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[[File:Rice distribution motivation.svg|thumb|300px|In the 2D plane, pick a fixed point at distance ''ν'' from the origin. Generate a distribution of 2D points centered around that point, where the ''x'' and ''y'' coordinates are chosen independently from a [[gaussian distribution]] 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|325px|Rice probability density functions σ = 1.0]]
  | cdf_image  = [[Image:Rice distributiona CDF.png|325px|Rice cumulative distribution functions σ = 1.0]]
  | parameters = ''ν''&thinsp;≥&thinsp;0 — distance between the reference point and the center of the bivariate distribution,<br>''σ''&thinsp;≥&thinsp;0
  | support    = ''x'' ∈ [0,&thinsp;+∞)
  | pdf        = <math>\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)}
{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)</math>
  | cdf        = <math>1-Q_1\left(\frac{\nu}{\sigma },\frac{x}{\sigma }\right)</math>
where ''Q''<sub>1</sub> is the [[Marcum Q-function]]
  | mean       = <math>\sigma  \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)</math>
  | median     =
  | mode       =
  | variance   = <math>2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)</math>
  | 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 circular bivariate normal random variable with potentially non-zero mean. It was named after [[Stephen O. Rice]].

==Characterization==
The probability density function is
:<math>
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),</math>
where ''I''<sub>0</sub>(''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'' <math>K = \frac{\nu^2}{2\sigma^2}</math>, defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the ''Scale parameter'' <math> \Omega = \nu^2+2\sigma^2 </math>, defined as the total power received in all paths.<ref>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</ref>

The [[Characteristic function (probability theory)|characteristic function]] of the Rice distribution is given as:<ref>[[#refLiu2007|Liu 2007 (in one of Horn's confluent hypergeometric functions with two variables).]]</ref><ref>[[#refAnnamalai2000|Annamalai 2000 (in a sum of infinite series).]]</ref>

:<math>
\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}
</math>

where <math>\Psi_2 \left( \alpha; \gamma, \gamma'; x, y \right)</math> is one of [[Horn function|Horn's confluent hypergeometric functions]] with two variables and convergent for all finite values of <math>x</math> and <math>y</math>. It is given by:<ref>[[#refErdelyi1953|Erdelyi 1953.]]</ref><ref>[[#refSrivastava1985|Srivastava 1985.]]</ref>

:<math>\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!},</math>

where

: <math>(x)_n = x(x+1)\cdots(x+n-1) = \frac{\Gamma(x+n)}{\Gamma(x)}</math>

is the [[rising factorial]].

==Properties==

===Moments===
The first few [[raw moments]] are:

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

and, in general, the raw moments are given by

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

Here ''L''<sub>''q''</sub>(''x'') denotes a [[Laguerre polynomials|Laguerre polynomial]]:

:<math>L_q(x)=L_q^{(0)}(x)=M(-q,1,x)=\,_1F_1(-q;1;x)</math>
where <math>M(a,b,z) = _1F_1(a;b;z)</math> 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:

:<math>
\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}
</math>

The second [[central moment]], the [[variance]], is
:<math>\mu_2= 2\sigma^2+\nu^2-(\pi\sigma^2/2)\,L^2_{1/2}(-\nu^2/2\sigma^2) .</math>

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

==Related distributions==
*<math>R \sim \mathrm{Rice}\left(|\nu|,\sigma\right)</math> has a Rice distribution if <math>R = \sqrt{X^2 + Y^2}</math> where <math>X \sim N\left(\nu\cos\theta,\sigma^2\right)</math> and <math>Y \sim N\left(\nu \sin\theta,\sigma^2\right)</math> are statistically independent normal random variables and <math>\theta</math> is any real number.
*Another case where <math>R \sim \mathrm{Rice}\left(\nu,\sigma\right)</math> comes from the following steps:

:1.  Generate <math>P</math> having a [[Poisson distribution]] with parameter (also mean, for a Poisson) <math>\lambda = \frac{\nu^2}{2\sigma^2}.</math>

:2.  Generate <math>X</math> having a [[chi-squared distribution]] with {{nowrap|2''P'' + 2}} degrees of freedom.

:3.  Set <math>R = \sigma\sqrt{X}.</math>

*If <math>R \sim \operatorname{Rice}(\nu,1)</math> then <math>R^2</math> has a [[noncentral chi-squared distribution]] with two degrees of freedom and noncentrality parameter <math>\nu^2</math>.
*If <math>R \sim \operatorname{Rice}(\nu,1)</math> then <math>R</math> has a [[noncentral chi distribution]] with two degrees of freedom and noncentrality parameter <math>\nu</math>.
*If <math>R \sim \operatorname{Rice}(0,\sigma)</math> then <math>R \sim \operatorname{Rayleigh}(\sigma)</math>, i.e., for the special case of the Rice distribution given by ν = 0, the distribution becomes the [[Rayleigh distribution]], for which the variance is <math>\mu_2= \frac{4-\pi}{2}\sigma^2</math>.
*If <math>R \sim \operatorname{Rice}(0,\sigma)</math> then <math>R^2</math> has an [[exponential distribution]].<ref>Richards, M.A., [http://users.ece.gatech.edu/mrichard/Rice%20power%20pdf.pdf Rice Distribution for RCS], Georgia Institute of Technology (Sep 2006)</ref>
*If <math>R \sim \operatorname{Rice}\left(\nu,\sigma\right)</math> then <math>1/R</math> has an Inverse Rician distribution.<ref>Jones, Jessica L., Joyce McLaughlin, and Daniel Renzi. [https://pdfs.semanticscholar.org/7201/a04ccad3b51849fe1036290eb0e11f89b5f1.pdf "The noise distribution in a shear wave speed image computed using arrival times at fixed spatial positions."], Inverse Problems 33.5 (2017): 055012.</ref>

==Limiting cases==
For large values of the argument, the Laguerre polynomial becomes<ref>Abramowitz and Stegun (1968) [http://www.math.sfu.ca/~cbm/aands/page_508.htm §13.5.1]</ref>

:<math>\lim_{x\rightarrow -\infty}L_\nu(x)=\frac{|x|^\nu}{\Gamma(1+\nu)}.</math>

It is seen that as ''ν'' becomes large or σ becomes small the mean becomes ''ν'' and the variance becomes σ<sup>2</sup>.

The transition to a Gaussian approximation proceeds as follows.  From Bessel function theory we have
:<math> I_\alpha(z) \rightarrow \frac{e^z}{\sqrt{2\pi z}} \left(1 - \frac{4 \alpha^2 - 1}{8z} + \cdots \right) \text { as } z \rightarrow \infty </math>
so, in the large <math> x\nu/\sigma^2 </math> region, an asymptotic expansion of the Rician distribution:

:<math> \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}
</math>

Moreover, when the density is concentrated around <math display="inline"> \nu </math> and <math display="inline">|x - \nu|  \ll \sigma </math> because of the Gaussian exponent, we can also write <math> \sqrt{ \frac{x}{\nu} } \approx 1 </math> and finally get the Normal approximation
: <math> 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
</math>
The approximation becomes usable for <math> \frac{\nu}{\sigma} > 3 </math>

==Parameter estimation (the Koay inversion technique)==
There are three different methods for estimating the parameters of the Rice distribution, (1) [[Method of moments (statistics)|method of moments]],<ref name=T>[[#RefTalukdar|Talukdar et al. 1991]]</ref><ref name=B>[[#RefBonny|Bonny et al. 1996]]</ref><ref name=S>[[#RefSijbers|Sijbers et al. 1998]]</ref><ref name=S2>[[#RefDekker2014|den Dekker and Sijbers 2014]]</ref>  (2) [[method of maximum likelihood]],<ref name=T/><ref name=B/><ref name=S/><ref name=V>[[#RefVaradarajan|Varadarajan and Haldar 2015]]</ref> 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 μ<sub>1</sub><sup>'</sup> and the sample standard deviation is an estimate of μ<sub>2</sub><sup>1/2</sup>.

The following is an efficient method, known as the "Koay inversion technique".<ref name=K>[[#refKoay2006|Koay et al. 2006 (known as the SNR fixed point formula).]]</ref> 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 (mathematics)|fixed point]] formula of [[Signal-to-noise ratio|SNR]]. Earlier works<ref name=T/><ref>[[#RefAbdi|Abdi 2001]]</ref> 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., <math>r=\mu^{'}_1/\mu^{1/2}_2</math>. The fixed point formula of SNR is expressed as

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

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

:<math> \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,</math>

where <math>I_0</math> and <math>I_1</math> are [[modified Bessel function of the first kind|modified Bessel functions of the first kind]].

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

:<math> \mu_2 = \xi{\left(\theta\right)} \sigma^2.\, </math>

To find the fixed point, <math> \theta^{*} </math>, of <math> g </math>, an initial solution is selected, <math> {\theta}_{0} </math>, that is greater than the lower bound, which is <math> {\theta}_{\mathrm{lower bound}} = 0 </math> and occurs when <math>r = \sqrt{\pi/(4-\pi)}</math><ref name="K"/> (Notice that this is the <math>r=\mu^{'}_1/\mu^{1/2}_2</math> 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 <math>\left|g^{i}\left(\theta_{0}\right)-\theta_{i-1}\right|</math> is less than some small positive value. Here, <math>g^{i}</math> denotes the composition of the same function, <math>g</math>, <math>i</math> times. In practice, we associate the final <math>\theta_{n}</math> for some integer <math>n</math> as the fixed point, <math>\theta^{*}</math>, i.e., <math>\theta^{*} =  g\left(\theta^{*}\right)</math>.

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

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

and

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

To speed up the iteration even more, one can use the Newton's method of root-finding.<ref name=K/> This particular approach is highly efficient.

==Applications==
*The [[Euclidean norm]] of a [[normal random vector|bivariate normally distributed random vector]].
*[[Rician fading]]
*Effect of sighting error on target shooting.<ref>{{cite web|title=Ballistipedia|url=http://ballistipedia.com/index.php?title=Closed_Form_Precision#How_many_sighter_shots_do_you_need.3F|accessdate=4 May 2014}}</ref>

==See also==
The multivariate Rician model is used in the analysis of diversity receivers in radio communications<ref>{{Cite journal|last=Beaulieu|first=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}}</ref><ref>{{Cite journal|last=Dharmawansa|first=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}}</ref>.
*[[Rayleigh distribution]]
*[[Stephen O. Rice]] (1907&ndash;1986)

==Notes==
{{Reflist}}

==References==
*Abramowitz, M. and Stegun, I. A.  (ed.), [[Abramowitz and Stegun|Handbook of Mathematical Functions]], National Bureau of Standards, 1964; reprinted Dover Publications, 1965. {{ISBN|0-486-61272-4}}
*[[Stephen O. Rice|Rice, S. O.]], Mathematical Analysis of Random Noise. Bell System Technical Journal 24 (1945) 46&ndash;156.
*{{cite journal | authors = I. Soltani Bozchalooi and Ming Liang | doi = 10.1016/j.jsv.2007.07.038 | title = A smoothness index-guided approach to wavelet parameter selection in signal de-noising and fault detection | journal = Journal of Sound and Vibration | volume = 308 | issue = 1–2 | date = 20 November 2007 | pages = 253–254 |ref=refBozchalooi2007| bibcode = 2007JSV...308..246B }}
*{{cite journal|doi=10.1016/j.jsv.2017.02.013 |title=On the distribution of the modulus of Gabor wavelet coefficients and the upper bound of the dimensionless smoothness index in the case of additive Gaussian noises: Revisited |journal=Journal of Sound and Vibration |volume=395 |pages=393–400 |year=2017 |last1=Wang |first1=Dong |last2=Zhou |first2=Qiang |last3=Tsui |first3=Kwok-Leung }}
*<cite id=refLiu2007>Liu, X. and Hanzo, L., [http://ieeexplore.ieee.org/Xplore/login.jsp?url=/iel5/7693/4350290/04350297.pdf?arnumber=4350297 A Unified Exact BER Performance Analysis of Asynchronous DS-CDMA Systems Using BPSK Modulation over Fading Channels], IEEE Transactions on Wireless Communications, Volume 6, Issue 10, October 2007, pp. 3504&ndash;3509.</cite>
*<cite id=refAnnamalai2000>Annamalai, A., Tellambura, C. and Bhargava, V. K., [http://ieeexplore.ieee.org/Xplore/login.jsp?url=/iel5/26/18877/00871398.pdf?temp=x Equal-Gain Diversity Receiver Performance in Wireless Channels], IEEE Transactions on Communications,Volume 48, October 2000, pp. 1732&ndash;1745.</cite>
*<cite id=refErdelyi1953>Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., [http://apps.nrbook.com/bateman/Vol1.pdf Higher Transcendental Functions, Volume 1.] McGraw-Hill Book Company Inc., 1953.</cite>
*<cite id=refSrivastava1985>Srivastava, H. M. and Karlsson, P. W., Multiple Gaussian Hypergeometric Series. Ellis Horwood Ltd., 1985.</cite>
*<cite id=refSijbers1998>Sijbers J., den Dekker A. J., Scheunders P. and Van Dyck D., [http://webh01.ua.ac.be/visielab/papers/sijbers/ieee98.pdf "Maximum Likelihood estimation of Rician distribution parameters"], IEEE Transactions on Medical Imaging, Vol. 17, Nr. 3, pp.&nbsp;357&ndash;361, (1998)</cite>
*<cite id=refVaradarajan2015>Varadarajan D. and Haldar J. P., [https://ieeexplore.ieee.org/abstract/document/7097060 "A Majorize-Minimize Framework for Rician and Non-Central Chi MR Images"], IEEE Transactions on Medical Imaging, Vol. 34, no. 10, pp. 2191&ndash;2202, (2015)</cite>
*{{cite journal | authors = den Dekker, A.J., and Sijbers, J | doi =10.1016/j.ejmp.2014.05.002 | pmid =25059432 | journal= Physica Medica | title = Data distributions in magnetic resonance images: a review | volume = 30 | issue = 7 | date = December 2014 | pages = 725–741|ref=RefDekker2014}}
*<cite id=refKoay2006> Koay, C.G. and Basser, P. J., [https://doi.org/10.1016/j.jmr.2006.01.016 Analytically exact correction scheme for signal extraction from noisy magnitude MR signals], Journal of Magnetic Resonance, Volume 179, Issue = 2, p. 317&ndash;322, (2006)</cite>
*<cite id=RefAbdi>Abdi, A., Tepedelenlioglu, C., Kaveh, M., and [[Georgios B. Giannakis|Giannakis, G.]] [http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=913150 On the estimation of the K parameter for the Rice fading distribution], IEEE Communications Letters, Volume 5, Number 3, March 2001, pp. 92&ndash;94.</cite>
*{{cite journal | authors = Talukdar, K.K., and Lawing, William D. | doi = 10.1121/1.400532 | title = Estimation of the parameters of the Rice distribution | journal = Journal of the Acoustical Society of America | volume = 89 | issue = 3 | date = March 1991 | pages = 1193–1197 |ref=RefTalukdar| bibcode = 1991ASAJ...89.1193T }}
*{{cite journal | authors = Bonny,J.M., Renou, J.P., and Zanca, M. | doi = 10.1006/jmrb.1996.0166 | title = Optimal Measurement of Magnitude and Phase from MR Data | journal = Journal of Magnetic Resonance, Series B | volume = 113 | issue = 2 | date = November 1996 | pages = 136–144 |ref=RefBonny| bibcode = 1996JMRB..113..136B }}

==External links==
*[http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=14237&objectType=FILE MATLAB code for Rice/Rician distribution] (PDF, mean and variance, and generating random samples)

{{ProbDistributions|continuous-semi-infinite}}

{{Use dmy dates|date=August 2019}}

{{DEFAULTSORT:Rice Distribution}}
[[Category:Continuous distributions]]

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