Applied Statistics Using SPSS, STATISTICA, MATLAB and R

10.3 The von Mises Distributions 383
Function pooledmean computes the pooled mean (see section 10.6.2) of
independent samples of circular or spherical observations, a. The last column of a
contains the group codes, starting with 1. The mean resultant length and the
weighted resultant length are returned through rw and rhow, respectively.
Function rotate returns the spherical data matrix v (standard format),
obtained by rotating a so that the mean direction maps onto the North Pole.
Function scattermx returns the scatter matrix t of the spherical data a (see
section 10.4.4).
Function dirdif returns the directional data of the differences of the unit
vectors corresponding to a and b (standard format).
The R functions behave in the same way as their equivalent MATLAB
functions. For instance, Example 10.5 is solved in R with:
j o o
[1] 0.6487324 1.4182647 -73.1138435 65.4200379
[5] 178.7780083 73.1304754
10.3 The von Mises Distributions
The importance of the von Mises distributions (see B.2.10) for directional data is
similar to the importance of the normal distribution for linear data. As mentioned
in B.2.10, several physical phenomena originate von Mises distributions. These
enjoy important properties, namely their proximity with the normal distribution as
mentioned in properties 3, 4 and 5 of B.2.10. The convolution of von Mises
distributions does not produce a von Mises distribution; however, it can be well
approximated by a von Mises distribution.
The generalised (p – 1)-dimensional von Mises density function, for a vector of
observations x, can be written as:
mµ κ , p ( x)
C p
κ µ ’
x
, = ( κ ) e , 10.10
where µ is the mean vector, κ is the concentration parameter, and Cp(κ) is a
normalising factor with the following values:
2
C 2 ( κ ) = 1/(
2π
I 0 ( κ )) , for the circle (p = 2);
C ( κ ) = κ /( 4π
sinh( κ )) , for the sphere (p = 3).
3
2
Ip denotes the modified Bessel function of the first kind and order p (see B.2.10).