Parameter estimation is performed using maximum likelihood methods. Q-Q plot of sample data vs von Mises distribution 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. 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. Note that in this case, the length of the mean vector, | r|=1-s 2, the circular variance. Wind Rose plot of sample wind data, 4500 measurements Summary statistics for this data are as follows: With combined speed and direction the magnitude of the directional vectors is also a factor in the computations. Note that the data provide direction information only, not speed.
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. A wind rose diagram of the data is shown below. The data consists of approximately 4500 directional measurements made at hourly intervals at a single location in mid-late 2002.
In order to illustrate its application, we show a sample wind dataset from the Oriana software package. Von Mises distribution, mean =0 (red),1 (blue),2(black),3 (green) Larger values of the concentration parameter yield more leptokurtic (peaky) distributions. In this example the R package, CircStats (dvm function), was used to create the graphs. īelow we provide graphs of the von Mises distribution for mean values 0,1,2 and 3, with the intensity or concentration parameter, κ=3. Graphically the von Mises distribution with a mean at π is essentially a (finite) Normal distribution with the x-axis being in the range. Where I 1( κ) is the modified Bessel function of the first kind, of order 1. The circular variance of this distribution is: I 0( κ) is the modified Bessel function of the first kind, of order 0. Where α is the mean direction of the distribution in the range, and κ>0 is a shape parameter known as the concentration (effectively equivalent to the standard deviation). The general form of the von Mises distribution is:
A non-commercial MATLab toolbox, CircStat, is available for download from the MATLab file exchange service and this includes many similar functions. 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, which includes a wide range of measures based on the topics covered in Jammalamadaka and SenGupta (2001 ). 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 ) and Mardia and Jupp (1999 ). 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. 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. % (default = 10).The von Mises distribution is a continuous distribution that is the equivalent of the Normal distribution for data defined with directional coordinates, i.e. % reps - number of repetitions with different initial conditions % bn_noise - allow for uniform background noise term ('T' or 'F', % (2009) 'Pattern Recognition and Machine Learning', Chapter 9. Algorithm follows steps outlined in Bishop % GM_EM(X,k) - fit a GMM to X, where X is N x n and k is the number of Usage: % GM_EM - fit a Gaussian mixture model to N points located in n-dimensional space. Plotting is provided automatically for 1D/2D cases with 5 GMs or less. Also requires at least MATLAB 7.9 (2009b) NOTE: This function requires the MATLAB Statistical Toolbox and, for plotting the ellipses, the function error_ellipse, available from. This is the standard EM algorithm for GMMs, presented in Bishop's book "Pattern Recognition and Machine Learning", Chapter 9, with one small exception, the addition of a uniform distribution to the mixture to pick up background noise/speckle data points which one would not want to associate with any cluster.