# A crossinggram for random fields on lattices

## TL;DR

This paper introduces a new crossinggram coefficient for max-stable random fields on lattices, which measures the tendency of the field's trajectories to oscillate around high levels, aiding in risk modeling.

## Contribution

The paper proposes a novel crossinggram coefficient that captures local oscillation tendencies of max-stable random fields, providing an interpretable and estimable summary measure.

## Key findings

- Crossinggram values range from 0 to 1, increasing with variable concordance.
- Higher crossinggram indicates smoother surface trajectories.
- The coefficient offers insights into local oscillations related to risk assessment.

## Abstract

The modeling of risk situations that occur in a space-time framework can be done using max-stable random fields on lattices. Although the summary coefficients for the spatial and temporal behaviour do not characterize the finite-dimensional distributions of the random field, they have the advantage of being immediate to interpret and easier to estimate. The coefficients that we propose give us information about the tendency of a random field for local oscillations of its values in relation to real valued high levels. It is not the magnitude of the oscillations that is being evaluated, but rather the greater or lesser number of oscillations, that is, the tendency of the trajectories to oscillate. We can observe surface trajectories more smooth over a region according to higher crossinggram value. It takes value in $[0,1]$ and increases with the concordance of the variables of the random field.