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## von Mises distribution |

The von Mises distribution is a continuous distribution that is the equivalent of the Normal distribution for data defined with directional coordinates, i.e. over the range [0,2π]. This distribution is not widely supported in standard software and general purpose packages, but is available in a number of more specialized libraries and toolsets. It is one of a number of distributions (others including the Uniform distribution and the wrapped Normal) that are used in the general field of directional statistics. The formulas used by the commercial Oriana software package, which is specifically designed for the analysis of directional data, follow those provided in Fisher (1993 [FIS1]) and Mardia and Jupp (1999 [MAR1]). Support for such analysis is not provided in the base package for "R" but is available in a number of submitted packages such as CircStats [BER1], which includes a wide range of measures based on the topics covered in Jammalamadaka and SenGupta (2001 [JAM1]). A non-commercial MATLab toolbox, CircStat, is available for download from the MATLab file exchange service and this includes many similar functions.

The general form of the von Mises distribution is:

where α is the mean direction of the distribution in the range [0,2π], and κ>0 is a shape parameter known as the concentration (effectively equivalent to the standard deviation). I0(κ) is the modified Bessel function of the first kind, of order 0. The circular variance of this distribution is:

where I1(κ) is the modified Bessel function of the first kind, of order 1. Graphically the von Mises distribution with a mean at π is essentially a (finite) Normal distribution with the x-axis being in the range [0,2π].

Below we provide graphs of the von Mises distribution for mean values 0,1,2 and 3, with the intensity or concentration parameter, κ=3. In this example the R package, CircStats (dvm function), was used to create the graphs. Larger values of the concentration parameter yield more leptokurtic (peaky) distributions.

von Mises distribution, mean =0 (red),1 (blue),2(black),3 (green)

Example: Wind dataset analysis

In order to illustrate its application, we show a sample wind dataset from the Oriana software package. The data consists of approximately 4500 directional measurements made at hourly intervals at a single location in mid-late 2002. A wind rose diagram of the data is shown below. The arrow shows the mean vector of the dataset and the areas of the stacked histogram are proportional to the grouped frequencies of the data in 15 degree intervals. Note that the data provide direction information only, not speed. With combined speed and direction the magnitude of the directional vectors is also a factor in the computations.

Summary statistics for this data are as follows:

•Mean Vector (µ): 226.719°

•Length of Mean Vector (r): 0.285

•Concentration, (κ): 0.594

•Circular Variance: 0.715

•Circular Standard Deviation: 90.844°

Wind Rose plot of sample wind data, 4500 measurements

Note that in this case, the length of the mean vector, |r|=1-s2, the circular variance. The cumulative frequency distribution of the source data can be plotted on a chart that is scaled to the von Mises distribution whose parameters have been estimated from the data. Essentially this is the same form of plot as a Normal distribution Q-Q plot, whereby the visual fit of the sample data to the von Mises distribution is assessed from it proximity to the diagonal line on the chart.

Q-Q plot of sample data vs von Mises distribution

Parameter estimation is performed using maximum likelihood methods. The sample (vector, r) mean direction (circular mean) is the maximum likelihood estimator of the distribution mean direction. The concentration parameter, κ, may be estimated from the square of the length of the mean vector |r|, r2, or from the circular variance. For small sample sizes (n<16 records) bias adjustment is recommended, of the form:

The two relations cited above show that

from which the concentration parameter can be derived numerically.

References

[BER1] Berens P, Velasco M J (2009) MATLab file exchange: CircStats: A circular statistics toolbox: https://www.mathworks.com/matlabcentral/fileexchange/10676-circular-statistics-toolbox-directional-statistics and

P. Berens (2009) CircStat: A Matlab Toolbox for Circular Statistics, Journal of Statistical Software, Volume 31, Issue 10

[FIS1] Fisher N I (1993) Statistical analysis of circular data. Cambridge University Press, Cambridge

[GAI1] Gaile G L, Burt J E (1980) Directional Statistics. CATMOG 25. Obtainable from: https://www.qmrg.org.uk/

[JAM1] Jammalamadaka S R, SenGupta A (2001) Topics in Circular Statistics. World Scientific Press, Singapore

[MAR1] Mardia K V, Jupp P E (1999) Directional statistics. 2nd ed., John Wiley, Chichester

[ZAR1] Zar J H (1999) Biostatistical Analysis. Prentice Hall, NJ, USA

Mathworld: von Mises distribution: https://mathworld.wolfram.com/vonMisesDistribution.html

Wikipedia: von Mises distribution: https://en.wikipedia.org/wiki/Von_Mises_distribution