C-Asscociation

In circular statistics , a c-association describes a correlation between a circular independent (θ) and a linear dependent variable y with the following properties:

• y has exactly one maximum and one minimum for θ in [0…2π] (full circle)
• the expected values and the slopes (first derivative) for y at θ = 0 and θ = 2π are identical, that is, the function is periodic.

Then the simplest function between y and θ is: ${\hat {y}}=a+b\cos(\theta -\theta _{0})=a+b\cos(\theta )+c\sin(\theta )$, where ${\hat {y}}$ is the calculated value for y, and $\theta _{0}$ the acrophase (angle where ${\hat {y}}$ is maximal).

The degree of c-association is measured by the U-test, which gives the probability of H₀: The variables θ and y are independent ($D_{n}=0$) against H₁: The variables θ and y are c-associated ($D_{n}>0$). $D_{n}$ is a quantity in [0…1], with values near 0 suggesting that there is no C-association. It can be considered a correlation coefficient.

References

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