Estimating Components of Univariate Gaussian Mixtures Using Prony's Method

Area of Science:

• Statistics
• Machine Learning
• Signal Processing

Background:

• Estimating parameters for Gaussian mixture models is crucial in various statistical and machine learning applications.
• Traditional method of moments struggles with analytical or reliable numerical solutions for more than two components due to nonlinear equations.
• Existing methods lack robustness for multi-component Gaussian mixture analysis.

Purpose of the Study:

• To develop a novel, analytically feasible, and numerically reliable technique for estimating parameters of univariate Gaussian mixture distributions.
• To address the limitations of existing methods in solving nonlinear moment equations for multi-component mixtures.
• To provide a robust solution for parameter estimation in mixtures with equal variances or equal means.

Main Methods:

• Utilizes the method of moments by equating sample and mixture moments.
• Transforms nonlinear moment equations into linear equations using Prony's method under specific conditions (equal variances or equal means).
• Applies the transformed linear equations for analytical and numerical solutions of component parameters.

Main Results:

• The proposed technique provides analytically feasible and numerically reliable solutions for parameter estimation in Gaussian mixtures.
• Successfully demonstrated the method's efficacy on two-, three-, and four-component mixtures.
• Overcomes the limitations of previous methods for mixtures with more than two components.

Conclusions:

• The new Prony's method-based approach offers a significant advancement in estimating parameters for Gaussian mixture models.
• This technique enhances the reliability and feasibility of analyzing complex multi-component distributions.
• The method is particularly effective for mixtures with components sharing equal variances or equal means.