Asymmetric Jones-Pewsey location-concentration-shape-asymmetry family

A general family implementing distributional regression for a circular (angular) response $y$ in radians under the asymmetric Jones-Pewsey law $$f(y) = c(\kappa, \psi, \nu)\,\bigl(\cosh(\kappa\psi) + \sinh(\kappa\psi)\cos g\bigr)^{1/\psi}, \qquad g = \phi + \nu\cos\phi,\ \ \phi = y - \xi,$$ the Jones-Pewsey family (jplss) with its angle warped forward by $g(\phi) = \phi + \nu\cos\phi$ (Abe, Pewsey & Shimizu 2013). It adds an asymmetry axis to the symmetric Jones-Pewsey umbrella: $\nu > 0$ and $\nu < 0$ tilt the density to opposite sides. (For $\psi \to 0$ the Jones-Pewsey kernel is the von Mises $\exp(\kappa\cos g)$.)

Usage

ajplss(link = list("tanhalf", "log", "identity", "tanh"))

Arguments

link: Four-element list of link names, for the location, concentration, shape and asymmetry. Currently only the defaults are available: "tanhalf" for the location $\xi$, "log" for the concentration $\kappa > 0$, "identity" for the shape $\psi \in \mathbb{R}$, and "tanh" for the asymmetry $\nu \in (-1, 1)$.

Value

An object of class c("general.family", "extended.family", "family") for use with gam (or its front end circ_gam).

Details

It has four linear predictors: the location $\xi$, the concentration $\kappa$, the shape $\psi$ and the asymmetry $\nu$ each get their own, any of which may contain smooth terms. Used with mgcv::gam and a list of four formulas; the first names the response and models $\xi$, then $\log\kappa$, then $\psi$, then $\nu$. The shape and asymmetry are most often held global, i.e. fitted intercept-only with ~ 1.

The shape $\psi$ and asymmetry $\nu$ index a family of symmetric and skewed circular laws:

$\nu = 0$: the symmetric Jones-Pewsey (jplss), whose $\psi$ in turn nests the von Mises ($\psi \to 0$), the cardioid ($\psi = 1$) and the wrapped Cauchy ($\psi \to -\infty$);
$\nu \ne 0$: an asymmetric (skewed) law;
$\kappa \to 0$: the circular uniform.

Mode anchor, not mean. As with ssjplss and ibslss, once $\nu \ne 0$ the location $\xi$ is the mode anchor of the asymmetric density, not its mean direction; fitted-direction summaries inherit that reading.

Forward warp – no implicit differentiation. The warp $g(\phi) = \phi + \nu\cos\phi$ is a monotone reparameterization of the angle ($g'(\phi) = 1 - \nu\sin\phi > 0$ for $|\nu| < 1$), so – unlike ibslss's inverse warps – the score needs no implicit differentiation: it reuses the Jones-Pewsey kernel terms evaluated at the warped angle $g$ and chains them through $g$ (with $g_\phi = 1 - \nu\sin\phi$ and $g_\nu = \cos\phi$).

Normalizer and cross terms. Unlike ssjplss's sine-skew factor – which integrates to 1 against the symmetric kernel and so leaves the Jones-Pewsey normalizer unchanged – the forward warp moves the normalizer: $c(\kappa, \psi, \nu)$ is the $u = g(\phi)$ substitution integrating the kernel against the warp Jacobian $1/g'$ over an asymmetry-aware Gauss-Legendre ladder. Consequently the $(\kappa,\nu)$ and $(\psi,\nu)$ cross-derivatives do not vanish (they do for ssjplss); the $\nu$-score and the full cross-parameter Hessian carry the $(\kappa, \psi, \nu)$ log-normalizer moments returned by that quadrature.

Optimizer. Only first- and second-order log-likelihood derivatives are provided, so available.derivs = 0 and the family is fitted by the extended Fellner-Schall optimizer rather than full Newton REML. gam selects optimizer = "efs" automatically; passing it explicitly is recommended so the fitted object is labelled correctly (which gam.check relies on). The shape $\psi$ and asymmetry $\nu$ are weakly identified when the response is diffuse; holding them global (~ 1) and keeping the concentration well identified is the robust default.

The location uses the Fisher-Lee tan-half link ($\xi \in (-\pi, \pi)$, antipode unrepresentable, winding number zero – see pnlss when the location must wind). The asymmetry rides the tanh link (shared with ssjplss, vmftlss and ibslss), bounded to $(-1, 1)$.

References

Jones, M. C. and Pewsey, A. (2005) A family of symmetric distributions on the circle. Journal of the American Statistical Association 100, 1422-1428.

Abe, T., Pewsey, A. and Shimizu, K. (2013) Extending circular distributions through transformation of argument. Annals of the Institute of Statistical Mathematics 65, 833-858.

Batschelet, E. (1981) Circular Statistics in Biology. Academic Press.

Wood, S. N. and Fasiolo, M. (2017) A generalized Fellner-Schall method for smoothing parameter optimization with application to Tweedie location, scale and shape models. Biometrics 73, 1071-1081.

Examples

library(mgcv)
set.seed(1)
n <- 300
x <- runif(n)
xi <- 2 * atan(sin(2 * pi * x))
kappa <- exp(1.0 + 0.4 * cos(2 * pi * x))
psi <- 0.5
nu <- 0.4       # asymmetry
# asymmetric Jones-Pewsey draws by inverse transform on the warped kernel:
# the forward warp g(phi) is closed-form, so gridding phi gives a
# (phi, density) grid to invert -- no root-finding needed for simulation.
phi <- seq(-pi, pi, length.out = 4001)
g <- phi + nu * cos(phi)
dev <- vapply(seq_len(n), function(i) {
  d <- (cosh(kappa[i] * psi) + sinh(kappa[i] * psi) * cos(g))^(1 / psi)
  d[d < 0] <- 0
  cdf <- cumsum((d[-1] + d[-length(d)]) / 2 * diff(phi))
  cdf <- c(0, cdf); cdf <- cdf / cdf[length(cdf)]
  approx(cdf, phi, runif(1), rule = 2)$y
}, numeric(1))
y <- atan2(sin(xi + dev), cos(xi + dev))
# smooth location and concentration; global (intercept-only) shape and asymmetry
b <- gam(list(y ~ s(x), ~ s(x), ~ 1, ~ 1), family = ajplss(), optimizer = "efs")
summary(b)