A $K$-function for inhomogeneous random measures with geometric features

TL;DR

This paper develops a $K$-function for inhomogeneous random measures with geometric marks, incorporating geometric features and density estimation, demonstrated on fiber pattern data and steel fibers in concrete.

Contribution

Introduces a $K$-function for geometric marked point processes with inhomogeneous density estimation, extending second-order analysis to geometric features.

Findings

Effective in analyzing fiber patterns and steel fiber data

Provides parametric density estimation methods

Enhances understanding of geometric features in random measures

Abstract

This paper introduces a $K$-function for assessing second-order properties of inhomogeneous random measures generated by marked point processes. The marks can be geometric objects like fibers or sets of positive volume, and the presented $K$-function takes into account geometric features of the marks, such as tangent directions of fibers. The $K$-function requires an estimate of the inhomogeneous density function of the random measure. We introduce parametric estimates for the density function based on parametric models that represent large scale features of the inhomogeneous random measure. The proposed methodology is applied to simulated fiber patterns as well as a three-dimensional data set of steel fibers in concrete.