The UU-test for Statistical Modeling of Unimodal Data

TL;DR

The paper introduces the UU-test, a new statistical method for determining whether one-dimensional data is unimodal, using the empirical CDF and modeling data segments as uniform distributions, with applications in clustering and data analysis.

Contribution

It proposes the UU-test, a novel approach that combines empirical CDF approximation with uniform mixture modeling to assess unimodality in data.

Findings

UU-test effectively decides unimodality compared to dip-test.

It provides a uniform mixture model for unimodal data.

The method's models are validated using log-likelihood and KS tests.

Abstract

Deciding on the unimodality of a dataset is an important problem in data analysis and statistical modeling. It allows to obtain knowledge about the structure of the dataset, ie. whether data points have been generated by a probability distribution with a single or more than one peaks. Such knowledge is very useful for several data analysis problems, such as for deciding on the number of clusters and determining unimodal projections. We propose a technique called UU-test (Unimodal Uniform test) to decide on the unimodality of a one-dimensional dataset. The method operates on the empirical cumulative density function (ecdf) of the dataset. It attempts to build a piecewise linear approximation of the ecdf that is unimodal and models the data sufficiently in the sense that the data corresponding to each linear segment follows the uniform distribution. A unique feature of this approach is that in the case of unimodality, it also provides a statistical model of the data in the form of a Uniform Mixture Model. We present experimental results in order to assess the ability of the method to decide on unimodality and perform comparisons with the well-known dip-test approach. In addition, in the case of unimodal datasets we evaluate the Uniform Mixture Models provided by the proposed method using the test set log-likelihood and the two-sample Kolmogorov-Smirnov (KS) test.