Wrapped normal distribution

In probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution that 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. The probability density function 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 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: 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 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 mean μ lies in the interval [−π, π), then Arg will be a (biased) estimator of the mean μ.