A Gibbs Sampler for Multivariate Linear Regression

Abstract

Kelly (2007, hereafter K07) described an efficient algorithm, using Gibbs sampling, for performing linear regression in the fairly general case where non-zero measurement errors exist for both the covariates and response variables, where these measurements may be correlated (for the same data point), where the response variable is affected by intrinsic scatter in addition to measurement error, and where the prior distribution of covariates is modeled by a flexible mixture of Gaussians rather than assumed to be uniform. Here I extend the K07 algorithm in two ways. First, the procedure is generalized to the case of multiple response variables. Second, I describe how to model the prior distribution of covariates using a Dirichlet process, which can be thought of as a Gaussian mixture where the number of mixture components is learned from the data. I present an example of multivariate regression using the extended algorithm, namely fitting scaling relations of the gas mass, temperature, and luminosity of dynamically relaxed galaxy clusters as a function of their mass and redshift. An implementation of the Gibbs sampler in the R language, called LRGS, is provided.

11 pages, 5 figures, 2 tables. Code is available on GitHub at https://github.com/abmantz/lrgs and from CRAN

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• Some Aspects of Measurement Error in Linear Regression of Astronomical Data
• LoCuSS: Comparison of Observed X-ray and Lensing Galaxy Cluster Scaling Relations with Simulations
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Cited by in corpus (17)

• Weighing the Giants V: Galaxy Cluster Scaling Relations
• Cosmology and Astrophysics from Relaxed Galaxy Clusters III: Thermodynamic Profiles and Scaling Relations
• X-ray scaling relations for a representative sample of Planck selected clusters observed with XMM-Newton
• CCCP and MENeaCS: (updated) weak-lensing masses for 100 galaxy clusters
• LoCuSS: Scaling relations between galaxy cluster mass, gas, and stellar content
• CoMaLit-V. Mass forecasting with proxies. Method and application to weak lensing calibrated samples
• XXL Survey groups and clusters in the Hyper Suprime-Cam Survey. Scaling relations between X-ray properties and weak lensing mass
• Center-Excised X-ray Luminosity as an Efficient Mass Proxy for Future Galaxy Cluster Surveys
• GalWeight Application: A publicly-available catalog of dynamical parameters of 1,800 galaxy clusters from SDSS-DR13, ($GalWCat19$)
• Coping with Selection Effects: A Primer on Regression with Truncated Data
• KLLR: A scale-dependent, multivariate model class for regression analysis
• The Evolution and Mass Dependence of Galaxy Cluster Pressure Profiles at 0.05 $\le z\le$ 0.60 and $4\times 10^{14}$ M${}_{\odot }$ $\le {\text{M}}_{500}\le 30\times 10^{14}$ M${}_{\odot }$
• Marginalised Normal Regression: Unbiased curve fitting in the presence of x-errors
• The XXL Survey: XXXVIII. Scatters and correlations of X-ray proxies in the bright XXL cluster sample
• Scaling relations of clusters and groups, and their evolution
• An approach to robust Bayesian regression in astronomy
• Forecasting the $Y_{500}-M_{500}$ scaling relation from the NIKA2 SZ Large Program