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:
“The greatest part of each day, each year, each lifetime is made up of small, seemingly insignificant moments. Those moments may be cooking dinner...relaxing on the porch with your own thoughts after the kids are in bed, playing catch with a child before dinner, speaking out against a distasteful joke, driving to the recycling center with a week’s newspapers. But they are not insignificant, especially when these moments are models for kids.”
—Barbara Coloroso (20th century)
“The amelioration of the world cannot be achieved by sacrifices in moments of crisis; it depends on the efforts made and constantly repeated during the humdrum, uninspiring periods, which separate one crisis from another, and of which normal lives mainly consist.”
—Aldous Huxley (1894–1963)
“Who among us has not, in moments of ambition, dreamt of the miracle of a form of poetic prose, musical but without rhythm and rhyme, both supple and staccato enough to adapt itself to the lyrical movements of our souls, the undulating movements of our reveries, and the convulsive movements of our consciences? This obsessive ideal springs above all from frequent contact with enormous cities, from the junction of their innumerable connections.”
—Charles Baudelaire (1821–1867)