Wrapped Normal Distribution

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The wrapped normal distribution (also called the wrapped up normal distribution) occurs via Brownian motion on a circle.  It is the circular analog of a normal distribution, achieved by wrapping the probability density function (PDF) of a real-valued, linear random variable to the circumference of a unit circle an infinite number of times [1].

PDF of the wrapped normal distribution.

Wrapped Normal Distribution Properties

The wrapped normal distribution is obtained by wrapping the (regular) normal distribution around the unit circle. All probability mass wrapped to the same point is added, which is equivalent to wrapped random variable X mod 2π [2].

The probability density function (PDF) of the wrapped normal distribution is [3]

wrapped normal distribution pdf


  • μ = mean of the unwrapped distribution,
  • σ = standard deviation of the unwrapped distribution,

Wrapped normal distribution alternate definitions

The density can also be defined as the series [4]

wrapped normal distribution

Where N is the normal distribution being wrapped and

  • μ = mean (taken as modulo 2 π)
  • σ = standard deviation.

The density is normalized because integrating it over 2 π is equivalent to integrating the normal distribution over the reals.

Early works define the distribution in different ways. For example, Haight’s 1958 reference work for the National Bureau of Statistics [5] defines the wrapped up normal distribution as:

wrapped up normal distribution

Stephens [6] made a brief mention of the wrapped up normal distribution in his 1958 paper Random Walk on a Circle. He defines it slightly differently, as “of the type”

alternate definition

Stephens also notes that when on a sphere, the wrapped up normal distribution is similar to the Riemann theta (presumably, the Riemann theta function).

Comparison to von Mises

The wrapped normal distribution PDF contains an infinite series, so many researchers choose the von Mises distribution instead. The two distributions are so similar that several hundred samples are needed to distinguish between them [7]. Thus, if working with small samples, the von Mises should be easier to work with, but for larger samples, use the wrapped normal distribution.

The first few terms of the series (density) may give a decent approximation for the wrapped normal. For example, Mardia and Jupp [8] suggest the wrapped normal density can be approximated by the first three terms when σ 2 > 2π. However, this should be taken as a rule of thumb as there is no theoretical justification for it [2].


[1] Mardia, Kantilal; Jupp, Peter E. (1999). Directional Statistics. Wiley. ISBN 978-0-471-95333-3.

[2] Kurz, G. etal. Efficient Evaluation of the Probability Density Function of a Wrapped Normal Distribution. Conference: IEEE ISIF Workshop on Sensor Data Fusion: Trends, Solutions, Applications (SDF 2014) At: Bonn, Germany

[3] Jorge , A. et al (Eds.) (2015). Machine Learning and Knowledge Discovery in Databases. European Conference, ECML PKDD 2015, Porto, Portugal, September 7-11, 2015, Proceedings, Part II · Part 2. Springer International.

[4] Pfaff, F. (2019). Multitarget Tracking Using Orientation Estimation for Optical Belt Sorting. KIT Scientific Publishing

[5] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[6] Stephens, M.A. (1963). Random Walk on a Circle. Biometrika Vol. 50, No. 3/4 (Dec.), pp. 385-390 (6 pages)

[7] D. Collett and T. Lewis, “Discriminating Between the Von Mises and
Wrapped Normal Distributions,” Australian Journal of Statistics, vol. 23,
no. 1, pp. 73–79, 1981.

[8] K. V. Mardia and P. E. Jupp, Directional Statistics, 1st ed. Wiley, 1999

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