The Zero Problem: Gaussian Process Emulators for Range-Constrained Computer Models

Abstract

We introduce a zero-censored Gaussian process as a systematic, model-based approach to building Gaussian process emulators for range-constrained simulator output. This approach avoids many pitfalls associated with modeling range-constrained data with Gaussian processes. Further, it is flexible enough to be used in conjunction with statistical emulator advancements such as emulators that model high-dimensional vector-valued simulator output. The zero-censored Gaussian process is then applied to two examples of geophysical flow inundation which have the constraint of nonnegativity.

Topographic Controls on Pyroclastic Density Current Hazard at Aluto Volcano (Ethiopia) Identified Using a Novel Zero‐Censored Gaussian Process Emulator

“…Nevertheless, for dilute flows to be incorporated into the hazard assessment, one would need to use a different PDC model, which accounted for settling of particles and entrainment of air into the flow (e.g., Esposti Ongaro et al, 2007;de' Michieli Vitturi et al, 2023). This said, the results obtained in this study illustrate the great potential and high versatility of the use of statistical emulators (Bayarri et al, 2009;Rutarindwa et al, 2019;Spiller et al, 2014Spiller et al, , 2020, particularly the zGP emulator (Spiller et al, 2023), to analyze, in a high and customizable level of detail, the particularities of PDC inundation and hazard at a given volcanic system of interest. Such analyses can expand on previous work based on selected scenarios, case studies and/or test runs (Charbonnier & Gertisser, 2012;Gueugneau et al, 2021;Ogburn & Calder, 2017;Ogburn et al, 2014).…”

“…Breard et al, 2023;Doronzo, 2012), more rapidly, and over shorter distances, for a given combination of flow size and mobility. Investigating such flow transitions following a systematic, ideally multi-model, approach (e.g., Gueugneau et al, 2021;Ogburn & Calder, 2017;Patra et al, 2020), and using probabilities of PDC inundation to fully capture uncertainty, could be a fruitful topic for future research, facilitated by the use of zGP emulators (Spiller et al, 2023).…”

“…positive output here taken as h > 0.1 m); and (b) 250 simulations that result in zero output at the road point. The number of zero-output design points used is similar to the number of positive-output design points, and chosen to strike a balance between: (a) the zGP emulators having enough design points at both sides of the zero-flow isoline as to capture it successfully (Figures 3b and 3c); and (b) the zGP construction not being overburdened, computationally, by having too many design points, which can compromise the inversion of large matrices required in the building of the emulators (Spiller et al, 2014(Spiller et al, , 2023. Regarding (a), it is important to note that, depending on the selection of design points, as well as on the use of either zGP emulators or classical GP emulators, the emulation results may be different (Figures 3b and 3c).…”

“…First, we apply a multi‐model framework (after Clarke, 2020; Clarke et al., 2020) to parameterize the TITAN2D model (Patra et al., 2005; Pitman et al., 2003) using flux sources, which better approximate a column‐collapse mechanism for the generation of PDCs, relevant for Aluto and many other volcanic systems (e.g., Rutarindwa et al., 2019; Tennant et al., 2021; Tierz et al., 2018). Second, we systematically apply a novel zero‐censored Gaussian Process (zGP) emulator (Spiller et al., 2023) to explore the links between topography and PDC hazard. The zGP emulators are built using sets of TITAN2D simulations and act as a substitute (or surrogate model, see Section 2.2) for the simulator.…”

“…They enable efficient quantification of the aleatory and epistemic uncertainties relevant for volcanic hazard assessment in a way that would be computationally unfeasible using TITAN2D alone (e.g., Bayarri et al., 2009; Spiller et al., 2014). The zGP emulators also represent an important advance in dealing with the “zero problem” in hazard assessment: a collection of issues including the fact that model outputs tend to be strictly positive or that “near‐hit” zero values are indistinguishable from those zero values far away from the zero‐/positive‐output envelope, within the model parameter space (Spiller et al., 2023). We use our novel TITAN2D‐zGP strategy to quantify uncertainty and explore the influence of topography on PDC hazard.…”