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:
“No man will ever bring out of that office the reputation which carries him into it. The honeymoon would be as short in that case as in any other, and its moments of ecstasy would be ransomed by years of torment and hatred.”
—Thomas Jefferson (1743–1826)
“The essence of the modern state is that the universal be bound up with the complete freedom of its particular members and with private well-being, that thus the interests of family and civil society must concentrate themselves on the state.... It is only when both these moments subsist in their strength that the state can be regarded as articulated and genuinely organized.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“There is no need to waste pity on young girls who are having their moments of disillusionment, for in another moment they will recover their illusion.”
—Colette [Sidonie Gabrielle Colette] (1873–1954)