Coherent Disaggregation and Uncertainty Quantification for Spatially Misaligned Data

Spatial misalignment arises when datasets are aggregated or collected at different spatial scales, leading to information loss. We develop a Bayesian disaggregation framework that links misaligned data to a continuous-domain model through an iteratively linearised integration scheme implemented with the Integrated Nested Laplace Approximation (INLA). The framework accommodates different ways of handling observations depending on the application, resulting in four variants: (i) \textit{Raster at Full Resolution}, (ii) \textit{Raster Aggregation}, (iii) \textit{Polygon Aggregation} (PolyAgg), and (iv) \textit{Point Values} (PointVal). The first three represent increasing levels of spatial averaging, while the last two address situations with incomplete covariate information. For PolyAgg and PointVal, we reconstruct the covariate field using three strategies – \textit{Value Plugin}, \textit{Joint Uncertainty}, and \textit{Uncertainty Plugin} – with the latter two propagating uncertainty. We illustrate the framework with an example motivated by landslide modelling, focusing on methodology rather than interpreting landslide processes. Simulations show that uncertainty-propagating approaches outperform \textit{Value Plugin} method and remain robust under model misspecification. Point-pattern observations and full-resolution covariates are therefore preferable, and when covariate fields are incomplete, uncertainty-aware methods are most reliable. The framework is well suited to landslide susceptibility modelling and other spatial mapping tasks, and integrates seamlessly with INLA-based tools.

💡 Research Summary

The paper addresses the pervasive problem of spatial misalignment, where datasets are collected at differing spatial resolutions, leading to a change‑of‑support issue that can cause loss of information and biased inference. The authors develop a coherent Bayesian disaggregation framework that links misaligned observations to a continuous‑domain intensity model using an iteratively linearised integration scheme implemented within the Integrated Nested Laplace Approximation (INLA) framework.

Four observation‑handling variants are defined: (i) Raster at Full Resolution, which retains the original raster grid; (ii) Raster Aggregation, which averages raster cells into coarser blocks; (iii) Polygon Aggregation (PolyAgg), which computes averages over user‑defined polygons such as slope units; and (iv) Point Values (PointVal), which uses the raw point‑pattern data directly. The first three represent increasing levels of spatial averaging, while the latter two are designed for situations where covariate information is incomplete.

For PolyAgg and PointVal the authors propose three strategies to reconstruct missing covariate fields: (a) Value Plugin, which simply plugs in observed covariate values and ignores their uncertainty; (b) Joint Uncertainty, which jointly estimates the covariate field and the intensity field in a single Bayesian model, thereby propagating uncertainty from both sources; and (c) Uncertainty Plugin, which first obtains a posterior distribution for the covariate field (e.g., via a separate INLA run) and then feeds sampled realizations into the intensity model. The covariate field itself is modelled as a Matérn Gaussian Random Field using the SPDE approach, with penalised complexity priors on range and marginal variance.

The core computational innovation is a first‑order Taylor expansion of the logarithmic integral of the intensity over each sub‑domain Ω_j. Both a continuous version M_j(u) and a discretised version m_j(u) are derived, with Jacobian and Hessian terms explicitly calculated. Theorem 3.1 shows that the approximation error is of order O(‖u‑u*‖³), allowing the authors to control accuracy by iteratively updating the linearisation point u* via line‑search within INLA. The domain is partitioned using a user‑specified mesh; the authors discuss mesh element shapes (triangles, squares, hexagons) and argue that hexagonal meshes minimise orientation bias while still tessellating the domain efficiently. Mesh resolution is guided by the spatial correlation length, typically choosing element diameters 5–10 times smaller than the range for accurate representation.

A simulation study, motivated by landslide susceptibility modelling, evaluates the four observation variants and the three covariate‑reconstruction strategies under various levels of model misspecification and covariate incompleteness. Results demonstrate that uncertainty‑propagating approaches (Joint Uncertainty and Uncertainty Plugin) consistently outperform the naive Value Plugin in terms of mean squared error, bias, and coverage of credible intervals. Point‑pattern observations (PointVal) combined with full‑resolution covariates yield the best performance, confirming that retaining the finest spatial information is advantageous. Moreover, the framework remains robust when the link function is misspecified (e.g., using a log‑link for a non‑log‑linear relationship) and when covariate fields are downscaled or partially missing.

In summary, the authors provide a flexible, computationally efficient, and statistically rigorous solution to spatial misalignment. By embedding the linearised integration within INLA, they avoid the high‑dimensional MCMC sampling traditionally required for hierarchical spatial models. The framework seamlessly integrates with the inlabru package, allowing practitioners to define meshes, specify observation types, and select covariate‑reconstruction strategies with minimal coding effort. The methodology is broadly applicable beyond landslide susceptibility, including environmental risk mapping, disease incidence modelling, and any spatial analysis where data are available at heterogeneous resolutions.