In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit n-sphere.
In one dimension, a wrapped distribution consists of points on the unit circle.
with probability density function (PDF)
on the line can be "wrapped" around the circumference of a circle of unit radius.
In most situations, a process involving circular statistics produces angles (
, and are described by an "unwrapped" probability density function
In other words, a measurement cannot tell whether the true angle
If we wish to calculate the expected value of some function of the measured angle it will be: We can express the integral as a sum of integrals over periods of
and exchanging the order of integration and summation, we have where
introduces an ambiguity into the expected value of
, similar to the problem of calculating angular mean.
has an unambiguous relationship to the true angle
: Calculating the expected value of a function of
will yield unambiguous answers: For this reason, the
parameter is preferred over measured angles
This concept can be extended to the multivariate context by an extension of the simple sum to a number of
sums that cover all dimensions in the feature space: where
A fundamental wrapped distribution is the Dirac comb, which is a wrapped Dirac delta function: Using the delta function, a general wrapped distribution can be written Exchanging the order of summation and integration, any wrapped distribution can be written as the convolution of the unwrapped distribution and a Dirac comb: The Dirac comb may also be expressed as a sum of exponentials, so we may write: Again exchanging the order of summation and integration: Using the definition of
yields a Laurent series about zero for the wrapped distribution in terms of the characteristic function of the unwrapped distribution: or Analogous to linear distributions,
is referred to as the characteristic function of the wrapped distribution (or more accurately, the characteristic sequence).
[2] This is an instance of the Poisson summation formula, and it can be seen that the coefficients of the Fourier series for the wrapped distribution are simply the coefficients of the Fourier transform of the unwrapped distribution at integer values.
in terms of the characteristic function and exchanging the order of integration and summation yields: From the residue theorem we have where
It follows that the moments are simply equal to the characteristic function of the unwrapped distribution for integer arguments: If
is a random variate drawn from a linear probability distribution
is a circular variate distributed according to the wrapped
is the angular variate distributed according to the wrapped
The information entropy of a circular distribution with probability density
[1] If both the probability density and its logarithm can be expressed as a Fourier series (or more generally, any integral transform on the circle), the orthogonal basis of the series can be used to obtain a closed form expression for the entropy.
are the Fourier coefficients for the Fourier series expansion of the probability density: If the logarithm of the probability density can also be expressed as a Fourier series: where Then, exchanging the order of integration and summation, the entropy may be written as: Using the orthogonality of the Fourier basis, the integral may be reduced to: For the particular case when the probability density is symmetric about the mean,
and the logarithm may be written: and and, since normalization requires that