Assessing the size of spatial extreme events using local coefficients based on excursion sets

TL;DR

The paper develops a framework of local excursion-set coefficients to quantify the spatial extent of threshold exceedances on regular grids, addressing limitations of pairwise extremal measures and accommodating nonstationarity and boundaries. By defining distance-like coefficients $\theta_k$ based on excursion sets conditioned on a reference site, and establishing their asymptotic behavior through a scaling function $\sigma(p)$ and index $\beta$, the authors enable extrapolation to very high thresholds. The approach is validated via simulations with Gaussian random fields and applied to French summer temperatures, revealing notable differences between Reanalysis and climate-model simulations in the spatial footprint of extremes and in asymptotic dependence properties. The results provide a versatile, interpretable toolset for climate risk assessment and open avenues for nonstationary tail modeling, spatiotemporal extensions, and connections to stochastic geometry and topological data analysis.

Abstract

Extreme events arising in georeferenced processes can take various forms, such as occurring in isolated patches or stretching contiguously over large areas, and can further vary with the spatial location and the extremeness of the events. We use excursion sets above threshold exceedances in data observed over a two-dimensional grid of rectangular pixels to propose a general family of coefficients that assess spatial-extent properties relevant for risk assessment, and study five candidate coefficients from this family. These coefficients are defined locally and interpreted as a spatial distance from a reference site where the threshold is exceeded. We develop statistical inference and discuss robustness to boundary effects and resolution of the pixel grid. To statistically extrapolate coefficients towards very high threshold levels, we formulate a semiparametric model and estimate a parameter characterizing how coefficients scale with the quantile level of the threshold. The utility of the new coefficients is illustrated through simulated data, as well as in an application to gridded daily temperature in continental France. We find notable differences in estimated coefficient maps between climate model simulations and observation-based reanalysis.

Figures (10)

• Figure 1: Boxplots of $\theta_k(\bm s; \mathcal{E}^{\mathcal{G}, X}_u, 0.5)$, for $k=1, \ldots, 4$ and $\theta_5(\bm s; \mathcal{E}^{\mathcal{G}, X}_u)$ for $\bm s = \bm 0$ (left) and $\bm s = (60,0)'$ (right). A gray horizontal line is drawn for each distance from $\bm s$ where there is a point in $\mathcal{G}$.
• Figure 2: Same as Figure \ref{['fig:boxplotsX']} but for $\mathcal{E}^{\mathcal{G}, Y}_u$ instead of $\mathcal{E}^{\mathcal{G}, X}_u$.
• Figure 3: Illustration of the estimation strategy (see Section \ref{['subsec:EstimationStrategy']}) carried out for the five considered local excursion set coefficients. The estimates $\widehat{\theta}_k$ are displayed by dots, the fitted weighted linear regressions by lines. Twice the value of the expected area of $\{\bm s\in [-60,60]^2 : X(\bm s) > u\}$ divided by the expected length of $\{\bm s\in [-60,60]^2 : X(\bm s) = u\}$ is shown as a dashed blue curve. Both panels display the same information but with different axes labels, so that the regression coefficients may be read off of the plot on the right.
• Figure 4: Estimated quantile thresholds $u_p(\bm{s})$ for $p = 0.86$ (left) and $p = 0.99$ (right) across mainland France based on the Reanalysis dataset.
• Figure 5: $\mathcal{E}_{u_p}^{\mathcal{F}, X}$, for several $p$, observed on four different days. This corresponds to displaying the temperature over France on uniform margins, binned using the chosen values of $p$.

Theorems & Definitions (17)

• Definition 1: Discrete domain
• Definition 2: Excursion set
• Definition 3: Distance-homogeneity
• Definition 4: Local excursion-set coefficient
• Definition 5: Connectivity
• Definition 6: Area and perimeter
• Remark 1
• Definition 7
• Remark 2: On Assumption \ref{['ass:rescalability']}
• Proposition 1