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On the spatial extent of extreme threshold exceedances

Ryan Cotsakis, Elena Di Bernardino, Thomas Opitz

Abstract

We introduce the extremal range, a local statistic for studying the spatial extent of extreme events in random fields on $\mathbb{R}^d$. Conditioned on exceedance of a high threshold at a location $s$, the extremal range at $s$ is the random variable defined as the smallest distance from $s\in\mathbb{R}^d$ to a location where there is a nonexceedance. We leverage tools from excursion-set theory, such as Lipschitz- Killing curvatures, to express distributional properties of the extremal range, including asymptotics for small distances and high thresholds. The extremal range captures the rate at which the spatial extent of conditional extreme events scales for increasingly high thresholds, and we relate its distributional properties with the well-known bivariate tail dependence coefficient and the extremal index of time series in Extreme-Value Theory. We calculate theoretical extremal-range properties for commonly used models, such as Gaussian or regularly varying random fields. Numerical studies illustrate that, when the extremal range is estimated from discretized excursion sets observed on compact observation windows, the distribution of the resulting estimators appropriately reproduces the theoretically derived links with the Lipschitz- Killing curvature densities.

On the spatial extent of extreme threshold exceedances

Abstract

We introduce the extremal range, a local statistic for studying the spatial extent of extreme events in random fields on . Conditioned on exceedance of a high threshold at a location , the extremal range at is the random variable defined as the smallest distance from to a location where there is a nonexceedance. We leverage tools from excursion-set theory, such as Lipschitz- Killing curvatures, to express distributional properties of the extremal range, including asymptotics for small distances and high thresholds. The extremal range captures the rate at which the spatial extent of conditional extreme events scales for increasingly high thresholds, and we relate its distributional properties with the well-known bivariate tail dependence coefficient and the extremal index of time series in Extreme-Value Theory. We calculate theoretical extremal-range properties for commonly used models, such as Gaussian or regularly varying random fields. Numerical studies illustrate that, when the extremal range is estimated from discretized excursion sets observed on compact observation windows, the distribution of the resulting estimators appropriately reproduces the theoretically derived links with the Lipschitz- Killing curvature densities.

Paper Structure

This paper contains 18 sections, 9 theorems, 28 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The extremal range and relevant notations
  3. Linking the extremal range and the Lipschitz-Killing curvatures
  4. Asymptotics for high thresholds and small distances
  5. Non-degenerate limit distributions of the extremal range
  6. Gaussian random fields
  7. Regularly varying fields
  8. Connections with the tail dependence coefficient
  9. Numerical studies
  10. Probability density of the extremal range near 0
  11. The conditional probability of exceedance as a function of the distance to the conditioning site
  12. Conclusion
  13. Technical definitions and examples
  14. Stationary and positive reach sets
  15. Further details about Assumption \ref{['assumtion']}
  16. ...and 3 more sections

Key Result

Proposition 1

Under Assumption assumtion, for any compact set $T\subset \mathbb{R}^d$ with $\mathcal{L}_d(T) > 0$, the distribution function of $R^{(u)}_0$ is given by for $r\geq 0$, and ${\mathbb P}(R^{(u)}_0 < 0) = 0$, where the subscript $-r$ denotes set erosion by a radius of $r$ (see Definition def:minkowski).

Figures (7)

  • Figure 1: Example of an excursion set $E_X(u)\cap T$ with the quantity $\widetilde{R}^{(u)}(s)$ shown for a chosen $s\in T \subset \mathbb{R}^2$.
  • Figure 2: Two amounts of erosion of the excursion set $E_X(u)\cap T$ from Figure \ref{['fig:excursion']}.
  • Figure 3: For each random field type, the quantity $\lim_{r\to 0^+}{\mathbb P}(R_0^{(u)} \leq r)/r$ is estimated for several values of the threshold $u$. These estimates and their 95% confidence intervals are shown along with the curves that correspond to the theoretical values.
  • Figure 4: See the caption of Figure \ref{['fig:thm1']}. The same information is displayed, but the random fields are normalized to have $\mathrm{Exp}(1)$ margins. That is, the threshold corresponding to $p\in (0,1)$ is $u_p$ as defined in Definition \ref{['def:u_p']}.
  • Figure 5: For each random field type, the quantity $f'_p(0)$ is estimated for several values of $p \in (0,1)$. These estimates and their 95% confidence intervals are shown along with the curves that correspond to the theoretical values.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Definition 1: Excursion and level sets
  • Definition 2: Upper bound
  • Definition 3: Extremal range
  • Remark 1
  • Definition 4: Erosion and dilation
  • Definition 5: $d$ and $(d-1)$-dimensional Lipschitz-Killing curvature densities
  • Remark 2: Discussion of Assumption \ref{['assumtion']}
  • Proposition 1
  • Lemma 1
  • Theorem 1
  • ...and 12 more