Visualizing directional dependence on cylinders and tori

Abstract

Directional data arise in many scientific settings where observations are measured on a repeating angular scale. Existing circular plots and joint density displays are useful, but they do not by themselves organize conditional estimands for cylindrical and toroidal sample spaces. We propose a conditional graphical grammar for visualizing directional dependence when an angular variable is paired with either a linear or another angular response. The main displays are circular topography, toroidal topography and conditional ridges. Each is tied to an explicit estimand, such as a conditional density or modal relation. Simulation scenarios and a buoy wind-wave example show how these displays reveal multimodality, seam-crossing structure and dependence that can be obscured by marginal or unfolded views. The methods are implemented in the R package ggcircular, which returns standard ggplot2 objects.