Mean of circular quantities

Mean of circular quantities

In mathematics, a mean of circular quantities is a mean which is suited for quantities like angles, daytimes, and fractional parts of real numbers. This is necessary since most of the usual means fail on circular quantities. For example, the arithmetic mean of 0° and 360° is 180°, although 0° would be clearly the better choice. [1] This is one of the simplest examples of statistics of non-Euclidean spaces.

Contents

Mean of angles

Since the arithmetic mean is not effective for angles, the following method can be used to obtain both a mean value and measure for the variance of the angles:

Convert all angles to corresponding points on the unit circle, e.g., α to (cos α,sin α). That is convert polar coordinates to Cartesian coordinates. Then compute the arithmetic mean of these points. The resulting point will lie on the unit disk. Convert that point back to polar coordinates. The angle is a reasonable mean of the input angles. The resulting radius will be 1 if all angles are equal. If the angles are uniformly distributed on the circle, then the resulting radius will be 0, and there is no circular mean. In other words, the radius measures the concentration of the angles.

Given the angles \alpha_1,\dots,\alpha_n the mean is computed by

\bar{\alpha} = \operatorname{atan2}\left(\frac{1}{n}\cdot\sum_{j=1}^n \sin\alpha_j, \frac{1}{n}\cdot\sum_{j=1}^n \cos\alpha_j\right)

using the atan2 variant of the arctangent function, or

\bar{\alpha} = \arg\left(\frac{1}{n}\cdot\sum_{j=1}^n \exp(i\cdot\alpha_j)\right)

using complex numbers.

Properties

The mean \bar{\alpha}

  • maximizes the likelihood of the mean parameter of the von Mises distribution and
  • minimizes the sum of a certain distance on the circle, more precisely
\bar{\alpha} = \underset{\beta}{\operatorname{argmin}} \sum_{j=1}^n d(\alpha_j,\beta), where d(φ,β) = 1 − cos(φ − β).
The distance d(φ,β) is equal to half the squared Euclidean distance between the two points on the unit circle associated with φ and β.

See also

References

  1. ^ Christopher M. Bishop: Pattern Recognition and Machine Learning (Information Science and Statistics), ISBN 0387310738

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Circular — is a basic geometric shape such as a Circle. Contents 1 Documents 2 Travel and transportation 3 Places …   Wikipedia

  • Mean — This article is about the statistical concept. For other uses, see Mean (disambiguation). In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). the expected value of a… …   Wikipedia

  • Circular dichroism — (CD) refers to the differential absorption of left and right circularly polarized light.[1][2] This phenomenon was discovered by Jean Baptiste Biot, Augustin Fresnel, and Aimé Cotton in the first half of the 19th century.[3] It is exhibited in… …   Wikipedia

  • Circular orbit — For other meanings of the term orbit , see orbit (disambiguation) A circular orbit is the orbit at a fixed distance around any point by an object rotating around a fixed axis. Below we consider a circular orbit in astrodynamics or celestial… …   Wikipedia

  • Directional 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. The… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • List of circle topics — This list of circle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or concretely in physical space. It does not include metaphors like inner circle or circular reasoning in… …   Wikipedia

  • List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

  • cosmos — /koz meuhs, mohs/, n., pl. cosmos, cosmoses for 2, 4. 1. the world or universe regarded as an orderly, harmonious system. 2. a complete, orderly, harmonious system. 3. order; harmony. 4. any composite plant of the genus Cosmos, of tropical… …   Universalium

  • mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”