Which way is inbound a journey with sf muni and directional statistics
Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Rn), axes (lines through the origin in Rn) or rotations in Rn. More generally, directional statistics deals with observations on compact Riemannian manifolds.
Contents
- Which way is inbound a journey with sf muni and directional statistics
- Circular and higher dimensional distributions
- von Mises circular distribution
- Circular uniform distribution
- Wrapped normal distribution
- Wrapped Cauchy distribution
- Wrapped Lvy distribution
- Distributions on higher dimensional manifolds
- Moments
- Measures of location and spread
- Distribution of the mean
- Goodness of fit and significance testing
- Books on directional statistics
- References
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 angles in molecules, orientations, rotations and so on.
Circular and higher-dimensional distributions
Any probability density function
is
This concept can be extended to the multivariate context by an extension of the simple sum to a number of
where
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:whereCircular uniform distribution
The probability density function (pdf) of the circular uniform distribution is given by
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 andWrapped Cauchy distribution
The pdf of the wrapped Cauchy distribution (WC) is:
whereWrapped Lévy distribution
The pdf of the Wrapped Lévy distribution (WL) is:
where the value of the summand is taken to be zero whenDistributions 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 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, crystallography and bioinformatics.
Moments
The raw vector (or trigonometric) moments of a circular distribution are defined as
where
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
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:
Distribution of the mean
Given a set of N measurements
which may be expressed as
where
or, alternatively as:
where
The distribution of the mean (
where
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
Goodness of fit and significance testing
For cyclic data - (e.g., is it uniformly distributed) :