Statistical Modeling of Univariate Multimodal Data

TL;DR

This paper introduces a non-parametric, hyperparameter-free method for partitioning univariate data into unimodal subsets using density valley detection, and models each subset with a Uniform Mixture Model to form a hierarchical unimodal mixture model.

Contribution

It proposes a novel recursive splitting method based on density valleys and constructs a hierarchical unimodal mixture model without requiring prior parameters.

Findings

Accurately partitions data into unimodal subsets.

Effectively models each subset with a UMM.

Demonstrates superior performance in clustering and density estimation.

Abstract

Unimodality constitutes a key property indicating grouping behavior of the data around a single mode of its density. We propose a method that partitions univariate data into unimodal subsets through recursive splitting around valley points of the data density. For valley point detection, we introduce properties of critical points on the convex hull of the empirical cumulative density function (ecdf) plot that provide indications on the existence of density valleys. Next, we apply a unimodal data modeling approach that provides a statistical model for each obtained unimodal subset in the form of a Uniform Mixture Model (UMM). Consequently, a hierarchical statistical model of the initial dataset is obtained in the form of a mixture of UMMs, named as the Unimodal Mixture Model (UDMM). The proposed method is non-parametric, hyperparameter-free, automatically estimates the number of unimodal subsets and provides accurate statistical models as indicated by experimental results on clustering and density estimation tasks.