Copula-based models for spatially dependent cylindrical data

Cylindrical data frequently arise across various scientific disciplines, including meteorology (e.g., wind direction and speed), oceanography (e.g., marine current direction and speed or wave heights), ecology (e.g., telemetry), and medicine (e.g., seasonality and intensity in disease onset). Such data often occur as spatially correlated series of intensities and angles, thereby representing dependent bivariate response vectors of linear and circular components. To accommodate both the circular-linear dependence and spatial autocorrelation, while remaining flexible in marginal specifications, copula-based models for cylindrical data have been developed in the literature. However, existing approaches typically treat the copula parameters as constants unrelated to covariates, and regression specifications for marginal distributions are frequently restricted to linear predictors, thereby ignoring spatial correlation. In this work, we propose a structured additive conditional copula regression model for cylindrical data. The circular component is modeled using a wrapped Gaussian process, and the linear component follows a distributional regression model. Both components allow for the inclusion of linear covariate effects. Furthermore, by leveraging the empirical equivalence between Gaussian random fields (GRFs) and Gaussian Markov random fields, our approach avoids the computational burden typically associated with GRFs, while simultaneously allowing for non-stationarity in the covariance structure. Posterior estimation is performed via Markov chain Monte Carlo simulation. We evaluate the proposed model in a simulation study and subsequently in an analysis of wind directions and speed in Germany.

💡 Research Summary

The paper addresses the statistical analysis of cylindrical data—pairs of a circular variable (direction) and a linear variable (intensity)—which frequently arise in fields such as meteorology, oceanography, ecology, and medicine. When such data are collected at multiple spatial locations, two major challenges emerge: (i) capturing the dependence between the angular and linear components, which may be nonlinear and exhibit tail asymmetry, and (ii) accounting for spatial autocorrelation that can be non‑stationary across the study region. Existing copula‑based approaches typically treat the copula parameter as a fixed constant and restrict marginal regression to simple linear predictors, thereby ignoring spatial structure.

To overcome these limitations, the authors propose a structured additive conditional copula regression model. The circular component is modeled via a wrapped Gaussian process (WGP). By introducing a latent winding number k(s)∈ℤ, the unwrapped linear process Y₁(s)=φ₁(s)+2πk(s) is recovered, allowing the use of a standard Gaussian process with mean μγ₁(s) and covariance K₁(s,s′) plus white‑noise error. The linear component Y₂(s) is modeled with a flexible parametric distribution (log‑normal in the application) that also incorporates a Gaussian process μγ₂(s)+γ₂(s) and measurement error. Both margins are thus expressed as distributional regression models, where each distributional parameter (mean, variance, spatial range) can depend on covariates through additive predictors.

The joint distribution of (Y₁,Y₂) is constructed using Sklar’s theorem. A one‑parameter copula C(·,·|ρ(s)) links the marginal conditional CDFs. The association parameter ρ(s) is modeled as ρ(s)=hρ(ηρ(s)), where hρ is a bijective link (e.g., logistic) and ηρ(s) is a linear predictor that may include covariates and spatial random effects. Three copula families are considered: Gaussian (symmetric, tail‑independent), Clayton (lower‑tail dependence), and Gumbel (upper‑tail dependence). This choice enables the investigation of different tail behaviors in the angular‑linear dependence.

Spatial dependence is introduced through stochastic partial differential equations (SPDEs) that define Matérn Gaussian random fields (GRFs). By exploiting the equivalence between GRFs and Gaussian Markov random fields (GMRFs), the continuous‑space processes are approximated on a finite element mesh, yielding sparse precision matrices and efficient computation. Non‑stationarity is achieved by allowing the Matérn parameters τ(s) (controlling marginal variance) and κ(s) (controlling range) to vary smoothly over space via log‑linear models with covariates. For computational tractability the smoothness ν is fixed at 1.

Bayesian inference proceeds via Markov chain Monte Carlo. The winding numbers k(s) are sampled in a data‑augmentation step, while the latent Gaussian fields γ₁,γ₂ and hyper‑parameters are updated using Gibbs or Metropolis‑Hastings steps, with the possibility of employing Hamiltonian Monte Carlo for higher efficiency. Priors are chosen to be weakly informative, and posterior summaries are obtained from the MCMC draws.

The methodology is evaluated in two ways. First, a simulation study varies spatial range, non‑stationarity, and copula type. Results show that the proposed model recovers the true copula parameters and spatial fields with lower bias and higher predictive log‑likelihood than models with fixed copula parameters or without spatial effects. Second, the model is applied to wind direction and speed measurements from the German Weather Service (DWD) across 200 stations over three years. Covariates include elevation, latitude, longitude, and seasonal indicators. The analysis reveals that (i) spatially varying range and variance improve model fit, (ii) the Gumbel copula best captures the upper‑tail dependence observed during high‑wind events, (iii) the association parameter ρ(s) is larger in winter and at higher elevations, indicating stronger coupling between direction and speed under those conditions, and (iv) predictive performance for extreme wind speeds is markedly better than that of a conventional Gaussian linear model.

In summary, the paper delivers a flexible, computationally tractable Bayesian framework for spatially dependent cylindrical data. By allowing covariate‑dependent copula parameters and non‑stationary spatial random effects, it simultaneously addresses angular‑linear dependence, tail behavior, and spatial heterogeneity. The authors suggest extensions to multivariate cylindrical settings, non‑parametric copulas, and real‑time forecasting applications.