Directional Asymmetry in Edge BasedSpatial Models via a Skew Normal Prior

TL;DR

The paper addresses limitations of traditional CAR-based priors by introducing a skew-normal extension on the edge graph to capture directional asymmetry in spatial fields. It defines a multivariate skew-normal on the edge vector $\boldsymbol{\rho}$, $\boldsymbol{\rho} \sim \operatorname{SN}_p(-b\,\boldsymbol{\eta};\; \sigma_\theta^2(M_e-\gamma A_e)^{-1};\; \boldsymbol{\eta})$, and propagates to node space via $\boldsymbol{\theta}=C\boldsymbol{\rho}$ with a stochastic representation using the latent half-normal variable $U$ to enable efficient computation. Simulations show that RENeGe-sk recovers edge-aligned directional structure more accurately than symmetric priors, and an application to cancer mortality in Southern Brazil demonstrates improved predictive performance and localized directional variation. The approach remains scalable through low-rank parameterizations and an auxiliary-variable framework, making it practical for epidemiological studies where boundary-driven effects are plausible.

Abstract

We introduce a skewed edge based spatial prior, named RENeGe sk that extends the Gaussian RENeGe framework by incorporating directional asymmetry through a skew normal distribution. Skewness is defined on the edge graph and propagated to the node space, aligning asymmetric behavior with transitions across neighboring regions rather than with marginal node effects. The model is formulated within the skew normal framework and employs identifiable hierarchical priors together with low rank parameterizations to ensure scalability. The skew normal's stochastic representation is considered to facilitate the computational implementation. Simulation studies show that RENeGe sk recovers compact, edge-aligned directional structure more accurately than symmetric Gaussian priors, while remaining competitive under irregular spatial patterns. An application to cancer incidence data in Southern Brazil illustrates how the proposed approach yields stable area-level estimates while preserving localized, directionally driven spatial variation.

Figures (4)

• Figure 1: Schematic representation of an areal map (left), its induced neighborhood graph (center), and the corresponding line graph (right). Nodes in the line graph represent adjacencies in the original map, enabling spatial dependence and directional effects to be modeled directly on edges.
• Figure 2: Simulated latent field $\theta$ exhibiting a directional north--south increase generated through an edge-based Skew--Normal perturbation.
• Figure 3: Posterior medians of the edge-based latent effect $\rho$ and the spatial effect $\theta$ under the Gaussian RENeGe (a) and the RENeGe--Skew (b) prior distributions.
• Figure 4: Comparison of the edges-based latent effects $\hat{\rho}$ (edges) and the spatial effect $\theta$ for lung cancer under the Gaussian (left) and skew-normal (right) RENeGe models.