In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or the 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 \theta } 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.
Definition The von Mises probability density function for the angle x is given by:
f ( x ∣ μ , κ ) = exp ( κ cos ( x − μ ) ) 2 π I 0 ( κ ) {\displaystyle f(x\mid \mu ,\kappa )={\frac {\exp(\kappa \cos(x-\mu ))}{2\pi I_{0}(\kappa )}}}
where I0(κ) is the modified Bessel function of the first kind of order 0, with this scaling constant chosen so that the distribution sums to unity: ∫ − π π exp ( κ cos x ) d x = 2 π I 0 ( κ ) . {\textstyle \int _{-\pi }^{\pi }\exp(\kappa \cos x)dx={2\pi I_{0}(\kappa )}.}
The parameters μ and 1/κ are analogous to μ and σ2 (the mean and variance) in the normal distribution:
μ is a measure of location (the distribution is clustered around μ), and κ is a measure of concentration (a reciprocal measure of dispersion, so 1/κ is analogous to σ2). If κ is zero, the distribution is uniform, and for small κ, it is close to uniform. If κ is large, the distribution becomes very concentrated about the angle μ with κ being a measure of the concentration. In fact, as κ increases, the distribution approaches a normal distribution in x with mean μ and variance 1/κ. The probability density can be expressed as a series of Bessel functions
f ( x ∣ μ , κ ) = 1 2 π ( 1 + 2 I 0 ( κ ) ∑ j = 1 ∞ I j ( κ ) cos [ j ( x − μ ) ] ) {\displaystyle f(x\mid \mu ,\kappa )={\frac {1}{2\pi }}\left(1+{\frac {2}{I_{0}(\kappa )}}\sum _{j=1}^{\infty }I_{j}(\kappa )\cos[j(x-\mu )]\right)}
where Ij(x) is the modified Bessel function of order j. The cumulative distribution function is not analytic and is best found by integrating the above series. The indefinite integral of the probability density is:
Φ ( x ∣ μ , κ ) = ∫ f ( t ∣ μ , κ ) d t = 1 2 π ( x + 2 I 0 ( κ ) ∑ j = 1 ∞ I j ( κ ) sin [ j ( x − μ ) ] j ) . {\displaystyle \Phi (x\mid \mu ,\kappa )=\int f(t\mid \mu ,\kappa )\,dt={\frac {1}{2\pi }}\left(x+{\frac {2}{I_{0}(\kappa )}}\sum _{j=1}^{\infty }I_{j}(\kappa ){\frac {\sin[j(x-\mu )]}{j}}\right).}
The cumulative distribution function will be a function of the lower limit of integration x0:
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