Wrapped distribution - AbsoluteAstronomy.com

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 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

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

.

Any probability density function

on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the pdf of the wrapped variable

in some interval of length

which is a periodic sum of period

. The preferred interval is generally

for which

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

Mean of circular quantities

In mathematics, a mean of circular quantities is a mean which is suited for quantities like angles, daytimes, and fractional parts of real numbers. This is necessary since most of the usual means fail on circular quantities...

. 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

Dirac comb

In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...

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

Characteristic function (probability theory)

In probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...

yields a Laurent series

Laurent series

In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...

about zero for the wrapped distribution in terms of the characteristic function of the unwrapped distribution:

By analogy with linear distributions, the

are referred to as the characteristic function of the wrapped distribution (or perhaps more accurately, the characteristic sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

). This is an instance of the Poisson summation formula

Poisson summation formula

In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples...

and it can be seen that the Fourier coefficients of the Fourier series for the wrapped distribution are just the Fourier coefficients of the Fourier transform of the unwrapped distribution at integer values.

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:

From the theory of residues

Residue (complex analysis)

In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...

we have

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:

Entropy

The information entropy of a circular distribution with probability density

is defined as:

where

is any interval of length

. If both the probability density and its logarithm can be expressed as a Fourier series

Fourier series

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

(or more generally, any integral transform on the circle) then the orthogonality property may be used to obtain a series representation for the entropy which may reduce to a closed form.

The moments of the distribution

are the Fourier coefficents 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

, the entropy may be written:

Theory

Expression in terms of characteristic functions

Moments

Entropy

See also