Circular Correlation Coefficient

Summary: A Circular Correlation Coefficient (CCC) measures the association between two angular variables.

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

A Circular Correlation Coefficient (CCC) measures the association between two angular variables. Angular variables, whose values are directions, are expressed in radians (or degrees) and they are characterized 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 defined as:

∑_i=1ⁿ⁻¹∑_j=i+1ⁿsin(a_i,−,a_j)sin(b_i,−,b_j)

\∑_i=1ⁿ⁻¹∑_j=i+1ⁿsin²(a_i,−,a_j)∑_i=1ⁿ⁻¹∑_j=i+1ⁿsin²(b_i,−,b_j)

(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 fixed, 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.