Wrapped Cauchy distribution
Encyclopedia
In probability theory
and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution
that results from the "wrapping" of the Cauchy distribution
around the unit circle
. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution.
The wrapped Cauchy distribution is often found in the field of spectroscopy where it is used to analyze diffraction patterns (e.g. see Fabry–Pérot interferometer)
is:
where
is the scale factor and 
is the peak position of the "unwrapped" distribution. Expressing
the above pdf in terms of the characteristic function
of the Cauchy distribution yields:
In terms of the circular variable
the circular moments of the wrapped Cauchy distribution are the characteristic function of the Cauchy distribution evaluated at integer arguments:
where
is some interval of length 
. The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:
The mean angle is
and the length of the mean resultant is
drawn from a wrapped Cauchy distribution may be used to estimate certain parameters of the distribution. The average of the series 
is defined as
and its expectation value will be just the first moment:
In other words,
is an unbiased estimator of the first moment. If we assume that the peak position 
lies in the interval 
, then Arg
will be a (biased) estimator of the peak position 
.
Viewing the
as a set of vectors in the complex plane, the 
statistic is the length of the averaged vector:
and its expectation value is
In other words, the statistic
will be an unbiased estimator of
, and 
will be a (biased) estimator of 
.
where
is any interval of length 
. The logarithm of the density of the wrapped Cauchy distribution may be written as a Fourier series
in
:
where
which yields:
(c.f. Gradshteyn and Rhyzik 4.224.15) and
(c.f. Gradshteyn and Rhyzik 4.397.6). The characteristic function representation for the wrapped Cauchy distribution in the left side of the integral is:
where
. Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written:
The series is just the Taylor expansion for the logarithm of
so the entropy may be written in closed form as:
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 Cauchy distribution is a wrapped probability distribution
Wrapped distribution
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...
that results from the "wrapping" of the Cauchy distribution
Cauchy distribution
The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...
around 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...
. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution.
The wrapped Cauchy distribution is often found in the field of spectroscopy where it is used to analyze diffraction patterns (e.g. see Fabry–Pérot interferometer)
Description
The pdf of the wrapped Cauchy distributionCauchy distribution
The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...
is:
where
is the scale factor and is the peak position of the "unwrapped" distribution. ExpressingWrapped distribution
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...
the above pdf in terms of the characteristic function
Characteristic function
In mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* In probability theory, the characteristic function of any probability distribution on the real line is given by...
of the Cauchy distribution yields:
In terms of the circular variable
the circular moments of the wrapped Cauchy distribution are the characteristic function of the Cauchy distribution evaluated at integer arguments:where
is some interval of length . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:The mean angle is
and the length of the mean resultant is
Estimation of parameters
A series of N measurements drawn from a wrapped Cauchy distribution may be used to estimate certain parameters of the distribution. The average of the series is defined asand its expectation value will be just the first moment:
In other words,
is an unbiased estimator of the first moment. If we assume that the peak position lies in the interval , then Arg will be a (biased) estimator of the peak position .Viewing the
as a set of vectors in the complex plane, the statistic is the length of the averaged vector:and its expectation value is
In other words, the statistic
will be an unbiased estimator of
, and will be a (biased) estimator of .Entropy
The information entropy of the wrapped Cauchy distribution is defined as:where
is any interval of length . The logarithm of the density of the wrapped Cauchy distribution may be written as a Fourier seriesFourier 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...
in
:where
which yields:
(c.f. Gradshteyn and Rhyzik 4.224.15) and
(c.f. Gradshteyn and Rhyzik 4.397.6). The characteristic function representation for the wrapped Cauchy distribution in the left side of the integral is:
where
. Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written:The series is just the Taylor expansion for the logarithm of
so the entropy may be written in closed form as:See also
- Wrapped distributionWrapped distributionIn 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...
- Dirac combDirac combIn mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...
- Wrapped normal distributionWrapped normal distributionIn probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution which results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownian motion and is a solution to the heat equation for...
- Circular uniform distributionCircular uniform distributionIn 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 pdf of the circular uniform distribution is:...
- McCullagh's parametrization of the Cauchy distributions