Boxplots and quartile plots for grouped and periodic angular data

TL;DR

This work advances angular data visualization by extending circular boxplots to concentric layouts for group comparisons and by adopting perception-informed boxwidth scaling $\propto 1/\sqrt{d}$. It introduces circular quartile plots for many groups and 3D toroidal displays to capture the periodicity of angular data over time, complemented by an R package CircularBoxplots. Through real-data examples in psychology, genomics, and meteorology, the methods demonstrate improved interpretability for grouped and temporally structured angular distributions. The work highlights perceptual considerations, provides quantitative assessments, and offers scalable tools for comprehensive angular data visualization with practical impact for diverse domains.

Abstract

Angular observations, or observations lying on the unit circle, arise in many disciplines and require special care in their description, analysis, interpretation and visualization. We provide methods to construct concentric circular boxplot displays of distributions of groups of angular data. The use of concentric boxplots brings challenges of visual perception, so we set the boxwidths to be inversely proportional to the square root of their distance from the centre. A perception survey supports this scaled boxwidth choice. For a large number of groups, we propose circular quartile plots. A three-dimensional toroidal display is also implemented for periodic angular distributions. We illustrate our methods on datasets in (1) psychology, to display motor resonance under different conditions, (2) genomics, to understand the distribution of peak phases for ancillary clock genes, and (3) meteorology and wind turbine power generation, to study the changing and periodic distribution of wind direction over the course of a year.

Figures (8)

• Figure 1: Circular boxplots of samples from three different two-component mixture of von Mises distributions. The mixing components have mean directions $95^\circ$ and $230^\circ$, and concentration parameters $5.0$ and $2.5$, and mixing proportions $\pi_1$ and $\pi_2$, with values as specified.
• Figure 2: (a, b) Two grouped circular boxplots for two groups of angular observations (A for green and B for orange). In each plot, both groups have the same circular spread ($\delta_A\approx\delta_B$) but a different mean direction. The mean directions of (a) are rotated clockwise by $144^\circ$ in (b). Further, in (a), the two boxplots have the same boxwidth while in (b) the boxwidths are inversely proportional to the square root of their distance from the centre of the circle. (c) Results of an anonymous survey where respondents were asked to compare the circular spreads of the distributions of the two groups in each of the two plots.
• Figure 3: Grouped circular boxplot of the three-groups data in Figure \ref{['fig1']}. We see that the medians and spreads of the three groups are easily compared (and distinct).
• Figure 4: Grouped circular boxplot of the motor resonance data. We notice that the implicit group is the most out-of-phase, with observations spanning a large arc around the plot, followed by explicit, then semi-implicit. The implicit group also has the most spread among groups.
• Figure 5: Distribution of peak phases for ancillary clock genes cycled in ad libitum and time-restricted feeding of mice.