Second-order properties for planar Mondrian tessellations

TL;DR

This paper derives explicit formulas for second-order properties of weighted planar Mondrian tessellations, which are relevant in machine learning, providing insights into their correlation functions and variance behavior.

Contribution

It introduces explicit formulas for second-order properties of weighted planar Mondrian tessellations, advancing understanding of their spatial correlation structures.

Findings

Explicit formulas for pair- and cross-correlation functions.

Asymptotic behavior of variances analyzed.

Applications in machine learning contexts.

Abstract

In this paper planar STIT tesselations with weighted axis-parallel cutting directions are considered. They are known also as weighted planar Mondrian tesselations in the machine learning literature, where they are used in random forest learning and kernel methods. Various second-order properties of such random tessellations are derived, in particular, explicit formulas are obtained for suitably adapted versions of the pair- and cross-correlation functions of the length measure on the edge skeleton and the vertex point process. Also, explicit formulas and the asymptotic behaviour of variances are discussed in detail.