Excursion and contour uncertainty regions for latent Gaussian models

An interesting statistical problem is to find regions where some studied process exceeds a certain level. Estimating such regions so that the probability for exceeding the level in the entire set is equal to some predefined value is a difficult problem that occurs in several areas of applications ranging from brain imaging to astrophysics. In this work, a method for solving this problem, as well as the related problem of finding uncertainty regions for contour curves, for latent Gaussian models is proposed. The method is based on using a parametric family for the excursion sets in combination with a sequential importance sampling method for estimating joint probabilities. The accuracy of the method is investigated using simulated data and two environmental applications are presented. In the first application, areas where the air pollution in the Piemonte region in northern Italy exceeds the daily limit value, set by the European Union for human health protection, are estimated. In the second application, regions in the African Sahel that experienced an increase in vegetation after the drought period in the early 1980s are estimated.

💡 Research Summary

This paper tackles two closely related but technically demanding problems that arise in spatial statistics: (1) identifying excursion sets—regions where a latent process exceeds a pre‑specified threshold—and (2) quantifying uncertainty for contour curves, i.e., the location of level sets of the process. Both problems require the joint probability that an entire spatial region satisfies a condition, a quantity that is analytically intractable for realistic models. The authors propose a unified solution within the framework of latent Gaussian models (LGMs), which are widely used in spatial and spatio‑temporal applications because they combine a Gaussian latent field with a possibly non‑Gaussian observation model and allow for full Bayesian inference.

The methodological core consists of two components. First, the authors restrict the class of possible excursion sets to a parametric family (e.g., unions of simple shapes, level‑set based regions, or thresholded posterior mean fields). By doing so, the problem of searching over an infinite collection of subsets becomes a finite‑dimensional optimization over the parameters that define the family. Second, they develop a sequential importance sampling (SIS) scheme to estimate the joint probability that the latent field exceeds the threshold on the candidate set. The SIS algorithm draws samples from the posterior Gaussian distribution of the latent field, evaluates whether each sample lies entirely inside the candidate set, and updates importance weights sequentially as the candidate set is enlarged. This sequential construction dramatically reduces variance compared with naïve Monte‑Carlo integration, especially in high‑dimensional settings where the latent field may have thousands of nodes.

For contour uncertainty, the authors introduce a “buffer zone” around a target level and compute the probability that the true contour lies within this zone using the same SIS machinery. The resulting contour uncertainty region can be visualized as a band around the estimated contour, providing a clear graphical representation of spatial confidence.

The paper validates the approach through extensive simulation studies. In a synthetic two‑dimensional Gaussian field with known covariance structure, the estimated excursion sets achieve the nominal coverage while requiring far fewer samples than standard Monte‑Carlo methods. A second simulation embeds the method in a spatio‑temporal LGM, demonstrating that the SIS estimator remains stable when the latent field evolves over time.

Two real‑world environmental applications illustrate the practical relevance. The first application estimates regions in the Piemonte region of northern Italy where daily PM10 concentrations exceed the European Union limit for human health. Using a hierarchical LGM that accounts for monitoring station locations, meteorological covariates, and spatial correlation, the method produces an excursion map together with a quantified probability that each pixel truly exceeds the limit. This information is directly useful for regulatory agencies and public‑health planners. The second application investigates vegetation recovery in the African Sahel after the severe drought of the early 1980s. Satellite‑derived NDVI data are modeled with a spatio‑temporal LGM; the excursion set corresponds to areas where the latent vegetation index shows a statistically significant increase relative to the drought baseline. The resulting map highlights regions of successful recovery and quantifies the uncertainty around the boundaries, offering valuable insight for ecological monitoring and policy.

Key contributions of the work are: (i) a coherent Bayesian framework that simultaneously addresses excursion set estimation and contour uncertainty; (ii) an efficient sequential importance sampling algorithm that makes joint probability estimation feasible for high‑dimensional latent Gaussian fields; (iii) thorough empirical validation showing both statistical accuracy and computational efficiency; and (iv) demonstrable applicability to pressing environmental problems where spatial risk assessment is essential.

The authors suggest several avenues for future research, including extensions to non‑Gaussian latent processes (e.g., using Gaussian copulas or transformed fields), incorporation of non‑linear observation models, and development of online SIS updates for streaming data. Overall, the paper provides a significant methodological advance for practitioners needing rigorous, probabilistic delineation of spatial regions in complex hierarchical models.