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Copula-based models for spatially dependent cylindrical data

Francesca Labanca, Anna Gottard, Nadja Klein

TL;DR

This work proposes a structured additive conditional copula regression model for cylindrical data that avoids the computational burden typically associated with GRFs, while simultaneously allowing for non-stationarity in the covariance structure.

Abstract

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.

Copula-based models for spatially dependent cylindrical data

TL;DR

This work proposes a structured additive conditional copula regression model for cylindrical data that avoids the computational burden typically associated with GRFs, while simultaneously allowing for non-stationarity in the covariance structure.

Abstract

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.
Paper Structure (40 sections, 1 theorem, 33 equations, 11 figures, 5 tables)

This paper contains 40 sections, 1 theorem, 33 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Bivariate conditional copula models with mixed circular-linear marginals
  3. Dependence structure
  4. Modeling the marginal distribution of $\varphi_1(\text{\boldmath$s$})$
  5. Modeling the marginal distribution of $Y_2(\text{\boldmath$s$})$
  6. Predictor specifications
  7. Modeling spatial effects
  8. Bayesian inference
  9. Prior specifications
  10. Prior scaling
  11. Likelihood
  12. Posterior estimation
  13. Model choice
  14. Simulation study
  15. Simulation design
  16. ...and 25 more sections

Key Result

Lemma 1

The margins of the $PW$ copula model defined in eq:pw are $p_2$ and the wrapped counterpart of $p_1$, i.e.,

Figures (11)

  • Figure 1: Graphical representation of the full hierarchical model. Shown are the spatially dependent covariates (grey), the model parameters and data $\{\varphi_1(\mathbf{s}_i)\}_{i=1}^n$ related to the wrapped circular random variable red), the model parameters and data $\{y_2(\mathbf{s}_i)\}_{i=1}^n$ related to the linear random variable (blue), and the copula-parameters (petrol).
  • Figure 2: Wind behaviour in Germany during a storm weather period (24-–29 January 2025). Left: station-wise average wind direction and mean wind speed (m/s), with arrow orientation indicating direction and colour scale representing speed. Right: wind rose summarizing the frequency distribution of directions the winds blew from across all stations, with colours denoting wind speed classes.
  • Figure 3: Left: credible intervals for wind vectors at test locations during a storm period. The rainbow arrows indicate the observed wind direction and speed, while the purple and violet arrows denote the $0.025$ and $0.975$ posterior quantiles, respectively. Right: posterior mean estimates of the copula parameter $\rho$ at all locations.
  • Figure S4: The three boxes show, from left to right, the covariates functions $z_\beta(\text{\boldmath$s$})=2\sin(2\pi \,s_1)\sin(4\pi\,s_2)$, $z_\kappa(\text{\boldmath$s$})= 1/2+\sin(2\pi \, s_1)\cos(4\pi\,s_2)$, and $z_\rho(\bm{s})=\sin(4*s_2+s_1)-\frac{1}{2}\exp(-64s_1^2)$ used in the simulation study.
  • Figure S5: Comparison of $\hbox{DIC}$ values under the correctly specified Gaussian copula model against the misspecified Clayton and Gumbel copula models. Columns correspond to sample sizes $n=250, 500, 750$, while rows display the two dependence scenarios: constant copula parameter $\rho$, and varying$\rho$ (covariate-dependent $\rho$).
  • ...and 6 more figures

Theorems & Definitions (2)

  • Lemma 1
  • proof