Singularities in bivariate normal mixtures

TL;DR

This paper classifies the singularities of bivariate normal mixture mappings, revealing three types with specific geometric properties and establishing upper bounds for the number of modes in each case.

Contribution

It provides a novel classification of bivariate normal mixture mappings using singularity theory and characterizes their geometric and statistical properties.

Findings

Identified three distinct types of mappings with specific geometric features.

Determined upper bounds for the number of modes in each mapping type.

Connected singularity classifications to statistical properties of mixtures.

Abstract

We investigate mappings $F = (f_1, f_2) \colon \mathbb{R}^2 \to \mathbb{R}^2 $ where $ f_1, f_2 $ are bivariate normal densities from the perspective of singularity theory of mappings, motivated by the need to understand properties of two-component bivariate normal mixtures. We show a classification of mappings $ F = (f_1, f_2) $ via $\mathcal{A}$-equivalence and characterize them using statistical notions. Our analysis reveals three distinct types, each with specific geometric properties. Furthermore, we determine the upper bounds for the number of modes in the mixture for each type.