# Kinematic Formula for Heterogeneous Gaussian Related Fields

## TL;DR

This paper generalizes the Gaussian Kinematic Formula to heterogeneous Gaussian-related fields, enabling better analysis of complex data like CMB polarization by decoupling expected Euler characteristics into manifold and distributional components.

## Contribution

It introduces a generalized GKF for non-Gaussian, heterogeneous Gaussian-related fields, including novel Gaussian Minkowski Functionals based on ellipsoidal tube volume expansions.

## Key findings

- Decouples expected Euler characteristic into LKCs and GMFs.
- Introduces Gaussian Minkowski Functionals for ellipsoidal tubes.
- Extends GKF to non-Gaussian, heterogeneous fields.

## Abstract

We provide a generalization of the Gaussian Kinematic Formula (GKF) in Taylor(2006) for multivariate, heterogeneous Gaussian-related fields. The fields under consideration are non-Gaussian fields built out of smooth, independent Gaussian fields with heterogeneity in distribution amongst the individual building blocks. Our motivation comes from potential applications in the analysis of Cosmological Data (CMB). Specifically, future CMB experiments will be focusing on polarization data, typically modeled as isotropic vector-valued Gaussian related fields with independent, but non-identically distributed Gaussian building blocks; this necessitates such a generalization. Extending results of Taylor(2006) to these more general Gaussian relatives with distributional heterogeneity, we present a generalized Gaussian Kinematic Formula (GKF). The GKF in this paper decouples the expected Euler characteristic of excursion sets into Lipschitz Killing Curvatures (LKCs) of the underlying manifold and certain Gaussian Minkowski Functionals (GMFs). These GMFs arise from Gaussian volume expansions of ellipsoidal tubes as opposed to the usual tubes in the Euclidean volume of classical tube formulae. The GMFs form a main contribution of this work that identifies this tubular structure and a corresponding volume of tubes expansion in which the GMFs appear.