Pewsey 2021 - Directional Statistics - Spherical, Toroidal and Other Topics

Summary

Beyond the circle: tori (pairs of angles), cylinders (angle + linear), spheres (von Mises-Fisher, Fisher-Bingham, Kent), and the associated machinery for density estimation, dimension reduction, classification, spatial processes and software. Copula-like constructions (“circulas”) bind marginal circular distributions into toroidal ones.

Overview

This note collects the parts of the survey that go beyond univariate circular models and regression: Secs. 3.2-3.4, 7.1, 8, 9, 11, 12, 13 and 14.

Main Content

Toroidal models (3.2)

Bivariate von Mises (Eq. 8, Mardia 1975)

$f({\theta }_1,{\theta }_2)\propto \exp \{{\kappa }_1\cos ({\theta }_1-{\mu }_1)+{\kappa }_2\cos ({\theta }_2-{\mu }_2)+(\cos ({\theta }_1-{\mu }_1),\sin ({\theta }_1-{\mu }_1))\mathbf{A}(\cos ({\theta }_2-{\mu }_2),\sin ({\theta }_2-{\mu }_2){)}^{\top }\}$, eight parameters (three more than the minimum five), maximum-entropy/exponential family, normalising constant a doubly infinite sum (Mardia 2010). Five-parameter submodels: sine (Singh et al. 2002), cosine (Mardia et al. 2007), hybrid (Kent et al. 2008); conditionals are vM but marginals are generally not (can be bimodal). Multivariate sine vM on $T^d$ (Mardia et al. 2008): conditionals vM, so pseudo-likelihood applies.

Binding densities and circulas (Eqs. 9-10)

Wehrly and Johnson (1980): $f({\theta }_1,{\theta }_2)=2\pi f_{{\Theta }_1}({\theta }_1)f_{{\Theta }_2}({\theta }_2)f_{\Omega }(2\pi [F_{{\Theta }_2}({\theta }_2)-q{F}_{{\Theta }_1}({\theta }_1)])$ with circular “binding” pdf $f_{\Omega }$ and $q=\pm 1$ (sign of dependence). Copula form (Shieh and Johnson 2005): $f=f_{{\Theta }_1}{f}_{{\Theta }_2}c(F_{{\Theta }_1},F_{{\Theta }_2})$. Jones et al. (2015): $f=4{\pi }^2{f}_{{\Theta }_1}{f}_{{\Theta }_2}{c}^{\circ }(2\pi F_{{\Theta }_1},2\pi F_{{\Theta }_2})$ where $c^{\circ }$ is a circula density (circular-uniform arguments, doubly periodic); (9) corresponds to $c^{\circ }({\varphi }_1,{\varphi }_2)=\frac{1}{2\pi }{f}_{\Omega }({\varphi }_2-q{\varphi }_1)$. Kato et al. (2018): Fourier-series circulas. Bivariate WC (Kato and Pewsey 2015) has all marginals and conditionals WC. Compare Factor Copulas - Overview for the linear-copula analogue.

Cylindrical models (3.3)

Mardia and Sutton (1978): conditioning a trivariate normal gives a six-parameter model with vM marginal for $\Theta$ and $X\mid \Theta =\theta$ normal (closed-form ML). Kato and Shimizu (2008): eight-parameter generalized vM extension. Marginal specification (Eq. 11): $f(\theta ,x)=2\pi f_{\Theta }(\theta )f_X(x)f_{\Omega }(2\pi [F_{\Theta }(\theta )-q{F}_X(x)])$ (Johnson and Wehrly 1978; wind-energy and ocean-engineering applications). Mastrantonio (2018): joint projected and skew-normal for multivariate cylindrical data.

Spherical models (3.4)

von Mises-Fisher and Fisher-Bingham (Eqs. 12-13)

$f_{\mathbf{X}}(\mathbf{x};\mathbf{\mu },\kappa )=\frac{{\kappa }^{(d-1)/2}}{(2\pi {)}^{(d+1)/2}{I}_{(d-1)/2}(\kappa )}\exp \{\kappa {\mathbf{x}}^{\top }\mathbf{\mu }\}$ on $S^d$; Fisher-Bingham: $f\propto \exp \{\kappa {\mathbf{x}}^{\top }\mathbf{\mu }+{\mathbf{x}}^{\top }\mathbf{Ax}\}$ ($\mathbf{A}$ symmetric $(d+1)\times (d+1)$) contains Kent, Fisher-Watson, Bingham-Mardia, Bingham, Watson (the last two for axial data). Normalising constants: Kume and Sei (2018). Simulation: Kent et al. (2018).

Other models: small-sphere distributions (Kim et al. 2019); spherical rotational-symmetry skew models (Ley and Verdebout 2017b; Jupp et al. 2016); projected normal family (Hernandez-Stumpfhauser et al. 2017); projected gamma on the positive orthant (Núñez-Antonio and Geneyro 2020); directional log-spline densities $f(\mathbf{x})=\exp \{c_0+{\sum }_j{c}_j{R}_d(\mathbf{x};{\mathbf{k}}_j)\}$ (Eq. 14, Ferreira et al. 2008).

Density estimation (7.1)

Kernel estimators (Eq. 17), series-based (needlets, Fourier) and Bayesian (Dirichlet process mixtures of vMF) approaches; see Pewsey 2021 - Circular Regression and Serial Dependence for the KDE formula.

Dimension reduction and clustering (8-9)

Manifold PCA: extrinsic (tangent space) vs intrinsic (geodesic), forward vs backward (Damon and Marron 2014). Principal geodesic analysis (Fletcher et al. 2004) centres at the intrinsic mean; its optimisation is hard (Sommer et al. 2014), so tangent PCA is an approximation. Classification: chordal-distance rules (SenGupta and Roy 2005), depth-vs-depth, Watson/vMF Bayes rules (Figueiredo and Gomes 2006; 2009), Dirichlet process mixture of vMF classifier (Bhattacharya and Dunson 2012), naive Bayes with vMF (López-Cruz et al. 2015). Clustering: mixtures with spherical components.

Spatial and spatio-temporal (11)

Hierarchical Bayesian models for directional fields fitted by MCMC: wrapped normal CAR for hurricane winds (Modlin et al. 2012), wrapped Gaussian spatial process for wave directions (Jona-Lasinio et al. 2012), projected normal processes for direction and height (Wang and Gelfand 2014), wrapped skew-normal process (Mastrantonio et al. 2016a), space-time extensions (Mastrantonio et al. 2016b); hidden-Markov models for sea-surface patterns (Lagona et al. 2015b) and hidden Markov random fields for cylindrical series (Lagona and Picone 2016; Ranalli et al. 2018).

Other topics (12)

Statistical depth (angular simplicial/Tukey depth; Agostinelli and Romanazzi 2013), design of experiments (Otieno and Anderson-Cook 2012; optimal designs by Dette and co-authors), order-restricted analysis, outlier detection, and compositional data analysis (12.5): Aitchison (1986) log-ratios vs Stephens (1982) square-root map from the simplex to the sphere $S^d$, developed by Scealy and Welsh (2011, 2014a,b, 2017); Cuesta-Albertos et al. (2009) link the two for uniformity testing. See Greenacre 2023 - Aitchison CoDA 40 Years On - Overview and Tsagris 2018 - Dirichlet Regression with Zeros - Discussion and Alternatives.

Software (13) and conclusions (14)

R: circular, Directional (general), bpDir, season, bReeze (plots), rgl, plot3D; other languages: CircStat (MATLAB), PyCircStat (Python), libDirectional, Mocapy++, geomstats. Expected future: more flexible models, high-dimensional/mixed-type data, Bayesian, nonparametric and resampling methods. A BibTeX file of over 1,700 references is at github.com/egarpor/DirectionalStatsBib.

Examples

Square-root map for shares

If $\mathbf{p}\in S_{\text{simplex}}^{D-1}$ then $\mathbf{x}=(\sqrt{p_1},\dots ,\sqrt{p_D})\in S_{\text{sphere}}^{D-1}$, with zeros landing on the sphere’s boundary (the orthant) rather than being forbidden — this is why Scealy and Welsh’s Kent regression handles zeros without imputation.

Connections

Compositional alternative to log-ratios; torus models relate to copulas (Factor Copulas - Overview).

See Also

Pewsey 2021 - Directional Statistics - Overview