Circular uniform distribution
Encyclopedia
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.
In terms of the circular variable
the circular moments of the circular uniform distribution all zero, except for 
:
where
is the Kronecker delta symbol.
The mean angle is undefined, and the length of the mean resultant is zero.
drawn from a circular uniform distribution is defined as:
where the average sine and cosine are:
and the average resultant length is:
and the mean angle is:
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:
where
consists of intervals of 
in the variables, subject to the constraint that 
and 
are constant, or, alternatively, that 
and 
are constant. The distribution of the angle
is uniform
and the distribution of
is given by:
where
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:
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:
where
or 
. 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, 
and 
, being the sum of a large number of i.i.d's, will be normally distributed with mean zero and variance 
. The mean resultant length 
, 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 
:
where
is any interval of length 
. This is the maximum entropy any circular distribution may have.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and directional statistics, a circular uniform distribution is a probability distribution on the unit circle whose density is uniform for all angles.
Description
The pdf of the circular uniform distribution is:In terms of the circular variable
the circular moments of the circular uniform distribution all zero, except for :where
is the Kronecker delta symbol.The mean angle is undefined, and the length of the mean resultant is zero.
Distribution of the mean
The sample mean of a set of N measurements drawn from a circular uniform distribution is defined as:where the average sine and cosine are:
and the average resultant length is:
and the mean angle is:
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:
where
consists of intervals of in the variables, subject to the constraint that and are constant, or, alternatively, that and are constant. The distribution of the angle is uniformand the distribution of
is given by:where
is the Bessel functionBessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
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:
For large N, the distribution of the mean can be determined from the central limit theorem for directional statistics
Central limit theorem for directional statistics
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed....
. Since the angles are uniformly distributed, the individual sines and cosines of the angles will be distributed as:
where
or . 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, and , being the sum of a large number of i.i.d's, will be normally distributed with mean zero and variance . The mean resultant length , 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 :Entropy
The differential information entropy of the uniform distribution is simplywhere
is any interval of length . This is the maximum entropy any circular distribution may have.