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# Circular Correlation Coefficient

Module by: Evaristo Rojas Mayoral. E-mail the author

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) α α m 2 for any direction αα, where mm 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:

r = i = 1 n - 1 j = i + 1 n sin a i - a j sin b i - b j i = 1 n - 1 j = i + 1 n sin 2 a i - a j i = 1 n - 1 j = i + 1 n sin 2 b i - b j r = i = 1 n - 1 j = i + 1 n sin a i - a j sin b i - b j i = 1 n - 1 j = i + 1 n sin 2 a i - a j i = 1 n - 1 j = i + 1 n sin 2 b i - b j
(1)

where rr is the CCC, nn is the number of data points, aa is a angular variable and bb is another angular variable.

The absolute value of rr 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 rr the relationship is of the type a=b+γ a b γ , while for a negative rr, the association has the form a=b+γ a b γ for a fixed, arbitrary direction γγ.

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.

## References

• FISHER, N. & LEE, A. (1983). A correlation coefficient for circular data.
Biometrika. 70 (2), 327-332. DOI: 10.1093/biomet/70.2.327

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