Cloud-Plots for Spherical Data With Applications to Glaucoma and Gait Cyclograms

In several biomedical contexts, the need for diagnostic analyses of data that is a set of directions in 2, 3, or higher dimensions arises, as for instance, when considering the relative movement of body parts like the limbs, or looking at anatomical orientations. Commonly used statistical tools, including graphical tools developed for data on the real line, or for dealing with multidimensional data taking values on Euclidean spaces, are quite inappropriate for analyzing such data, due to major differences in their underlying geometry. Taking clues from the Box-plot, which was introduced by Tukey [1977] for linear data, here we propose what we label a "Cloud-plot", which is a visualization tool and diagnostic framework for plotting data on 3-dimensional directions, which take values on the surface of a unit sphere, the ${𝕊}^2$ . The analysis is based on "geodesic distances" between any 2 observations, which are the shortest paths between these two points on the surface of the sphere. Using these distances, we construct robust analogues of the spherical median and the quartiles. This is followed by a definition of the "Spherical Median Absolute Deviation" ( $SMAD$ ). We use these quantities to define outliers, and the performance of the proposed method is evaluated through a comprehensive simulation study that compares true and false positive rates given by several existing discordancy tests. Our extensive results indicate that the approach based on our Cloud-plot provides a very successful and competitive performance for detecting outliers, and remains stable across a range of sample sizes and contamination settings. The cloud-plot is illustrated using two interesting medical applications viz. (i) the identification of structural biomarkers of glaucoma, and (ii) the analysis of functional joint instability in gait cyclograms. These examples demonstrate the potential of the Cloud-plots to detect directional anomalies that may not be captured otherwise by tools commonly used on Euclidean spaces.