A transformation-based approach to Gaussian mixture density estimation for bounded data

TL;DR

This paper introduces a transformation-based Gaussian mixture modeling approach for bounded data, addressing bias issues in standard models by estimating densities on transformed data and jointly optimizing parameters with EM.

Contribution

It proposes a novel transformation-based method for Gaussian mixture density estimation tailored for bounded variables, improving accuracy over traditional models.

Findings

Effective on simulated data

Applicable to real-world bounded data

Reduces bias in density estimation

Abstract

Finite mixture of Gaussian distributions provide a flexible semi-parametric methodology for density estimation when the variables under investigation have no boundaries. However, in practical applications variables may be partially bounded (e.g. taking non-negative values) or completely bounded (e.g. taking values in the unit interval). In this case the standard Gaussian finite mixture model assigns non-zero densities to any possible values, even to those outside the ranges where the variables are defined, hence resulting in severe bias. In this paper we propose a transformation-based approach for Gaussian mixture modelling in case of bounded variables. The basic idea is to carry out density estimation not on the original data but on appropriately transformed data. Then, the density for the original data can be obtained by a change of variables. Both the transformation parameters and the parameters of the Gaussian mixture are jointly estimated by the Expectation-Maximisation (EM) algorithm. The methodology for partially and completely bounded data is illustrated using both simulated data and real data applications.