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:
“It is time to provide a smashing answer for those cynical men who say that a democracy cannot be honest, cannot be efficient.... We have in the darkest moments of our national trials retained our faith in our own ability to master our own destiny.”
—Franklin D. Roosevelt (1882–1945)
“The moments of the past do not remain still; they retain in our memory the motion which drew them towards the future, towards a future which has itself become the past, and draw us on in their train.”
—Marcel Proust (1871–1922)
“There are moments of existence when time and space are more profound, and the awareness of existence is immensely heightened.”
—Charles Baudelaire (1821–1867)