[Paper Review] Pointless Continuous Spatial Surface Reconstruction

Katherine Wilson, Jon Wakefield|arXiv (Cornell University)|Sep 27, 2017

3D Shape Modeling and Analysis23 references3 citations

TL;DR

This paper proposes a Bayesian hierarchical model using stochastic partial differential equations (SPDEs) to reconstruct a continuous spatial surface from mixed point and areal data, enabling efficient inference via integrated nested Laplace approximations (INLA) for Gaussian outcomes and Hamiltonian Monte Carlo (HMC) or empirical Bayes for non-Gaussian outcomes. The method effectively handles the change of support problem and reduces ecological bias by modeling a latent continuous surface, with strong performance even when only areal data are available.

ABSTRACT

The analysis of area-level aggregated summary data is common in many disciplines including epidemiology and the social sciences. Typically, Markov random field spatial models have been employed to acknowledge spatial dependence and allow data-driven smoothing. In this paper, we exploit recent theoretical and computational advances in continuous spatial modeling to carry out the reconstruction of an underlying continuous spatial surface. In particular, we focus on models based on stochastic partial differential equations (SPDEs). We also consider the interesting case in which the aggregate data are supplemented with point data. We carry out Bayesian inference, and in the language of generalized linear mixed models, if the link is linear, an efficient implementation of the model is available via integrated nested Laplace approximations. For nonlinear links, we present two approaches: a fully Bayesian implementation using a Hamiltonian Monte Carlo algorithm, and an empirical Bayes implementation, that is much faster, and is based on Laplace approximations. We examine the properties of the approach using simulation, and then estimate an underlying continuous risk surface for the classic Scottish lip cancer data.

Motivation & Objective

• To address the change of support problem in spatial modeling when data are observed at multiple resolutions, including point locations and aggregated areal units.
• To develop a unified framework for reconstructing a continuous spatial surface that avoids arbitrary administrative boundaries and reduces ecological bias.
• To enable efficient Bayesian inference for both Gaussian and non-Gaussian outcomes using SPDE-based models.
• To evaluate the performance of the method in simulation and real-world settings, particularly with sparse or coarse areal data.
• To provide a computationally efficient alternative to MCMC for complex spatial models, especially for non-Gaussian likelihoods.

Proposed method

• Models the latent spatial surface using a Gaussian Markov random field (GMRF) derived from a stochastic partial differential equation (SPDE), enabling fast computation via sparse precision matrices.
• Uses the SPDE-approach to map the continuous spatial field to both point-level and areal-level observations through appropriate linear projections.
• Applies integrated nested Laplace approximations (INLA) for fast inference under Gaussian likelihoods, leveraging the GMRF approximation.
• For non-Gaussian outcomes (e.g., Poisson), implements a fully Bayesian approach using Hamiltonian Monte Carlo (HMC) and an empirical Bayes approach using Laplace approximations.
• Combines point data (e.g., survey locations) and areal data (e.g., census aggregates) under a single hierarchical model with a shared latent continuous surface.
• Employs a hierarchical Bayesian framework where the observed data are conditionally distributed given the latent continuous surface, with proper priors on hyperparameters.

Experimental results

Research questions

• RQ1 Can a continuous spatial surface be accurately reconstructed from mixed point and areal data using SPDE-based models?
• RQ2 How does the inclusion of point data affect the accuracy and smoothness of the reconstructed surface compared to areal data alone?
• RQ3 To what extent does the SPDE-based model reduce ecological bias compared to discrete areal models like ICAR?
• RQ4 How do computational methods like INLA and HMC compare in terms of speed and accuracy for non-Gaussian outcomes in this context?
• RQ5 How sensitive is the model to spatial resolution and the number of areal units when only areal data are available?

Key findings

• The SPDE-based continuous surface model produced relative risk estimates nearly identical to those from a discrete ICAR model when applied to the Scottish lip cancer data, despite the absence of point data.
• Even with only areal data, the model maintained good accuracy, though spatial parameter estimates became more variable when aggregating to coarser regions (e.g., from 47 counties to 8 provinces).
• For Poisson outcomes, the fully Bayesian HMC approach took approximately one week to fit on a cluster, while the empirical Bayes approach reduced computation to minutes, with HMC taking about ten minutes after empirical Bayes initialization.
• The model effectively mitigated ecological bias by allowing for point-level covariate data, avoiding the ecological fallacy inherent in areal-level analyses.
• The method demonstrated robustness in simulation, with minimal loss in accuracy when point data were absent, especially under strong spatial dependence.
• The use of SPDEs enabled efficient computation for Gaussian outcomes (under 2 minutes per model on a laptop), while non-Gaussian models required more advanced MCMC or hybrid approaches.