stat.MEOct 26, 2026
SaveDirectional Statistics on Circles and Spheres: A New Angle on Data
Organizations: Bristol Myers Squibb, 5 Giralda Farms, Madison, NJ 07940, USA
Abstract
Two compass bearings near north can have an ordinary arithmetic mean pointing south: the difficulty is that angles live on a circle, not a straight line. Using real wind observations, diffusion-MRI orientations, and geometric examples from machine learning, this article introduces circular and axial averages, directional variability, and probability models on circles and spheres. The guiding principle is to identify the geometry and symmetries of the observations before choosing a statistical method.
Figures & tables
| Area | Example | Geometric issue |
|---|---|---|
| Weather / ocean | Wind, current, wave direction | Circular wraparound; multimodality |
| Ecology / chronobiology | Migration bearings; circadian phase | Preferred direction; 24-hour periodicity |
| Microscopy / medicine | Cell, collagen, or fiber orientation | Often axial: |
| Geology / astronomy | Faults, paleomagnetism, sky directions | Circular, spherical, or axial |
| Robotics / computer vision | Heading, attitude, surface normals | Periodic angles and 3-D orientation |
| Machine learning / AI | Unit-normalized features and embeddings | High-dimensional sphere; angular similarity |
Table 1: Directional data across science and technology. The appropriate analysis depends on geometry, symmetry, and the scientific meaning of the measurement; representative applications are discussed in standard directional-statistics, biology, diffusion-MRI, and spherical-clustering references [ 1 , 3 , 4 , 5 ] .
Figure 1: A geometry-first decision guide. First check periodicity, then whether vector magnitude matters, and finally whether opposite vectors describe the same physical state. These are diagnostic questions, not a unique model-selection rule.
Figure 2: Real hourly wind directions at JFK Airport on 15 January 2013. (A) A straight scale places the two equivalent labels for north, and , at opposite ends. (B) An outlined wind rose restores their adjacency; the circular mean is about and . (C) Replacing the label by the equivalent label changes the arithmetic mean from to , while the circular mean remains unchanged. In panel C, the solid, dashed, and dotted arrows represent , , and , respectively. Data: nycflights13::weather [ 6 ] .
Figure 3: Properly normalized von Mises densities. Increasing concentration sharpens the peak around zero.
Figure 4: From an MRI image to an axial statistical problem. (A) A vector-drawn, schematic MRI-like slice illustrates the voxelized image on which diffusion measurements are made; diffusion MRI acquires many volumes under different gradient directions. (B) Within one voxel, the directional signal measurements are used to fit a local diffusion tensor, whose long principal axis summarizes orientation. (C) The orientations and describe the same physical axis; treating antipodal observations as ordinary arrows can therefore cause artificial cancellation. All panels are schematic rather than patient data [ 4 , 9 ] .
Figure 5: An illustrative document embedding: cosine similarity between A and B is , corresponding to an angle of approximately . The vectors are constructed, not outputs of a trained model.