insight - Algorithms and Data Structures - # Parameter Estimation for One-dimensional Gaussian Mixture Models

Core Concepts

A novel algorithm is proposed to estimate the parameters, including the number of components, variance, means, and weights, in one-dimensional Gaussian mixture models by leveraging the Hankel structure in the Fourier data obtained from i.i.d. samples. The algorithm does not require prior knowledge of the number of components or good initial guesses, and it demonstrates superior performance in estimation accuracy and computational cost compared to classic methods like the method of moments and maximum likelihood.

Abstract

The paper introduces a novel algorithm for estimating the parameters of one-dimensional Gaussian mixture models (GMMs). The key highlights are:
The algorithm takes advantage of the Hankel structure inherent in the Fourier data obtained from independent and identically distributed (i.i.d) samples of the mixture.
For GMMs with a unified variance, a singular value ratio (SVR) functional using the Fourier data is introduced and used to resolve the variance and component number simultaneously. The consistency of the estimator is derived.
Compared to classic algorithms such as the method of moments and the maximum likelihood method, the proposed algorithm does not require prior knowledge of the number of Gaussian components or good initial guesses.
Numerical experiments demonstrate the superior performance of the proposed algorithm in estimation accuracy and computational cost compared to the EM algorithm.
The paper also reveals a fundamental limit to the problem of estimating the number of Gaussian components or model order in the mixture model if the number of i.i.d samples is finite. It is shown that the model order can be successfully estimated only if the minimum separation distance between the component means exceeds a certain threshold value, referred to as the computational resolution limit.

Stats

The variance of the Gaussian noise term W(ω) is of the order O(1/√n), where n is the number of i.i.d. samples.
The minimum separation distance between the component means is denoted as dmin.
The minimum weight of the components is denoted as πmin.

Quotes

"The purpose of this paper is twofold. First, we propose a novel algorithm for estimating parameters in one-dimensional Gaussian mixture models (GMMs). The algorithm takes advantage of the Hankel structure inherent in the Fourier data obtained from independent and identically distributed (i.i.d) samples of the mixture."
"Second, we reveal that there exists a fundamental limit to the problem of estimating the number of Gaussian components or model order in the mixture model if the number of i.i.d samples is finite."

Key Insights Distilled From

by Xinyu Liu,Ha... at **arxiv.org** 04-22-2024

Deeper Inquiries

To extend the proposed algorithm to handle Gaussian mixtures with unequal variances, we can modify the algorithm to estimate the variances separately for each component. Instead of assuming a unified variance for all components, we can introduce a variance parameter for each Gaussian component in the mixture model. This would involve adjusting the Hankel matrix construction and the singular value ratio functional to account for multiple variances. The algorithm would then iterate over each variance parameter to estimate the optimal values for all variances simultaneously. By incorporating this modification, the algorithm can effectively handle Gaussian mixtures with unequal variances.

The computational resolution limit has significant implications for the practical applications of Gaussian mixture models. This limit sets a threshold on the minimum separation distance between the means of the Gaussian components for accurate model order estimation. When the separation distance falls below this threshold, the model order estimation may fail, leading to inaccurate parameter estimates. This limitation highlights the importance of considering the sample size, variance, and number of components when applying Gaussian mixture models in real-world scenarios. Practitioners need to be aware of this computational resolution limit to ensure reliable and accurate model selection in their applications.

The ideas behind Fourier-based parameter estimation for Gaussian mixture models can be extended to other types of mixture models beyond Gaussian distributions. By leveraging the Fourier data obtained from samples of the mixture model, similar algorithms can be developed for estimating parameters in mixture models with different distributions, such as Poisson mixtures, exponential mixtures, or even non-parametric mixtures. The key lies in adapting the Hankel matrix structure and the singular value ratio functional to suit the characteristics of the specific distribution in the mixture model. This approach can offer a novel and efficient way to estimate parameters in a wide range of mixture models, enhancing the applicability of Fourier-based methods in statistical modeling and analysis.

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