[Paper Review] Calculating probabilistic excursion sets and related quantities using excursions

David Bolin, Finn Lindgren|Edinburgh Research Explorer (University of Edinburgh)|Dec 13, 2016

Soil Geostatistics and Mapping20 references3 citations

TL;DR

This paper presents the R package *excursions*, which efficiently computes probabilistic excursion sets, contour credible regions, and simultaneous confidence bands for latent Gaussian processes using sparse precision matrices and advanced Gaussian integral computation. The method enables uncertainty quantification in spatial and spatiotemporal modeling, with key contributions including fast computation via Markov properties and robust tools for contour map quality assessment.

ABSTRACT

The R software package excursions contains methods for calculating probabilistic excursion sets, contour credible regions, and simultaneous confidence bands for latent Gaussian stochastic processes and fields. It also contains methods for uncertainty quantification of contour maps and computation of Gaussian integrals. This article describes the theoretical and computational methods used in the package. The main functions of the package are introduced and two examples illustrate how the package can be used.

Motivation & Objective

• To provide a computational framework for quantifying uncertainty in latent Gaussian processes, particularly in spatial and spatiotemporal applications.
• To address the need for rigorous selection of contour map levels by enabling uncertainty quantification of contour curves and maps.
• To develop efficient algorithms for computing high-dimensional Gaussian integrals using Markov properties in latent Gaussian models.
• To extend the functionality beyond excursion sets to include simultaneous credible bands and quality measures for contour maps.
• To support flexible model input via direct parameter specification, INLA outputs, or Monte Carlo simulations.

Proposed method

• Uses the latent Gaussian model (LGM) framework, where the latent field follows a Gaussian Markov random field (GMRF) with sparse precision matrices.
• Employs sparse Cholesky factorization to accelerate high-dimensional Gaussian integral computations, crucial for excursion set and credible region estimation.
• Applies the theory of excursion sets and level sets from Bolin & Lindgren (2015, 2016b) to compute regions where the process exceeds a threshold with posterior probability.
• Implements the *QC* (quasi-Monte Carlo) method for approximating Gaussian integrals in high dimensions, improving computational efficiency.
• Introduces the *tricontourmap* and *contourmap.inla* functions to generate contour maps with uncertainty-aware level selection.
• Uses the *P2* quality measure to evaluate contour map credibility, enabling data-driven selection of the optimal number of contour levels.

Experimental results

Research questions

• RQ1 How can probabilistic excursion sets be efficiently computed for large-scale latent Gaussian processes with Markov structure?
• RQ2 What is the optimal number of contour levels in a contour map that reflects the underlying uncertainty in a spatial surface estimate?
• RQ3 How can simultaneous credible bands be constructed for latent Gaussian processes using sparse precision matrices?
• RQ4 To what extent can uncertainty in contour curves be quantified and used to guide the design of contour maps?
• RQ5 How can Gaussian integrals over high-dimensional regions be computed efficiently when the precision matrix is sparse?

Key findings

• The *excursions* package enables fast and accurate computation of excursion sets using sparse precision matrices and quasi-Monte Carlo integration.
• The P2 quality measure successfully evaluates contour map credibility, with results showing that two contours for log-precipitation had a credibility of approximately 0.5, indicating a balanced choice.
• Using one contour resulted in a P2 value of 1.0, indicating over-credibility, while three contours yielded a P2 of 0.0048, indicating under-credibility.
• The package supports multiple input formats, including direct parameter specification, INLA outputs, and Monte Carlo simulations, enhancing usability across modeling workflows.
• The implementation is computationally efficient due to exploitation of Markov properties in the latent field, enabling scalable inference in large spatial and spatiotemporal models.
• Future developments will extend the package to handle large-scale models where sparse Cholesky factorization is infeasible and support spatial extreme value theory applications.