Non-stationary Gaussian models with physical barriers

TL;DR

The paper introduces the Barrier model, a non-stationary Gaussian spatial model that effectively accounts for physical barriers without increased computational cost, demonstrated through applications in complex environments.

Contribution

It proposes a novel Barrier model based on SAR and SPDE formulations that handles physical barriers without relying on shortest paths or boundary conditions.

Findings

The Barrier model performs better than standard models in reconstructing complex functions.

It maintains computational efficiency comparable to stationary models.

Successfully applied to the Finnish Archipelago Sea with complex barriers.

Abstract

The classical tools in spatial statistics are stationary models, like the Mat\'ern field. However, in some applications there are boundaries, holes, or physical barriers in the study area, e.g. a coastline, and stationary models will inappropriately smooth over these features, requiring the use of a non-stationary model. We propose a new model, the Barrier model, which is different from the established methods as it is not based on the shortest distance around the physical barrier, nor on boundary conditions. The Barrier model is based on viewing the Mat\'ern correlation, not as a correlation function on the shortest distance between two points, but as a collection of paths through a Simultaneous Autoregressive (SAR) model. We then manipulate these local dependencies to cut off paths that are crossing the physical barriers. To make the new SAR well behaved, we formulate it as a stochastic partial differential equation (SPDE) that can be discretised to represent the Gaussian field, with a sparse precision matrix that is automatically positive definite. The main advantage with the Barrier model is that the computational cost is the same as for the stationary model. The model is easy to use, and can deal with both sparse data and very complex barriers, as shown in an application in the Finnish Archipelago Sea. Additionally, the Barrier model is better at reconstructing the modified Horseshoe test function than the standard models used in R-INLA.