[Paper Review] Mixture Models in Astronomy

Michael A. Kuhn, Eric D. Feigelson|arXiv (Cornell University)|Nov 29, 2017

Bayesian Methods and Mixture Models44 references9 citations

TL;DR

This paper advocates for the use of mixture models—particularly Gaussian and astrophysically informed distributions—in astronomy for cluster analysis, classification, and density estimation. It demonstrates their effectiveness in handling overlapping populations, spatial distributions, and complex data with heteroscedastic errors, using real-world examples from SDSS and LSST surveys, and highlights their role in improving object classification and model selection over heuristic methods.

ABSTRACT

Mixture models combine multiple components into a single probability density function. They are a natural statistical model for many situations in astronomy, such as surveys containing multiple types of objects, cluster analysis in various data spaces, and complicated distribution functions. This chapter in the CRC Handbook of Mixture Analysis is concerned with astronomical applications of mixture models for cluster analysis, classification, and semi-parametric density estimation. We present several classification examples from the literature, including identification of a new class, analysis of contaminants, and overlapping populations. In most cases, mixtures of normal (Gaussian) distributions are used, but it is sometimes necessary to use different distribution functions derived from astrophysical experience. We also address the use of mixture models for the analysis of spatial distributions of objects, like galaxies in redshift surveys or young stars in star-forming regions. In the case of galaxy clustering, mixture models may not be the optimal choice for understanding the homogeneous and isotropic structure of voids and filaments. However, we show that mixture models, using astrophysical models for star clusters, may provide a natural solution to the problem of subdividing a young stellar population into subclusters. Finally, we explore how mixture models can be used for mathematically advanced modeling of data with heteroscedastic uncertainties or missing values, providing two example algorithms, the measurement error regression model of Kelly (2007) and the Extreme Deconvolution model of Bovy et al. (2011). The challenges presented by astronomical science, aided by the public availability of catalogs from major surveys and missions, are a rich area for collaboration between statisticians and astronomers.

Motivation & Objective

• To demonstrate the utility of mixture models in addressing complex astronomical classification and clustering problems.
• To show how mixture models improve upon heuristic and subjective classification methods in multi-dimensional data spaces.
• To promote the use of parametric mixture models with model selection criteria (e.g., BIC) for stable, reproducible clustering in large astronomical surveys.
• To highlight the integration of statistical methodology with astrophysical knowledge for modeling spatial and photometric distributions.
• To encourage collaboration between statisticians and astronomers by showcasing publicly available data and real applications in modern surveys.

Proposed method

• Application of finite Gaussian mixture models to classify astronomical objects in multi-dimensional parameter spaces, such as galaxy emission line properties.
• Use of model selection criteria like the Bayesian Information Criterion (BIC) to objectively determine the number of components in a mixture model.
• Adaptation of astrophysically motivated distributions (e.g., log-normal, gamma, Pareto) for modeling physical quantities like stellar masses and galaxy luminosities.
• Implementation of measurement error regression models (Kelly, 2007) and Extreme Deconvolution (Bovy et al., 2011) to handle heteroscedastic uncertainties in astronomical data.
• Use of spatial mixture models to subdivide young stellar populations into subclusters using astrophysical models of star cluster formation.
• Leveraging large-scale survey data (e.g., SDSS, LSST) and public catalogs to validate and apply mixture modeling techniques.

Experimental results

Research questions

• RQ1 How can mixture models improve the classification of galaxies based on emission line properties compared to traditional heuristic methods?
• RQ2 In what ways can mixture models provide stable and reproducible clustering results in the presence of overlapping populations in astronomical surveys?
• RQ3 How can mixture models be adapted to incorporate astrophysical priors when modeling distributions of physical quantities such as stellar masses or galaxy luminosities?
• RQ4 What role do mixture models play in analyzing spatial distributions of stars and galaxies, particularly in young star-forming regions or redshift surveys?
• RQ5 How can advanced mixture modeling techniques handle measurement errors and missing data in large astronomical datasets?

Key findings

• Gaussian mixture models successfully refined a traditional 3-cluster classification of galaxies in the Baldwin-Phillips-Terlevich diagram into a more accurate 4-cluster structure.
• Model selection via BIC led to more stable and reproducible component counts than subjective or nonparametric clustering methods like 'friends-of-friends'.
• Mixture models using astrophysical priors enabled natural subdivision of young stellar populations into subclusters, improving the analysis of star formation regions.
• The application of Extreme Deconvolution and measurement error regression models allowed for robust inference in data with heteroscedastic uncertainties, common in astronomical photometry and spectroscopy.
• Despite widespread use, astronomers often do not recognize mixture models by name, indicating a need for greater methodological awareness and collaboration with statisticians.
• Publicly available catalogs from surveys like SDSS and LSST provide rich, large-scale datasets ideal for testing and advancing mixture modeling techniques.