlier worksAbdi 2001 on the method of moments usually use a root-finding method to solve the problem, which is not efficient.

First, the ratio of the sample mean to the sample standard deviation is defined as r, i.e., $r=\mu^{'}_1/\mu^{1/2}_2$ . The fixed point formula of SNR is expressed as

: $g(\theta) = \sqrt{ \xi{(\theta)} \left[ 1+r^2\right] - 2},$

where $\theta$ is the ratio of the parameters, i.e., $\theta = {\nu}/{\sigma}$ , and $\xi{\left(\theta\right)}$ is given by:

: $\xi{\left(\theta\right)} = 2 + \theta^2 - \frac{\pi}{8} \exp{(-\theta^2/2)}\left[ (2+\theta^2) I_0 (\theta^2/4) + \theta^2 I_1(\theta^{2}/4)\right]^2,$

where $I_0$ and $I_1$ are modified Bessel functions of the first kind.

Note that $\xi{\left(\theta\right)}$ is a scaling factor of $\sigma$ and is related to $\mu_{2}$ by:

: $\mu_2 = \xi{\left(\theta\right)} \sigma^2.$

To find the fixed point, $\theta^{*}$ , of $g$ , an initial solution is selected, ${\theta}_{0}$ , that is greater than the lower bound, which is ${\theta}_{\text{lower bound}} = 0$ and occurs when $r = \sqrt{\pi/(4-\pi)}$ (Notice that this is the $r=\mu^{'}_1/\mu^{1/2}_2$ of a Rayleigh distribution). This provides a starting point for the iteration, which uses functional composition,{{clarify|reason=is this worth saying if meaning is not defined|date=June 2012}} and this continues until $\left|g^{i}\left(\theta_{0}\right)-\theta_{i-1}\right|$ is less than some small positive value. Here, $g^{i}$ denotes the composition of the same function, $g$ , $i$ times. In practice, we associate the final $\theta_{n}$ for some integer $n$ as the fixed point, $\theta^{*}$ , i.e., $\theta^{*} = g\left(\theta^{*}\right)$ .

Once the fixed point is found, the estimates $\nu$ and $\sigma$ are found through the scaling function, $\xi{\left(\theta\right)}$ , as follows:

: $\sigma = \frac{\mu^{1/2}_2}{\sqrt{\xi\left(\theta^{*}\right)}},$

and

: $\nu = \sqrt{\left( \mu^{'~2}_1 + \left(\xi\left(\theta^{*}\right) - 2\right)\sigma^2 \right)}.$

To speed up the iteration even more, one can use the Newton's method of root-finding. This particular approach is highly efficient.

Applications

The Euclidean norm of a bivariate circularly-symmetric normally distributed random vector.
Rician fading (for multipath interference))
Effect of sighting error on target shooting.{{cite web|title=Ballistipedia|url=http://ballistipedia.com/index.php?title=Closed_Form_Precision#How_many_sighter_shots_do_you_need.3F |access-date=4 May 2014}}
Analysis of diversity receivers in radio communications.{{Cite journal|last1=Beaulieu|first1=Norman C|last2=Hemachandra|first2=Kasun|date=September 2011 |title=Novel Representations for the Bivariate Rician Distribution|journal=IEEE Transactions on Communications|volume=59|issue=11|pages=2951–2954 |doi=10.1109/TCOMM.2011.092011.090171|s2cid=1221747 }}{{Cite journal|last1=Dharmawansa|first1=Prathapasinghe| last2=Rajatheva|first2=Nandana| last3=Tellambura|first3=Chinthananda| date=March 2009|title=New Series Representation for the Trivariate Non-Central Chi-Squared Distribution| url=http://www.ece.ualberta.ca/~chintha/resources/papers/2009/4799042.pdf|journal=IEEE Transactions on Communications|volume=57 |issue=3|pages=665–675| doi=10.1109/TCOMM.2009.03.070083|citeseerx=10.1.1.582.533|s2cid=15706035 }}
Distribution of eccentricities for models of the inner Solar System after long-term numerical integration.{{Cite journal |last=Laskar |first=J. |date=2008-07-01 |title=Chaotic diffusion in the Solar System |url=https://www.sciencedirect.com/science/article/pii/S0019103508001097 |journal=Icarus |language=en |volume=196 |issue=1 |pages=1–15 |doi=10.1016/j.icarus.2008.02.017 |arxiv=0802.3371 |bibcode=2008Icar..196....1L |s2cid=11586168 |issn=0019-1035}}

References

External links

[https://au.mathworks.com/matlabcentral/fileexchange/14237-rice-rician-distribution MATLAB code for Rice/Rician distribution] (PDF, mean and variance, and generating random samples)

Category:Continuous distributions

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