probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

and

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, a circular distribution or polar distribution is a

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

of a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

whose values are angles, usually taken to be in the range A circular distribution is often a

continuous probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

, and hence has a

probability density In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values ...

, but such distributions can also be

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...

, in which case they are called circular lattice distributions. Circular distributions can be used even when the variables concerned are not explicitly angles: the main consideration is that there is not usually any real distinction between events occurring at the opposite ends of the range, and the division of the range could notionally be made at any point.

Graphical representation

If a circular distribution has a density :

p(\phi)   \qquad  \qquad (0\le\phi<2\pi),\,

it can be graphically represented as a closed

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

(\phi),y(\phi) = (\phi)\cos\phi, \, r(\phi)\sin\phi \,

where the radius

r(\phi)\,

is set equal to :

r(\phi) = a+b p(\phi), \,

and where ''a'' and ''b'' are chosen on the basis of appearance.

Examples

By computing the probability distribution of angles along a handwritten ink trace, a lobe-shaped polar distribution emerges. The main direction of the lobe in the first quadrant corresponds to the

slant Slant can refer to: Bias *Bias or other non- objectivity in journalism, politics, academia or other fields Technical * Slant range, in telecommunications, the line-of-sight distance between two points which are not at the same level * Slant ...

of handwriting (see:

graphonomics Graphonomics is the interdisciplinary field directed towards the scientific analysis of the handwriting process, product, and other graphic skills., Researchers in handwriting recognition, forensic handwriting examination, kinesiology, psych ...

). An example of a circular lattice distribution would be the probability of being born in a given month of the year, with each calendar month being thought of as arranged round a circle, so that "January" is next to "December".

References

Sources

External links

Circular Values Math and Statistics with C++11
A C++11 infrastructure for circular values (angles, time-of-day, etc.) mathematics and statistics {{ProbDistributions, families Types of probability distributions Directional statistics Statistical charts and diagrams

Graphical representation

Examples

See also

References

Sources

External links