# VonMisesDistribution

VonMisesDistribution[μ,κ]

represents a von Mises distribution with mean μ and concentration κ.

# Background & Context

• VonMisesDistribution[μ,κ] represents a continuous statistical distribution supported over the interval and parametrized by a real number μ (the mean of the distribution) and by a non-negative real number κ (its concentration), which together determine the overall behavior of its probability density function (PDF). In general, the PDF of a von Mises distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and the horizontal location of its maximum) is determined by the values of μ and κ. The von Mises distribution is sometimes referred to as the circular normal distribution or as the Tikhonov distribution.
• The von Mises distribution was first proposed in the early 1900s and was later introduced as a statistical model in a 1918 paper by German mathematician and statistician Richard von Mises as a tool to help model the distribution of atomic weights of elements known at the time. The von Mises distribution is the circular analog of the NormalDistribution defined on the real line and is among the most studied distributions in the field of circular statistics. It has been generalized and applied as a modeling tool in a number of different contexts. In particular, the von Mises distribution has been used to model phenomena including the spread of diseases, protein data, interference alignment in signal processing, and privacy-preserving algorithms in machine learning.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a von Mises distribution. Distributed[x,VonMisesDistribution[μ,κ]], written more concisely as xVonMisesDistribution[μ,κ], can be used to assert that a random variable x is distributed according to a von Mises distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions for von Mises distributions may be given using PDF[VonMisesDistribution[μ,κ],x] and CDF[VonMisesDistribution[μ,κ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a von Mises distribution, EstimatedDistribution to estimate a von Mises parametric distribution from given data, and FindDistributionParameters to fit data to a von Mises distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic von Mises distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic von Mises distribution.
• TransformedDistribution can be used to represent a transformed von Mises distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a von Mises distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving von Mises distributions.
• VonMisesDistribution is related to a number of other distributions. VonMisesDistribution is an immediate generalization of UniformDistribution, in the sense that the PDF of VonMisesDistribution[μ,0] is precisely the same as that of UniformDistribution[{μ-π,μ+π}] and also limits to NormalDistribution in the sense that the PDF of a VonMisesDistribution tends to that of a NormalDistribution as κ. VonMisesDistribution is also closely related to WignerSemicircleDistribution, LogNormalDistribution, HalfNormalDistribution, BinormalDistribution, and InverseGaussianDistribution.

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function does not have closed form, but can be evaluated numerically:

Mean:

Circular mean is given by the following:

Median:

## Scope(5)

Generate a sample of pseudorandom numbers from a von Mises distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:

Hazard function:

Quantile function:

Use dimensionless Quantity to define VonMisesDistribution:

Find the mean angle:

## Applications(2)

Generate random points on a unit circle:

Scaled density function on the unit circle:

Generate random points on a unit circle with different concentrations around :

## Properties & Relations(3)

Von Mises distribution is closed under translation:

Relationships to other distributions:

With zero concentration, a von Mises distribution becomes UniformDistribution:

## Neat Examples(1)

PDFs for different κ values with CDF contours:

Wolfram Research (2010), VonMisesDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/VonMisesDistribution.html (updated 2016).

#### Text

Wolfram Research (2010), VonMisesDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/VonMisesDistribution.html (updated 2016).

#### CMS

Wolfram Language. 2010. "VonMisesDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/VonMisesDistribution.html.

#### APA

Wolfram Language. (2010). VonMisesDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VonMisesDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_vonmisesdistribution, author="Wolfram Research", title="{VonMisesDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/VonMisesDistribution.html}", note=[Accessed: 19-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_vonmisesdistribution, organization={Wolfram Research}, title={VonMisesDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/VonMisesDistribution.html}, note=[Accessed: 19-July-2024 ]}