Scaled von Mises–Fisher Distributions and Regression Models for Paleomagnetic Directional Data

Abstract

We propose a new distribution for analyzing paleomagnetic directional data, that is, a novel transformation of the von Mises–Fisher distribution. The new distribution has ellipse-like symmetry, as does the Kent distribution; however, unlike the Kent distribution the normalizing constant in the new density is easy to compute and estimation of the shape parameters is straightforward. To accommodate outliers, the model also incorporates an additional shape parameter, which controls the tail-weight of the distribution. We also develop a general regression model framework that allows both the mean direction and the shape parameters of the error distribution to depend on covariates. The proposed regression procedure is shown to be equivariant with respect to the choice of coordinate system for the directional response. To illustrate, we analyses paleomagnetic directional data from the GEOMAGIA50.v3 database. We predict the mean direction at various geological time points and show that there is significant heteroscedasticity present. It is envisaged that the regression structures and error distribution proposed here will also prove useful when covariate information is available with (i) other types of directional response data; and (ii) square-root transformed compositional data of general dimension. Supplementary materials for this article are available online. Code submitted with this article was checked by an Associate Editor for Reproducibility and is available as an online supplement.