Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics
that deals with directions
s in R''n''
through the origin in R''n''
) or rotation
s in R''n''
. More generally, directional statistics deals with observations on compact Riemannian manifold
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 (in this case, angular data). Other examples of data that may be regarded as directional include statistics involving temporal periods (e.g. time of day, week, month, year, etc.), compass directions, dihedral angle
s in molecules, orientations, rotations and so on.
Any probability density function (pdf)
on the line can be "wrapped"
around the circumference of a circle of unit radius. That is, the pdf of the wrapped variable
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:
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 (Fisher, 1993).
The pdf of the von Mises distribution is:
is the modified Bessel function
of order 0.
Circular uniform distribution
The probability density function (pdf) 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'' (WN) is:
:where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively and
is the Jacobi theta function
Wrapped Cauchy distribution
The pdf of the ''wrapped Cauchy distribution'' (WC) is:
is the scale factor and
is the peak position.
Wrapped Lévy distribution
The pdf of the ''wrapped Lévy distribution'' (WL) 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
(such as the Kent distribution
), the ''N''-dimensional sphere
(the von Mises–Fisher distribution
) or the torus
(the bivariate von Mises distribution
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 (''N'' − 1)-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 (which is the one-dimensional sphere) 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
The raw vector (or trigonometric) moments of a circular distribution are defined as
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.
[Fisher, NI., ''Statistical Analysis of Circular Data'', Cambridge University Press, 1993. ]
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:
* The . For the sample the circular variance is defined as:
:and for the population
:Both will have values between 0 and 1.
:with values between 0 and infinity. This definition of the standard deviation (rather than the square root of the variance) is useful because for a wrapped normal distribution, it is an estimator of the standard deviation of the underlying normal distribution. It will therefore allow the circular distribution to be standardized as in the linear case, for small values of the standard deviation. This also applies to the von Mises distribution which closely approximates the wrapped normal distribution. Note that for small
, we have
:with values between 0 and infinity. This measure of spread is found useful in the statistical analysis of variance.
Distribution of the mean
Given a set of ''N'' measurements
the mean value of ''z'' is defined as:
which may be expressed as
or, alternatively as:
The distribution of the mean (
) for a circular pdf ''P''(''θ'') will be given by:
is over any interval of length
and the integral is subject to the constraint that
are constant, or, alternatively, that
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. (main article: Central limit theorem for directional statistics
). It can be shown
that the distribution of