Pewsey 2021 - Circular Regression and Serial Dependence

Summary

Regression taxonomy: circular-circular, circular-linear (angular response), linear-circular (harmonic/trigonometric-polynomial predictors) and spherical responses. For an angular response, link-based models ($2{\tan }^{-1}$) are hard to estimate by ML (Presnell et al. 1998), so projected-normal regression and Bayesian priors are favoured. Circular time series use projected, wrapped, linked-ARMA and circular-AR constructions.

Overview

Notation $Y\text{-}X$: response-covariate. Nonparametric regression is in Sec. 7.2.

Main Content

Circular-circular

Inverse-tangent link model (Eq. 16, Polsen and Taylor 2015)

$\Psi =\mathrm{Arg}\{g_1(\Theta ;\mathbf{\eta })+i{g}_2(\Theta ;\mathbf{\eta })\}+\epsilon (\mathrm{mod}2\pi )$, with conditional mean direction given by the Arg term, $g_1,g_2$ not uniquely identifiable, $\epsilon$ a circular error. Special cases: decentred model (Rivest 1997, vM errors), Möbius models (Downs and Mardia 2002 with vM; Kato et al. 2008 with WC; Kato and Jones 2010). Sarma and Jammalamadaka (1993) bivariate model with trigonometric-polynomial $g_1,g_2$.

Circular-linear (angular response)

Link-function regression (Fisher and Lee 1992)

$E(\Theta \mid \mathbf{X}=\mathbf{x})=2{\tan }^{-1}({\mathbf{\beta }}^{\top }\mathbf{x})$ (the “tan-half” link): maps the origin of $\mathbb{R}$ to angle 0 and both extremes to $\pm \pi$. Difficulties in ML estimation (Presnell et al. 1998); semi-parametric Bayes (George and Ghosh 2006); weakly informative priors (Mulder and Klugkist 2017).

Projected-normal regression (Presnell et al. 1998)

Latent bivariate linear response ${\mathbf{W}}_i\sim N_2({\mathbf{B}}^{\top }{\mathbf{x}}_i,\mathbf{\Sigma })$ and ${\Theta }_i=\mathrm{Arg}(W_{i1}+i{W}_{i2})$. Popular for Bayesian work (Núñez-Antonio et al. 2011; Wang and Gelfand 2013; Hernandez-Stumpfhauser et al. 2016); interpretation of predictor effects: Cremers et al. (2018). Extensions: mixed models (Núñez-Antonio and Gutiérrez-Peña 2014), finite mixtures with correlated random coefficients (Maruotti 2016), hidden Markov heterogeneity (Maruotti et al. 2016).

Other approaches: Lund (2002) tree-based; longitudinal models by estimating equations (Artes et al. 2000), variance components (D’Elia 2001), correlated sine multivariate vM (Lagona 2016, with Bayesian extension Mulder and Klugkist 2017); animal orientation models with vM errors and consensus mean (Rivest et al. 2016); random-effects circular regression (Rivest and Kato 2019).

Linear-circular

Trigonometric (harmonic) regression

Regress a linear response on sums of trigonometric polynomials of the circular covariate (Johnson and Wehrly 1978), by least squares: $E[Y\mid \theta ]={\beta }_0+{\sum }_{m=1}^M\{a_m\cos (m\theta )+b_m\sin (m\theta )\}$ (standard form). Bayesian versions: Bhattacharya and SenGupta (2009a); SenGupta and Bhattacharya (2015). Cylindrical responses: Cremers et al. (2020).

Spherical responses

Chang (1986) first; Downs (2003) Möbius/stereographic with vMF; Hinkle et al. (2014) polynomial manifold-linear; Rosenthal et al. (2014) projective linear transformations; Cornea et al. (2017) semi-parametric intrinsic; Paine et al. (2020) general $S^2$ response with anisotropic errors; Scealy and Wood (2019) $S^d$-${\mathbb{R}}^q$ heteroscedastic scaled vMF.

Nonparametric (Sec. 7)

Spherical KDE and Nadaraya-Watson (Eqs. 17-20)

$\hat{f}(\mathbf{x};h)=\frac{c_L(h)}{n}{\sum }_{j=1}^{n}L\left(\frac{1-{\mathbf{x}}^{\top }{\mathbf{X}}_j}{h^2}\right)$. With vMF kernel $L(r)=e^{-r}$ on $S^1$: $\hat{f}(\theta ;\kappa )=\frac{1}{2\pi I_0(\kappa )n}{\sum }_j\exp \{\kappa \cos (\theta -{\Theta }_j)\}$, $\kappa =1/h^2$. Nadaraya-Watson: $\hat{m}(\theta ;\kappa )=\frac{{\sum }_j{Y}_j\exp \{\kappa \cos (\theta -{\Theta }_j)\}}{{\sum }_j\exp \{\kappa \cos (\theta -{\Theta }_j)\}}$. Using a WC kernel instead of vM worsens performance (Tsuruta and Sagae 2017a). Local polynomial versions: Di Marzio et al. (2009).

Serial dependence (Sec. 10)

Circular time-series constructions

• Projected: ${\Theta }_t=\mathrm{Arg}(X_t+i{Y}_t)$ for a stationary bivariate normal process.
• Wrapped: ${\Theta }_t=X_t\mathrm{mod}2\pi$ (e.g. wrapped AR, Breckling 1989).
• Linked ARMA: ${\Theta }_t=\mu +g(X_t)$ with link $g:\mathbb{R}\to (-\pi ,\pi )$ and $X_t=g^{-1}({\Theta }_t)$ an ARMA(p,q) process.
• CAR (circular autoregressive): ${\Theta }_t\mid {\Theta }_{t-1}={\theta }_{t-1},\dots$ is vM with mean ${\mu }_t=\mu +g[{\alpha }_1{g}^{-1}({\theta }_{t-1}-\mu )+\cdots +{\alpha }_p{g}^{-1}({\theta }_{t-p}-\mu )]$ and concentration $\kappa$.
• Markov processes from bivariate circular densities (Wehrly and Johnson 1980; Hughes 2007; Kato 2010). Continuous time: vM diffusion random walks (Hill and Häder 1997); toroidal Langevin diffusions (García-Portugués et al. 2019); filtering with vMF, Bingham, wrapped normal; particle filters for WC ship tracking.

Examples

Harmonic seasonal predictor

Day-of-year $d$ with $\theta =2\pi d/365.25$: $\mu (d)={\beta }_0+a_1\cos \theta +b_1\sin \theta +a_2\cos 2\theta +b_2\sin 2\theta$. Amplitude of harmonic $m$ is $\sqrt{a_m^2+b_m^2}$, phase $atan2(b_m,a_m)$. This is the E81 “harmonic circular predictor” and the same basis as periodic Fourier seasonality terms in MMM; see Local Linear Trend and Seasonality.

Connections

Pewsey 2021 - Circular Models in PyMC, Generalized Linear Models (the mean-direction models are non-canonical-link GLM analogues).

See Also

Pewsey 2021 - Directional Statistics - Overview