# Circular correlation coefficient, college study notes - A circular correlation coefficient (ccc) measures the association between two angular variables, Study notes for Physics. Michigan Technological University (MI)

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Connexions module: m22974 1

Circular Correlation Coefficient ∗

Evaristo Rojas Mayoral

This work is produced by The Connexions Project and licensed under the

Abstract

A Circular Correlation Coecient (CCC) measures the association between two angular variables.

A correlation coecient is a number that describes some characteristics of the relationship between two variables. There are several correlation coecients, each used to analyze a particular data type.

A Circular Correlation Coecient (CCC) measures the association between two angular variables. Angular variables, whose values are directions, are expressed in radians (or degrees) and they are character- ized by α = α+m (2π) for any direction α, where m is any integer.

For example, to study the relationship between the direction of a marine current and the direction of the wind on the sea surface at some point, it would be convenient to use a CCC.

The CCC proposed by Fisher and Lee (1983) is dened as:

r =

∑n−1 i=1

∑n j=i+1 sin (ai − aj) sin (bi − bj)√∑n−1

i=1

∑n j=i+1 sin

2 (ai − aj) ∑n−1

i=1

∑n j=i+1 sin

2 (bi − bj) (1)

where r is the CCC, n is the number of data points, a is a angular variable and b is another angular variable. The absolute value of r takes on values within the interval from zero to one, where zero indicates that

there is no relationship between the variables and one represents the strongest association possible. For positive r the relationship is of the type a = b + γ, while for a negative r, the association has the form a = −b+ γ for a xed, arbitrary direction γ.

If you need to analyze the relationship between two angular variables, I recommend that you use this CCC, which is easily calculated using my circcorrcoef MATLAB function.