Circular uniform distribution

In probability theory and directional statistics, a circular uniform distribution is a probability distribution on the unit circle whose density is uniform for all angles.

Description

The probability density function (pdf) of the circular uniform distribution is:

f_{{UC}}(\theta )={\frac {1}{2\pi }}.

In terms of the circular variable $z=e^{i\theta }$ the circular moments of the circular uniform distribution all zero, except for $m_{0}$ :

\langle z^{n}\rangle =\delta _{n}

where $\delta _{n}$ is the Kronecker delta symbol.

The mean angle is undefined, and the length of the mean resultant is zero.

R=|\langle z^{n}\rangle |=0\,

Distribution of the mean

The sample mean of a set of N measurements $z_{n}=e^{i\theta _{n}}$ drawn from a circular uniform distribution is defined as:

\overline {z}={\frac {1}{N}}\sum _{{n=1}}^{N}z_{n}=\overline {C}+i\overline {S}=\overline {R}e^{{i\overline {\theta }}}

where the average sine and cosine are:

\overline {C}={\frac {1}{N}}\sum _{{n=1}}^{N}\cos(\theta _{n})\qquad \qquad \overline {S}={\frac {1}{N}}\sum _{{n=1}}^{N}\sin(\theta _{n})

and the average resultant length is:

{\overline {R}}^{2}=|{\overline {z}}|^{2}={\overline {C}}^{2}+{\overline {S}}^{2}

and the mean angle is:

{\overline {\theta }}=\mathrm {Arg} ({\overline {z}}).\,

The sample mean for the circular uniform distribution will be concentrated about zero, becoming more concentrated as N increases. The distribution of the sample mean for the uniform distribution is given by:^[1]

{\frac {1}{(2\pi )^{N}}}\int _{\Gamma }\prod _{{n=1}}^{N}d\theta _{n}=P(\overline {R})P(\overline {\theta })\,d\overline {R}\,d\overline {\theta }

where $\Gamma \,$ consists of intervals of $2\pi$ in the variables, subject to the constraint that ${\overline {R}}$ and ${\overline {\theta }}$ are constant, or, alternatively, that ${\overline {C}}$ and ${\overline {S}}$ are constant. The distribution of the angle $P(\overline {\theta })$ is uniform

P(\overline {\theta })={\frac {1}{2\pi }}

and the distribution of ${\overline {R}}$ is given by:^[1]

P_{N}(\overline {R})=N^{2}\overline {R}\int _{0}^{\infty }J_{0}(N\overline {R}\,t)J_{0}(t)^{N}t\,dt

A 10,000 point Monte Carlo simulation of the distribution of the sample mean of a circular uniform distribution for N = 3

where $J_{0}$ is the Bessel function of order zero. There is no known general analytic solution for the above integral, and it is difficult to evaluate due to the large number of oscillations in the integrand. A 10,000 point Monte Carlo simulation of the distribution of the mean for N=3 is shown in the figure.

For certain special cases, the above integral can be evaluated:

P_{2}({\overline {R}})={\frac {2}{\pi {\sqrt {1-{\overline {R}}^{2}}}}}.

For large N, the distribution of the mean can be determined from the central limit theorem for directional statistics. Since the angles are uniformly distributed, the individual sines and cosines of the angles will be distributed as:

P(u)du={\frac {1}{\pi }}\,{\frac {du}{{\sqrt {1-u^{2}}}}}

where $u=\cos \theta _{n}\,$ or $\sin \theta _{n}\,$ . It follows that they will have zero mean and a variance of 1/2. By the central limit theorem, in the limit of large N, $\overline {C}\,$ and $\overline {S}\,$ , being the sum of a large number of i.i.d's, will be normally distributed with mean zero and variance $1/2N$ . The mean resultant length $\overline {R}\,$ , being the square root of the sum of two normally distributed variables, will be Chi-distributed with two degrees of freedom (i.e.Rayleigh-distributed) and variance $1/2N$ :

\lim _{N\rightarrow \infty }P_{N}({\overline {R}})=2N{\overline {R}}\,e^{-N{\overline {R}}^{2}}.

Entropy

The differential information entropy of the uniform distribution is simply

H_{U}=-\int _{\Gamma }{\frac {1}{2\pi }}\ln \left({\frac {1}{2\pi }}\right)\,d\theta =\ln(2\pi )

where $\Gamma$ is any interval of length $2\pi$ . This is the maximum entropy any circular distribution may have.

References

1 2 Jammalamadaka, S. Rao; Sengupta, A. (2001). Topics in Circular Statistics. World Scientific Publishing Company. ISBN 978-981-02-3778-3. Retrieved 2010-03-03.

Probability distributions

List

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 Lomax 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 Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's 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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Circular uniform distribution

Description

Distribution of the mean

Entropy

See also

References