A general family implementing distributional regression for a circular
(angular) response \(y\) in radians under the wrapped Cauchy law
$$f(y) = \frac{1 - \rho^2}{2\pi\,(1 + \rho^2 - 2\rho\cos(y - \mu))},$$
with both the mean direction \(\mu\) and the mean resultant length
\(\rho\) getting their own linear predictor, each of which may contain
smooth terms. Used with mgcv::gam and a list of two formulas: the
first names the response and models \(\mu\), the second models
\(\mathrm{logit}(\rho)\).
Usage
wclss(link = list("tanhalf", "logit"))Value
An object of class c("general.family", "extended.family", "family")
for use with gam (or its front end circ_gam).
Details
The wrapped Cauchy is the heavy-tailed counterpart of the von Mises
(vmlss): sharply peaked with fat circular tails, so it is
the more robust choice when the data contain angular outliers. Its
trigonometric moments are simply \(\rho^p\), which gives clean
residual conventions: Pearson residuals standardize by
\(\mathrm{Var}\{\sin(y-\mu)\} = (1-\rho^2)/2\).
The mean direction uses the Fisher-Lee tan-half link
(\(\mu \in (-\pi, \pi)\), antipode unrepresentable, winding number
zero – see pnlss when the mean direction must wind).
Log-likelihood derivatives up to fourth order are implemented, so the family
supports full Newton REML (method = "REML"); optimizer = "efs"
also works.
Internally the density denominator is computed in the
cancellation-free form \((1-\rho)^2 + 4\rho\sin^2((y-\mu)/2)\) so the
log-likelihood stays exact as \(\rho \to 1\).