Circular Random Variable
- Values represent directions or angles and lie on a circle rather than a linear scale.
- The variable is periodic (values repeat over ranges such as every 360 degrees or every 180 degrees) and requires circular-specific methods.
- Modeled using circular distributions (e.g., von Mises, wrapped Cauchy, wrapped normal) and analyzed with circular statistics.
Definition
Section titled “Definition”A circular random variable is a type of random variable that takes on values that lie on the circumference of a circle. This type of variable is commonly used in situations where the values of the variable are angles or directions.
Explanation
Section titled “Explanation”Circular random variables describe quantities measured as angles or directions relative to a fixed reference point. Their key characteristic is periodicity: values repeat over a defined range (for example, every 360 degrees for wind direction or every 180 degrees for certain orientations). Standard linear statistical methods do not account for this periodic nature, so specialized tools—collectively called circular statistics—are used.
These variables are typically modeled with probability distributions defined on the circle, known as circular distributions. Common circular distributions include the von Mises distribution, the wrapped Cauchy distribution, and the wrapped normal distribution. Such distributions express the probability of different angular values and support prediction and inference for circular data.
Examples
Section titled “Examples”Wind direction
Section titled “Wind direction”The variable takes values that represent the direction the wind is blowing (e.g., north, south, east, or west). Values are determined by the angle at which the wind is blowing relative to a fixed reference point, such as north.
Orientation of a spinning top
Section titled “Orientation of a spinning top”The variable takes values that represent the direction the top is pointing (e.g., up, down, left, or right). Values are determined by the angle at which the top is pointing relative to a fixed reference point, such as up.
Use cases
Section titled “Use cases”- Meteorology
- Physics
- Engineering
Notes or pitfalls
Section titled “Notes or pitfalls”- Circular variables have a periodic nature that must be accounted for in analysis (examples: every 360 degrees for wind direction; every 180 degrees for some orientations).
- Use circular statistics and circular distributions rather than linear methods when analyzing these variables.
Related terms
Section titled “Related terms”- Circular statistics
- Circular distributions
- von Mises distribution
- Wrapped Cauchy distribution
- Wrapped normal distribution
- Angle
- Direction