Weight-aware Gaussian location-scale family

A Gaussian location-scale family for distributional regression, modelling a real-valued response \(y\) with $$y \sim N(\mu, \sigma^2),$$ where the mean \(\mu\) and the precision \(\tau = 1/\sigma\) each get their own linear predictor, which may contain smooth terms. It is a weight-aware, metadata-carrying adaptation of mgcv's gaulss: unlike gaulss it honours prior weights (needed for a weighted MLE and for EM mixtures), and it carries the circlss parameter metadata so circ_gam treats it as a first-class location-scale family with named, response-scale output.

Usage

gausslss(link = list("identity", "logb"), b = 0.01)

Arguments

link

Two-element list of link names for the mean and the precision, following gaulss: "identity" (or "log", "inverse", "sqrt") for the mean and "logb" for the precision.

b

The logb link's offset, as in gaulss.

Value

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

Details

In the circlss regression trio this is the linear-circular (l~c) family: a real-valued response over a circular covariate – a level that varies around a cycle (time of day, season, phase) – fitted with a cyclic smooth,

circ_gam(list(y ~ s(phi, bs = "cc"), ~ s(phi, bs = "cc")),
         family = gausslss())

circ_gam then places the fitted mean on the "can" (an upright cylinder: the circular covariate wraps the ring, the linear response is the height).

The parameterization follows gaulss exactly: the mean uses an identity link and the second parameter is the precision \(\tau = 1/\sigma\) on the logb link, so the second fitted column is \(1/\sigma\), not the standard deviation. Log-likelihood derivatives up to fourth order are implemented, so the family supports full Newton REML (method = "REML"); optimizer = "efs" also works. At unit weights the fit matches gaulss; integer prior weights reproduce a row-replicated fit.

This family adapts GPL-licensed code from mgcv; see the package's inst/COPYRIGHTS.

See also

gammalss, circ_gam, gaulss

Examples

library(mgcv)
set.seed(1); n <- 300
phi <- runif(n, -pi, pi)                       # circular covariate (radians)
y <- 2 + 1.5 * sin(phi) + 0.8 * cos(2 * phi) + rnorm(n) * 0.3
b <- circ_gam(list(y ~ s(phi, bs = "cc"), ~ s(phi, bs = "cc")),
              data = data.frame(y, phi), family = gausslss())
head(predict(b, type = "response"))            # columns named mu, tau
plot(b, view = "both")                         # the l~c "can" + the mean panel