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In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or Tikhonov distribution) is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution. A freely diffusing angle θ {displaystyle heta } on a circle is a wrapped normally distributed random variable with an unwrapped variance that grows linearly in time. On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i.e. with a preferred orientation. The von Mises distribution is the maximum entropy distribution for circular data when the real and imaginary parts of the first circular moment are specified. The von Mises distribution is a special case of the von Mises–Fisher distribution on the N-dimensional sphere. The von Mises probability density function for the angle x is given by: where I0( κ {displaystyle kappa } ) is the modified Bessel function of order 0. The parameters μ and 1/ κ {displaystyle kappa } are analogous to μ and σ2 (the mean and variance) in the normal distribution: The probability density can be expressed as a series of Bessel functions where Ij(x) is the modified Bessel function of order j.

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