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Recent advances in directional statistics

Arthur Pewsey, Eduardo García-Portugués

arXiv:2005.06889v3stat.ME

TL;DR

Directional statistics addresses data on compact Riemannian manifolds where standard Euclidean methods can be misleading. This paper reviews developments since Mardia and Jupp (1999), spanning exploratory analysis, models, inference, applications, software, and future directions.

  • Problem

    Directional data on compact manifolds such as circles and spheres require methods adapted to their non-Euclidean supports because standard statistical methods can be misleading.

  • Method

    The paper provides a concise, comprehensive review of developments in directional statistics since Mardia and Jupp (1999), covering methods, applications, software, and future research.

  • Results

    The review organizes recent advances across exploratory analysis, distributional models, inference, testing, regression, curve estimation, dimension reduction, classification, clustering, and temporal and spatial modelling.

  • Takeaways & Limitations

    Future development is expected to emphasize more flexible models, high-dimensional and complex directional data, Bayesian, nonparametric, and resampling methods, alongside software implementation.

  • Takeaways & Limitations

    Principal geodesic analysis on spheres can force great circles to cross at the data centre and fail to describe some forms of variation.

Abstract

from arXiv · show

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.

1 Introduction

Directional statistics addresses data supported on manifolds such as circles, spheres, tori, and cylinders, where standard Euclidean methods can be inappropriate. This review surveys developments since Mardia and Jupp (1999), spanning methods, applications, software, and future directions.

  • Data types and supports: Directional data include circular, spherical, toroidal, and cylindrical observations supported on manifolds such as S1, Sd, and S1 × R.Examples include angles, star positions, paired wind directions, and joint wind direction and velocity.
  • Data types and supports: Standard statistical methods designed for Euclidean spaces can be inappropriate or misleading for data on compact manifolds.The paper motivates specialized directional methods because the geometry of the support differs from that of Rd.
  • Review scope: The review defines recent developments as literature since Mardia and Jupp (1999), aiming for a concise and broadly exhaustive account within length constraints.It directs readers to original sources for more detailed treatment and acknowledges that some developments may have been omitted.
  • Applications: Recent directional-statistics research has been stimulated by applications in bioinformatics, astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, machine learning, and environmental modelling.The cited applications include wildfire and sea-condition modelling.
  • Review contents: The paper covers exploratory analysis, distributional models, inference, testing, regression, nonparametric estimation, dimension reduction, classification, clustering, and temporal and spatial modelling.It also reviews available software for directional data analysis.

3 Distributional models

Recent circular and spherical modelling developments extend classical constructions into more flexible, multimodal, and skewed families, while toroidal and other manifold-supported models use analogous transformation-based approaches.

  • Circular models: Six established constructions generate circular models: wrapping, projection, perturbation, conditioning, diffusion, and characterisations such as maximum likelihood or maximum entropy.The von Mises model can be derived using five of these constructions.
  • Circular models: Wrapping maps linear variables modulo 2π, preserving their characteristic-function values as circular Fourier coefficients or trigonometric moments.The wrapped density is generally an infinite sum, although the wrapped Cauchy is an important closed-form exception.
  • Circular models: Projection and perturbation provide alternative circular constructions, with projection supporting asymmetric or bimodal projected-normal densities and perturbation introducing controlled skewness.Sine-skewed families retain the base density’s normalising constant but model only moderate departures from symmetry and are not necessarily unimodal.
  • Circular models: Recent circular families broaden flexibility through overarching symmetric unimodal classes, trigonometric-moment extensions, Möbius transformations, Brownian-motion variants, and transformations of argument.Inverse Batschelet distributions can adopt Laplace-like shapes, unlike smooth unimodal Kato–Jones models.
  • Circular models: Finite mixtures, especially mixtures with von Mises components, remain the most popular approach for modelling multimodal circular data.Mixture models with von Mises components have received renewed attention.
  • Spherical models: Spherical modelling includes Fisher–Bingham families, efficient simulation and normalising-constant methods, and rotationally symmetric extensions such as multimodal spherical logistic distributions.The Fisher–Bingham normalising constant is challenging to evaluate, while holonomic-gradient and acceptance-rejection methods address evaluation and simulation.

4 General approaches to inference

Inference for directional models remains predominantly frequentist and likelihood-based, but recent work expands rank-based, high-concentration, Bayesian, robust, and manifold-mean methods while exposing non-Euclidean asymptotic caveats.

  • Classical and asymptotic inference: Frequentist inference for directional models commonly uses trigonometric moments or likelihood, with maximum-likelihood estimators in full exponential families coinciding with moment estimators.Closed-form maximum-likelihood estimators exist for models including von Mises and some cylindrical distributions.
  • Classical and asymptotic inference: Large-sample likelihood inference generally relies on regularity conditions and asymptotic normality, while the delta method yields asymptotic distributions for circular location, concentration, skewness, and kurtosis measures.Boundary parameters can require special treatment.
  • Modern inferential approaches: Le Cam’s local asymptotic normality framework has been adapted to directional problems, including optimal rank-based estimators and inference for spherical location under high concentration.These developments extend asymptotic decision-theoretic methods to rotationally symmetric spherical distributions.
  • Modern inferential approaches: Bayesian directional inference has expanded through MCMC methods on embedded Riemannian manifolds and applications to von Mises, wrapped Cauchy, von Mises–Fisher, and mixture models.A general manifold MCMC approach was illustrated for the Fisher–Bingham distribution.
  • Modern inferential approaches: Robust estimators have been developed for von Mises, wrapped normal, von Mises–Fisher, and other circular distributions.The cited work covers both individual models and broader series of circular-distribution estimators.
  • Manifold-specific inference: Intrinsic and extrinsic means have manifold-specific asymptotics, including smeariness on circles and spheres that can slow rates below n^-1/2.Under smeariness, quantile-based tests may be inappropriate, whereas suitable bootstrap tests remain valid.
  • Manifold-specific inference: The choice of reference system for circular distributions can affect inference, motivating explicit examination of reference-system sensitivity.This issue was explored by Mastrantonio et al. (2019).

5 Hypothesis testing

This section reviews directional hypothesis tests for uniformity, symmetry, parameter hypotheses, and goodness-of-fit across circular, spherical, and other manifold-valued data. Calibration commonly uses asymptotic theory, with resampling methods especially relevant for small or moderate samples.

  • Uniformity: Uniformity, meaning no preferred direction, is the central dividing hypothesis in directional statistics.The review covers tests on circular and spherical supports, including high-dimensional settings.
  • Uniformity: Sobolev tests form the broadest class of uniformity tests on S^d, with weighting choices controlling local optimality, consistency, and power against alternatives.Rayleigh and Bingham tests arise from particular choices of the Sobolev weights.
  • Uniformity: Data-driven truncation simplifies Sobolev-test computation and asymptotic calibration, while increased truncation on S1 and S2 yields a normal rather than weighted chi-squared limit.The normal-limit approach was proposed by Jammalamadaka et al. (2020).
  • Uniformity: Recent work extends uniformity testing through projected empirical cdfs, spherical harmonics, non-Sobolev statistics, Bayesian procedures, and simulation comparisons across supports and sampling conditions.These developments include tests for grouped circular data and alternatives based on vMF, Watson, and related models.
  • Other testing problems: The broader testing literature addresses symmetry, rotational symmetry, model parameters, and goodness-of-fit, often using likelihood, rank, bootstrap, permutation, or transformation-based procedures.For circular data, fully specified goodness-of-fit can be transformed into circular uniformity testing; other manifolds lack a canonical transformation to uniformity.

6 Correlation and regression

Recent directional regression work covers relationships among circular, linear, spherical, cylindrical, and toroidal variables. Parametric link-based, projected, Bayesian, tree-based, and random-effects approaches address increasingly varied response and predictor structures.

  • Regression types: Directional regression encompasses circular-circular, circular-linear, linear-circular, spherical, cylindrical, and toroidal response-predictor relationships.The review distinguishes parametric regression from nonparametric regression discussed separately.
  • Circular-circular regression: Circular-circular models use an inverse-tangent link in which the first term gives the conditional mean direction and the error is circular.Several established models are special cases of this general formulation, with different circular error distributions.
  • Circular-linear regression: Circular-linear regression commonly uses the 2 tan^-1 link, mapping linear predictors to a circular conditional mean, but maximum-likelihood estimation can face practical difficulties.Semi-parametric Bayesian and covariance-analysis approaches were proposed to address related inferential issues.
  • Alternative models: Alternative regression strategies include projected-normal models, tree-based prediction, hierarchical Bayesian models for bimodal repeated measures, and random-effects models for clustered circular data.These approaches avoid or extend standard link-based formulations in different settings.
  • Spherical regression: Spherical regression uses Möbius transformations, stereographic projections, link functions, polynomial models, projective transformations, and small-circle fitting for manifold-valued responses.These methods include models for S^2-to-S^2 and general S^d-to-S^d regression.

7 Nonparametric curve estimation

Nonparametric curve estimation for directional data has expanded from kernel density estimation to regression, splines, needlets, deconvolution, Bayesian mixtures, and other manifold-adapted smoothers. The methods address circular, spherical, toroidal, product-manifold, and mixed-support data.

  • Density estimation: Kernel density estimation has been extended to circular cdfs, conditional densities, compact Riemannian manifolds, product supports, and sequentially updated estimators.These extensions retain the basic bandwidth-and-kernel framework while increasing geometric scope.
  • Alternative density estimators: Alternatives to standard KDE include nearest-neighbour, tangent-space, trigonometric-moment, Fourier, spherical-harmonic, needlet, and deconvolution estimators.Some moment- and Fourier-based estimators can become negative, whereas constraints or alternative constructions can address that issue.
  • Regression estimation: Nonparametric regression methods include Nadaraya-Watson and local-polynomial estimators using circular or spherical kernels, tangent-normal expansions, local means, and product kernels.These methods cover mixed circular-linear predictors and responses as well as S^d-to-S^q regression.
  • Other nonparametric methods: Additional approaches include needlet regression, thin-plate and Bayesian splines, quantile regression, circular processes, and Dirichlet-process mixture models for directional densities.Applications include functional data, neural firing rates, and pairs of dihedral angles on the torus.

8 Dimension reduction methods

Dimension reduction for directional data adapts PCA and nonlinear embedding to manifold geometry, with methods differing by intrinsic or extrinsic construction and forward or backward computation. Recent work also develops manifold-specific approaches for spheres and tori.

  • General manifold methods: Manifold PCA methods are organized by extrinsic versus intrinsic geometry and forward versus backward construction.These distinctions apply to manifolds such as spheres and tori.
  • General manifold methods: Principal geodesic analysis generalizes PCA through geodesics centered at the intrinsic mean, but its optimization is complex and its geodesics impose restrictive variation patterns.On S^2, great circles are forced to cross at the intrinsic mean and cannot represent certain forms of variation.
  • General manifold methods: GPCA, horizontal component analysis, PSSA, barycentric subspace analysis, probabilistic PGA, anisotropic normal models, and principal flows provide alternatives to standard PGA.These methods relax crossing constraints, change the subspace construction, or model variation through local tangent covariance.
  • Sphere-specific methods: Principal arc analysis and principal nested spheres use small circles and tangent-normal decompositions to improve flexibility beyond geodesic approaches on spheres.PNS generalizes the sphere-specific idea from S^2 to S^d.
  • Toroidal methods: Toroidal PCA methods respond to the pathological behavior of geodesics on tori, using angle transformations, complex representations, circular means, deformations, or model-selected geodesics.T-PCA deforms the torus into a sphere through a data-driven cut, contraction, and reconnection.
  • Nonlinear methods: Nonlinear dimension reduction has been adapted through t-SNE variants with WC or vMF neighbourhoods and multidimensional scaling designed for spherical distances.Applications include mapping 3D-object textures onto spheres and modelling normalized time-warping functions.

9 Classification and clustering

Directional-data classification spans circular, toroidal, cylindrical, spherical, and axial settings, while clustering is dominated by spherical k-means and directional mixture models. Recent work extends these foundations through nonparametric, Bayesian, distributionally flexible, and computationally adaptive methods.

  • Classification: Classification methods cover circular, toroidal, cylindrical, spherical, and axial data using distance rules, likelihood tests, logistic models, Bayesian algorithms, and directional mixture classifiers.Approaches include chordal-distance discrimination, generalized likelihood-ratio tests, KDE, local logistic regression, depth-based classification, and vMF-related models.
  • Clustering: Spherical k-means and finite mixtures with von Mises–Fisher components are the two most popular approaches to clustering data on S^d.Spherical k-means maximizes cosine similarity between observations and cluster centroids, while vMF mixtures provide a model-based alternative.
  • Clustering: Expectation-Maximisation algorithms fit vMF mixtures, with spherical k-means arising as a particular limiting case.Alternative fitting strategies include fuzzy-partition embeddings, variational inference, and Bayesian graphical modelling.
  • Clustering: Clustering flexibility extends through Kent, inverse-stereographic-normal, wrapped-normal, Watson, and Bingham mixture models.These models address spherical and axial data with distributions beyond vMF components.
  • Clustering: Nonparametric and modified partitioning methods include spherical fuzzy and possibilistic c-means, computationally efficient adaptations, and kernel mean shift with intrinsic or extrinsic formulations.Time-varying bandwidths have also been adapted for spherical data.

10 Modelling serial dependence

Serial dependence models for directional data include projected, wrapped, linked, autoregressive, Markov, hidden-Markov, state-space, and filtering approaches. The literature covers circular, cylindrical, toroidal, and spherical processes, with applications including forecasting, protein structures, and biological movement.

  • Circular time series: Circular time-series models include projected normal, wrapped, linked ARMA, and circular autoregressive formulations.Linked processes map real-valued processes to angles through a link function, while wrapped processes reduce real-valued processes modulo 2π.
  • Circular time series: Linked autoregressive models use transformed lagged angles and can be extended with covariates and distributions beyond the von Mises model.Four ARMA-based approaches have also been used for short-term forecasting of wind speed and direction.
  • Markov models: Transition densities derived from bivariate circular distributions provide a route to stationary Markov processes.This construction has been applied using sine and cosine bivariate von Mises models and circular regression models.
  • Hidden Markov models: Hidden Markov models provide flexible serial dependence by using mixtures of distributions to represent different underlying regimes.They have been developed for circular, cylindrical, and toroidal series, including protein-backbone dihedral angles.
  • Further developments: Recent alternatives include circular state-space models based on wrapped Gaussian processes, linked processes allowing long-range dependence, continuous-time toroidal diffusions, and recursive filtering algorithms.Filtering analogues of the Kalman filter use vMF and Bingham distributions, while wrapped-normal filtering has been studied on S^1.
  • Nonparametric trends: Nonparametric kernel methods estimate trends in circular time series and normalized symmetric linear estimators address trends in S^d-valued series.These methods broaden serial-dependence analysis beyond parametric time-series models.

11 Spatial and spatio-temporal modelling

Spatial and spatio-temporal directional modelling has advanced through hierarchical Bayesian processes, hidden Markov models, and random-field methods. Applications include winds, waves, environmental regimes, wildfires, animal movement, and spherical random fields.

  • Spatial modelling: Directional spatial modelling has seen major recent advances, especially through hierarchical Bayesian models fitted with MCMC for meteorological data.These models address spatial dependence in directional observations and vector fields.
  • Spatial modelling: Wrapped-normal, wrapped-Gaussian, and projected-normal processes have been used to model hurricane winds, wave directions, and other spatial directional fields.The models differ in their process construction and accommodate spatial dependence in directional quantities.
  • Spatio-temporal modelling: An HMM-based alternative models temporal evolution through time-varying circular-linear patterns generated by latent environmental conditions and fitted by pseudo-likelihood.This approach avoids the specific prior assumptions and ad hoc MCMC fitting required by the referenced Bayesian hierarchical models.
  • Spatio-temporal modelling: Wrapped skew-normal and other extended processes add asymmetric marginal distributions, space-time dependence, and space- and time-varying covariate information.These developments extend wrapped-normal and projected-normal approaches to spatio-temporal data.
  • Spatial segmentation: Hidden Markov random fields support segmentation of latent environmental conditions in cylindrical spatial series, including wildfire spatial segmentation.The wildfire model extends an earlier hidden Markov random-field approach using a mixture model.
  • Animal movement: Animal-orientation models use Bayesian mixtures of random walks and hidden behavioural states to represent movement paths with random step lengths and turning angles.Each movement step and turn is assigned to a random walk associated with a hidden behavioural state.
  • Random fields: Research on spherical random fields includes high-frequency limits, Gaussianity and isotropy tests, needlet-coefficient functionals, matrix-valued covariance functions, and covariance-process construction.Applications include analysis of the cosmic microwave background.
  • Filtering: Cylindrical filtering has been reviewed for spatio-temporal nonlinear filtering and illustrated with battlespace data.

12 Other topics

Other developments address directional depth, experimental design, circular regression and order restrictions, outlier detection, and the relationship between compositional and directional data. These contributions extend directional methodology to inference, design, biological phase analysis, data quality, and simplex-valued observations.

  • Depth methods: Directional depth methods include angular simplicial and Tukey depths, projection-quantile depths, and computationally tractable distance-based depths for location estimation and classification.
  • Experimental design: Optimal designs for linear-spherical regression have been established on S^1, S^2, and S^d with d > 2 using Fourier series and spherical harmonics.
  • Regression and order restrictions: Circular regression methods estimate gene-expression phase angles, infer their relative order, test specified orderings, and handle order restrictions on von Mises mean directions.
  • Outlier detection: Outlier detection procedures for circular data use circular distances, sums of distances, or gaps, while spherical methods use minimum-distance parameter estimation.
  • Outlier detection: Additional outlier tests target cylindrical, simple circular regression, circular time-series, and axial data under Watson-distribution assumptions.
  • Compositional data: Compositional data can be related to directional data through a square-root transformation from the unit simplex to S^d.This alternative to the Aitchison approach has been further developed for compositional analysis.

13 Software

Recent R packages and related tools have substantially broadened software support for directional data across manifolds, models, inferential tasks, and applications.

  • Core R packages: The circular and Directional R packages provide core functionality for circular, toroidal, cylindrical, and spherical data.Both support S1, T2, and S1 × R; Directional additionally includes routines for S^d.
  • Core R packages: The circular package covers descriptive analysis, density estimation, distributions, hypothesis tests, change points, ANOVA, regression, and datasets for S1.Its capabilities include S1-S1 and S1-R regression and functions for simulation and estimation of circular distributions.
  • Core R packages: Directional supports spherical visualization, transformations, density estimation, distributional inference, correlation, regression, ANOVA, classification, and clustering on S^d.Its spherical functionality includes maximum-likelihood estimation and simulation for various spherical distributions.
  • Organization: The software overview organizes packages and functionality according to the themes developed in the paper’s preceding sections.This structure links implementation resources to the review’s broader methodological coverage.
  • Specialized functionality: Additional packages address animal orientation, toroidal diffusions, spatial and spatio-temporal models, spherical random fields, cosmic microwave background data, order restrictions, outliers, and depth.These tools extend software coverage from specialized applications to methodological tasks on S1 and S2.
  • Specialized functionality: The wider R ecosystem provides specialized tools for manifolds, visualization, mixtures, Bayesian models, symmetry tests, nonparametric methods, clustering, and time-series analysis.Examples include RiemBase, rgl, plot3D, movMF, BAMBI, sphunif, rotasym, NPCirc, DirStats, nprotreg, mgcv, shapes, skmeans, and NHMSAR.

14 Conclusions and future developments

The paper presents a comprehensive overview of two decades of directional-statistics developments and identifies likely near-term directions while emphasizing uncertainty about longer-term evolution. Future progress is expected to depend on flexible methods, complex data, software, and new applications.

  • Conclusions: The review aims to provide seasoned and new researchers with a concise, comprehensive, and useful overview of directional-statistics developments over the last two decades.The authors note that many developments arose independently and without coordination across researchers and groups.
  • Future developments: Flexible models and methods for high-dimensional, complex, and mixed-type directional data are identified as probable short-term developments.The authors also highlight Bayesian, nonparametric, and resampling methods as consistent with current trends.
  • Future developments: Progress across the field’s covered areas is expected to evolve in response to new applications and the requirements of Riemannian directional-data supports.The authors frame this as a broad possibility rather than a precise long-term forecast.
  • Software and application: Continued software development is described as crucial for the wider and proper application of new directional-statistics techniques.The paper links implementation resources to the broader adoption of methods adapted to Riemannian supports.

Supplementary materials

The supplementary materials include a BibTeX file containing more than 1700 references related to directional statistics.

  • Reference resource: The DirectionalStats.bib file contains entries for over 1700 references related to directional statistics.The file is available through the paper’s GitHub repository as a resource for researchers.
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