Geometric modelling of spatial extremes

Abstract

Recent developments in extreme value statistics have established the so-called geometric approach as a powerful modelling tool for multivariate extremes. We tailor these methods to the case of spatial modelling and examine their efficacy at inferring extremal dependence and performing extrapolation. The geometric approach is based around a limit set described by a gauge function, which is a key target for inference. We consider a variety of spatially-parameterised gauge functions and perform inference on them by building on the framework of Wadsworth and Campbell (2024), where extreme radii are modelled via a truncated gamma distribution. We also consider spatial modelling of the angular distribution, for which we propose two candidate models. Estimation of extreme event probabilities is possible by combining draws from the radial and angular models respectively. We compare our method with two other established frameworks for spatial extreme value analysis and show that our approach generally allows for unbiased, albeit more uncertain, inference compared to the more classical models. We illustrate the methodology on a space weather dataset of daily geomagnetic field fluctuations.

Figures (15)

• Figure 1: Examples of bivariate limit sets linked to the gauge functions introduced in Section \ref{['sec:spat gauges']}. Limit sets of (a) generalised Gaussian gauges with correlation parameter $\rho=0.5$, and $\nu=0.5,1,2,3.5$ in yellow, red, blue and green, respectively; (b) HW gauges with an underlying generalised Gaussian gauge, correlation parameter $\rho=0.5$, $\nu=0.5,2,3.5$ in yellow, blue and green, respectively, and mixing parameter $\delta=0.45$; (c) HW gauges with an underlying generalised Gaussian gauge, correlation parameter $\rho=0.5$, $\nu=0.5,2,3.5$ in yellow, blue and green, respectively, and $\delta=0.55$; (d) IBR gauge with semivariogram parameter $\gamma = 0.3,0.5,0.7$ in blue, red and yellow, respectively.
• Figure 2: Boxplots of pairwise $\chi_u$ estimates for the pair $\left(X(\bm{s}_2), X(\bm{s}_3)\right)$ at distance $5.78$ apart, calculated for a range of $u$ values and using all $200$ simulated datasets. $X(\bm{s})$ is simulated as specified in (\ref{['fig:box1_mvn']}) - (\ref{['fig:box1_hw6']}) with parameter $\bm{\theta}_1$. Boxplots on the left correspond to $d=5$, $d=10$ in the middle and $d=20$ on the right. Estimates in dark blue (labelled am0), turquoise blue (am1) and light blue (am2) are obtained via the empirical angular distribution, the angular distribution in \ref{['eq:am1']} and the angular distribution in \ref{['eq:am2']}, respectively. Estimates in red come from cG fits, while estimates in orange from HW fits. The pink horizontal line corresponds to the simulated truth.
• Figure 3: Boxplots of pairwise $\chi_u$ estimates for the pair $\left(X(\bm{s}_2), X(\bm{s}_3)\right)$ at distance $5.78$ apart, calculated for a range of $u$ values and using all $200$ simulated datasets. $X(\bm{s})$ is simulated as specified in (\ref{['fig:box2_mvn']}) - (\ref{['fig:box2_hw6']}) with parameter $\bm{\theta}_2$. Boxplots on the left correspond to $d=5$, $d=10$ in the middle and $d=20$ on the right. Estimates in dark blue (labelled am0), turquoise blue (am1) and light blue (am2) are obtained via the empirical angular distribution, the angular distribution in \ref{['eq:am1']} and the angular distribution in \ref{['eq:am2']}, respectively. Estimates in red come from cG fits, while estimates in orange from HW fits. The pink horizontal line corresponds to the simulated truth.
• Figure 4: (a) Pairwise scatter plots of $\bm{X}|R'>1$ data simulated using the process-based (black) and gauge-based (red, green, blue) angular models in \ref{['eq:am1']} and \ref{['eq:am2']}, respectively. The dashed line corresponds to the diagonal line, $x_j=x_k$, where $j=20,15,8$ and $k=3, 11, 18$. (b) Pairwise $\chi_u(\bm{s}_j,\bm{s}_k)$ estimates computed for all $d\choose2$ pairs and plotted over distance, for $d=20$. Estimates in light blue stem from the gauge-based angular model, while empirical $\chi_u$ are shown in pink. Differently coloured points correspond to estimates obtained from the $(X_j,X_k)$ pairs presented in Figure \ref{['fig:am2perfplots']}(\ref{['fig:am2_scatter']}) and are coloured accordingly.
• Figure 5: Pairwise $\chi(u)$ estimates over (geodesic) distance for $u=0.95$ (left) and $u=0.98$ (right). Pink points correspond to empirical estimates, dark blue to estimates with resampled angles and turquoise points to estimates from the angular model in \ref{['eq:am1']}. Loess curves are fitted to each of these point clouds.