Wrapped normal distribution
Encyclopedia
In 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 periodic boundary conditions. It is closely approximated by the von Mises distribution, which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics.
of the wrapped normal distribution is
where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively. Expressing
the above density function in terms of the characteristic function
of the normal distribution yields:
where
is the Jacobi theta function, given by
and 
The wrapped normal distribution may also be expressed in terms of the Jacobi triple product:
where
and 
the circular moments of the wrapped Normal distribution are the characteristic function of the Normal 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
The circular standard deviation, which is a useful measure of dispersion for the wrapped Normal distribution and its close relative, the von Mises distribution is given by:
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 mean μ lies in the interval[ −π, π) , then Arg will be a (biased) estimator of the mean μ.
Viewing the zn as a set of vectors in the complex plane, the 2 statistic is the square of the length of the averaged vector:
and its expected value is:
In other words, the statistic
will be an unbiased estimator of e−σ2, and ln(1/Re2) will be a (biased) estimator of σ2
where
is any interval of length 
. Defining 
and 
, the Jacobi triple product representation for the wrapped normal is:
where
is the Euler function. The logarithm of the density of the wrapped normal distribution may be written:
Using the series expansion for the logarithm:
the logarithmic sums may be written as:
so that the logarithm of density of the wrapped normal distribution may be written as:
which is essentially a a Fourier series
in
. Using the characteristic function representation for the wrapped normal distribution in the left side of the integral:
the entropy may be written:
which may be integrated to yield:
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 normal 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...
which results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
and is a solution to the heat equation for periodic boundary conditions. It is closely approximated by the von Mises distribution, which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics.
Definition
The probability density functionProbability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
of the wrapped normal distribution is
where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively. Expressing
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...
the above density function 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 normal distribution yields:
where
is the Jacobi theta function, given by and The wrapped normal distribution may also be expressed in terms of the Jacobi triple product:
where
and Moments
In terms of the circular variable the circular moments of the wrapped Normal distribution are the characteristic function of the Normal 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
The circular standard deviation, which is a useful measure of dispersion for the wrapped Normal distribution and its close relative, the von Mises distribution is given by:
Estimation of parameters
A series of N measurements zn = e iθn drawn from a wrapped normal 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 mean μ lies in the interval
Viewing the zn as a set of vectors in the complex plane, the 2 statistic is the square of the length of the averaged vector:
and its expected value is:
In other words, the statistic
will be an unbiased estimator of e−σ2, and ln(1/Re2) will be a (biased) estimator of σ2
Entropy
The information entropy of the wrapped normal distribution is defined as:where
is any interval of length . Defining and , the Jacobi triple product representation for the wrapped normal is:where
is the Euler function. The logarithm of the density of the wrapped normal distribution may be written:Using the series expansion for the logarithm:
the logarithmic sums may be written as:
so that the logarithm of density of the wrapped normal distribution may be written as:
which is essentially a 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...
in
. Using the characteristic function representation for the wrapped normal distribution in the left side of the integral:the entropy may be written:
which may be integrated to yield: