Moments
The moments of the von Mises distribution are usually calculated as the moments of z = eix rather than the angle x itself. These moments are referred to as "circular moments". The variance calculated from these moments is referred to as the "circular variance". The one exception to this is that the "mean" usually refers to the argument of the circular mean, rather than the circular mean itself.
The nth raw moment of z is:
where the integral is over any interval of length 2π. In calculating the above integral, we use the fact that zn = cos(nx) + i sin(nx) and the Bessel function identity (See Abramowitz and Stegun §9.6.19):
The mean of z is then just
and the "mean" value of x is then taken to be the argument μ. This is the "average" direction of the angular random variables. The variance of z, or the circular variance of x is:
Read more about this topic: Von Mises Distribution
Famous quotes containing the word moments:
“One of the few moments of happiness a man knows in Australia is that moment of meeting the eyes of another man over the tops of two beer glasses.”
—Anonymous. Quoted by Bruce Chatwin in “From the Notebooks,” ch. 30, The Songlines (1987)
“There are moments when all anxiety and stated toil are becalmed in the infinite leisure and repose of nature.”
—Henry David Thoreau (1817–1862)
“There are moments when, faced with our lack of success, I wonder whether we are failures, proud but impotent. One thing reassures me as to our value: the boredom that afflicts us. It is the hall-mark of quality in modern men.”
—Edmond De Goncourt (1822–1896)