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 will consist of points on the unit circle.
Any probability density function on the line can be "wrapped" around the circumference of a circle of unit radius.[1] That is, the pdf of the wrapped variable
in some interval of length
is
Theory
In most situations, a process involving circular statistics produces angles () which lie in the interval from negative infinity to positive infinity, and are described by an "unwrapped" probability density function . However, a measurement will yield a "measured" angle which lies in some interval of length (for example ). In other words, a measurement cannot tell if the "true" angle has been measured or whether a "wrapped" angle has been measured where a is some unknown integer. That is:
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 (e.g. 0 to ):
Changing the variable of integration to and exchanging the order of integration and summation, we have
where is the pdf of the "wrapped" distribution and a' is another unknown integer (a'=a+k). It can be seen that the unknown integer a' introduces an ambiguity into the expectation value of . A particular instance of this problem is encountered when attempting to take the mean of a set of measured angles. If, instead of the measured angles, we introduce the parameter it is seen that z has an unambiguous relationship to the "true" angle since:
Calculating the expectation value of a function of z will yield unambiguous answers:
and it is for this reason that the z parameter is the preferred statistical variable to use in circular statistical analysis rather than the measured angles . This suggests, and it is shown below, that the wrapped distribution function may itself be expressed as a function of z so that:
where is defined such that . 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 is the th Euclidean basis vector.
Expression in terms of characteristic functions
A fundamental wrapped distribution is the Dirac comb which is a wrapped 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 , the characteristic function of yields a Laurent series about zero for the wrapped distribution in terms of the characteristic function of the unwrapped distribution[2]:
or
By analogy with linear distributions, the are referred to as the characteristic function of the wrapped distribution[2] (or perhaps more accurately, the characteristic sequence).
Moments
The moments of the wrapped distribution are defined as:
Expressing in terms of the characteristic function and exchanging the order of integration and summation yields:
where is the Kronecker delta function. It follows that the moments are simply equal to the characteristic function of the unwrapped distribution for integer arguments: