Pewsey 2021 - Directional Statistics - Overview

Summary

Pewsey and García-Portugués (2021, TEST) review developments in directional statistics since Mardia and Jupp (1999): data on circles (angles, times of day), spheres, tori and cylinders. Standard linear methods can be “thoroughly misleading” because supports are compact manifolds. The paper surveys distributions, inference, testing, regression, smoothing, dimension reduction, clustering, time series, spatial models and software.

Overview

Question. What has changed in modelling and inference for directional data since 1999?

Contribution. A concise, near-exhaustive survey with pointers to sources, 14 sections, ~300+ references. Models are no longer insisted to be exponential-family; flexible skewness/peakedness, mixtures, Bayesian and score-matching inference are now common.

Key result (for applied use). The von Mises is derivable five ways and closely approximated by the wrapped normal; regression for angular responses is done through (a) link functions such as $2{\tan }^{-1}$ or (b) projecting a latent bivariate normal, with Bayesian estimation avoiding the ML pathologies of link-based models.

Motivating PyMC examples

E81 “Circular data: directions, times of day and seasons” (Santa Barbara hourly wind 2021-23; Barro Colorado camera traps; Oklahoma quakes as a null) — mean resultant vector, von Mises / wrapped / projected normal, tan-half vs canonical link, wrapped-prior and step-mode failures, harmonic circular predictor — at https://redam94.github.io/pymc-challenges/examples.html. See Pewsey 2021 - Circular Models in PyMC.

Main Content — Source Map

Paper section	Covered in
1 Introduction (S^1, S^2, tori, cylinders, axial data, applications)	this note
2 Exploratory data analysis (rose diagrams, circular boxplot, CircSiZer, WiZer)	Pewsey 2021 - Circular Distributions and Constructions
3.1 Circular models (wrapped, projected, perturbed; vM, WC, cardioid, sine-skewed, Jones-Pewsey, Kato-Jones, Möbius, mixtures, generalized vM)	Pewsey 2021 - Circular Distributions and Constructions
3.2 Toroidal, 3.3 cylindrical, 3.4 spherical models	Pewsey 2021 - Directional Statistics - Spherical, Toroidal and Other Topics
4 General approaches to inference	Pewsey 2021 - Inference, Uniformity Tests and Smeariness
5 Hypothesis testing (uniformity, symmetry, GOF)	Pewsey 2021 - Inference, Uniformity Tests and Smeariness
6 Correlation and regression	Pewsey 2021 - Circular Regression and Serial Dependence
7 Nonparametric curve estimation (KDE, Nadaraya-Watson)	Pewsey 2021 - Circular Regression and Serial Dependence
8-9 Dimension reduction, classification, clustering	Pewsey 2021 - Directional Statistics - Spherical, Toroidal and Other Topics
10 Serial dependence	Pewsey 2021 - Circular Regression and Serial Dependence
11 Spatial and spatio-temporal	Pewsey 2021 - Directional Statistics - Spherical, Toroidal and Other Topics
12 Other topics (depth, experiments, outliers, compositional)	Pewsey 2021 - Directional Statistics - Spherical, Toroidal and Other Topics
13 Software; 14 Conclusions	Pewsey 2021 - Directional Statistics - Spherical, Toroidal and Other Topics
(concept translation)	Pewsey 2021 - Circular Models in PyMC

Relevance to marketing measurement

Time-of-day and day-of-week/season effects (ad exposure hours, purchase timing, seasonal campaign response) are periodic variables; treating hour as linear splits 23:59 from 00:01. Circular distributions and harmonic predictors respect the wrap-around. See also Local Linear Trend and Seasonality.

Connections

Compositional data connect to the sphere through the square-root transform (Greenacre 2023 - Aitchison CoDA 40 Years On - Overview).

See Also

Hilbert Space Gaussian Processes (periodic GPs for the same seasonal effects)