Bivariate von Mises distribution
Encyclopedia
In probability theory
and statistics
, the bivariate von Mises distribution is a probability distribution
describing values on a torus
. It may be thought of as an analogue on the torus of the bivariate normal distribution. The distribution belongs to the field of directional statistics. The general bivariate von Mises distribution was first proposed by Kanti Mardia in 1975. One of its variants is today used in the field of bioinformatics
to formulate a probabilistic model of protein structure
in atomic detail.
defined on the torus
,
in 
.
The probability density function of the general bivariate von Mises distribution for the angles
is given by
where
and 
are the means for 
and 
, 
and 
their concentration and the matrix 
is related to their correlation.
Two commonly used variants of the bivariate von Mises distribution are the sine and cosine variant.
The cosine variant of the bivariate von Mises distribution has the probability density function
where
and 
are the means for 
and 
, 
and 
their concentration and 
is related to their correlation. 
is the normalization constant.
The sine variant has the probability density function
where the parameters have the same interpretation.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the bivariate von Mises distribution is a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
describing values on a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
. It may be thought of as an analogue on the torus of the bivariate normal distribution. The distribution belongs to the field of directional statistics. The general bivariate von Mises distribution was first proposed by Kanti Mardia in 1975. One of its variants is today used in the field of bioinformatics
Bioinformatics
Bioinformatics is the application of computer science and information technology to the field of biology and medicine. Bioinformatics deals with algorithms, databases and information systems, web technologies, artificial intelligence and soft computing, information and computation theory, software...
to formulate a probabilistic model of protein structure
Protein structure
Proteins are an important class of biological macromolecules present in all organisms. Proteins are polymers of amino acids. Classified by their physical size, proteins are nanoparticles . Each protein polymer – also known as a polypeptide – consists of a sequence formed from 20 possible L-α-amino...
in atomic detail.
Definition
The bivariate von Mises distribution is a probability distributionProbability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
defined on the torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
,
in .The probability density function of the general bivariate von Mises distribution for the angles
is given bywhere
and are the means for and , and their concentration and the matrix is related to their correlation.Two commonly used variants of the bivariate von Mises distribution are the sine and cosine variant.
The cosine variant of the bivariate von Mises distribution has the probability density function
where
and are the means for and , and their concentration and is related to their correlation. is the normalization constant.The sine variant has the probability density function
where the parameters have the same interpretation.
See also
- Von Mises distribution, a similar distribution on the one-dimensional unit circle
- Kent distribution, a related distribution on the two-dimensional unit sphere
- von Mises-Fisher distribution
- Directional statistics