The Circular copula of the coordinates \((x, y)\) of a point chosen at random on the unit circle (Nelsen, 2006, p. 56) is given by $$\mathbf{C}_{\mathrm{CIRC}}(u,v) = \mathbf{M}(u,v) \mathrm{\ for\ }|u-v| > 1/2\mathrm{,}$$ $$\mathbf{C}_{\mathrm{CIRC}}(u,v) = \mathbf{W}(u,v) \mathrm{\ for\ }|u+v-1| > 1/2\mathrm{,\ and}$$ $$\mathbf{C}_{\mathrm{CIRC}}(u,v) = \frac{u+v}{2} - \frac{1}{4} \mathrm{\ otherwise\ }\mathrm{.}$$
The coordinates of the unit circle are given by $$\mathrm{CIRC}(x,y) = \biggl(\frac{\mathrm{cos}\bigl(\pi(u-1)\bigr)+1}{2}, \frac{\mathrm{cos}\bigl(\pi(v-1)\bigr)+1}{2}\biggr)\mathrm{.}$$
CIRCcop(u, v, para=NULL, as.circ=FALSE, ...)Value(s) for the copula are returned.
Nonexceedance probability \(u\) in the \(X\) direction;
Nonexceedance probability \(v\) in the \(Y\) direction;
Optional parameter list argument that can contain the logical as.circ instead;
A logical, if true, to trigger the transformation \(u = 1 - \mathrm{acos}(2x - 1) / \pi\) and \(v = 1 - \mathrm{acos}(2y - 1) / \pi\) to convert \((X,Y)\) coordinates of a uniform unit circle to the \((U,V)\) in nonexceedance probability; and
Additional arguments to pass, if ever needed.
W.H. Asquith
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.