Directional statistics


Directional statistics is the subdiscipline of statistics that deals with directions, axes or rotations in Rn. More generally, directional statistics deals with observations on compact Riemannian manifolds.
can be parameterized as a sequence of points on the unit sphere. Shown are two views of the spherical histogram of such points for a large collection of protein structures. The statistical treatment of such data is in the realm of directional statistics.
The fact that 0 degrees and 360 degrees are identical angles, so that for example 180 degrees is not a sensible mean of 2 degrees and 358 degrees, provides one illustration that special statistical methods are required for the analysis of some types of data. Other examples of data that may be regarded as directional include statistics involving temporal periods, compass directions, dihedral angles in molecules, orientations, rotations and so on.

Circular and higher-dimensional distributions

Any probability density function on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the pdf of the wrapped variable
is
This concept can be extended to the multivariate context by an extension of the simple sum to a number of sums that cover all dimensions in the feature space:
where is the th Euclidean basis vector.
The following sections show some relevant circular distributions.

von Mises circular distribution

The von Mises distribution is a circular distribution which, like any other circular distribution, may be thought of as a wrapping of a certain linear probability distribution around the circle. The underlying linear probability distribution for the von Mises distribution is mathematically intractable; however, for statistical purposes, there is no need to deal with the underlying linear distribution. The usefulness of the von Mises distribution is twofold: it is the most mathematically tractable of all circular distributions, allowing simpler statistical analysis, and it is a close approximation to the wrapped normal distribution, which, analogously to the linear normal distribution, is important because it is the limiting case for the sum of a large number of small angular deviations. In fact, the von Mises distribution is often known as the "circular normal" distribution because of its ease of use and its close relationship to the wrapped normal distribution.
The pdf of the von Mises distribution is:
where is the modified Bessel function of order 0.

Circular uniform distribution

The probability density function of the circular uniform distribution is given by
It can also be thought of as of the von Mises above.

Wrapped normal distribution

The pdf of the wrapped normal distribution is:

Wrapped Cauchy distribution

The pdf of the wrapped Cauchy distribution is:

Wrapped Lévy distribution

The pdf of the Wrapped Lévy distribution is:
where the value of the summand is taken to be zero when, is the scale factor and is the location parameter.

Distributions on higher-dimensional manifolds

There also exist distributions on the two-dimensional sphere, the N-dimensional sphere or the torus.
The matrix von Mises–Fisher distribution is a distribution on the Stiefel manifold, and can be used to construct probability distributions over rotation matrices.
The Bingham distribution is a distribution over axes in N dimensions, or equivalently, over points on the -dimensional sphere with the antipodes identified. For example, if N = 2, the axes are undirected lines through the origin in the plane. In this case, each axis cuts the unit circle in the plane at two points that are each other's antipodes. For N = 4, the Bingham distribution is a distribution over the space of unit quaternions. Since a unit quaternion corresponds to a rotation matrix, the Bingham distribution for N = 4 can be used to construct probability distributions over the space of rotations, just like the Matrix-von Mises–Fisher distribution.
These distributions are for example used in geology, crystallography and bioinformatics.

Moments

The raw vector moments of a circular distribution are defined as
where is any interval of length, is the PDF of the circular distribution, and. Since the integral is unity, and the integration interval is finite, it follows that the moments of any circular distribution are always finite and well defined.
Sample moments are analogously defined:
The population resultant vector, length, and mean angle are defined in analogy with the corresponding sample parameters.
In addition, the lengths of the higher moments are defined as:
while the angular parts of the higher moments are just. The lengths of all moments will lie between 0 and 1.

Measures of location and spread

Various measures of location and spread may be defined for both the population and a sample drawn from that population. The most common measure of location is the circular mean. The population circular mean is simply the first moment of the distribution while the sample mean is the first moment of the sample. The sample mean will serve as an unbiased estimator of the population mean.
When data is concentrated, the median and mode may be defined by analogy to the linear case, but for more dispersed or multi-modal data, these concepts are not useful.
The most common measures of circular spread are:
Given a set of N measurements the mean value of z is defined as:
which may be expressed as
where
or, alternatively as:
where
The distribution of the mean for a circular pdf P will be given by:
where is over any interval of length and the integral is subject to the constraint that and are constant, or, alternatively, that and are constant.
The calculation of the distribution of the mean for most circular distributions is not analytically possible, and in order to carry out an analysis of variance, numerical or mathematical approximations are needed.
The central limit theorem may be applied to the distribution of the sample means.. It can be shown that the distribution of approaches a bivariate normal distribution in the limit of large sample size.

Goodness of fit and significance testing

For cyclic data – :