Constructing a multivariate distribution function with a vine copula: toward multivariate luminosity and mass functions

TL;DR

This paper introduces a systematic method using vine copulas to construct high-dimensional multivariate distribution functions, exemplified by a galaxy mass function involving stellar, atomic, and molecular gas.

Contribution

It extends the copula method to higher dimensions with vine copulas, providing a flexible, systematic approach for constructing complex multivariate distributions from marginal data.

Findings

Successfully constructed a 3D galaxy mass function using vine copulas.

Demonstrated maximum likelihood estimation and model selection for the vine copula-based mass function.

Showed the flexibility and extendability of vine copulas for high-dimensional distribution modeling.

Abstract

The need for a method to construct multidimensional distribution function is increasing recently, in the era of huge multiwavelength surveys. We have proposed a systematic method to build a bivariate luminosity or mass function of galaxies by using a copula. It allows us to construct a distribution function when only its marginal distributions are known, and we have to estimate the dependence structure from data. A typical example is the situation that we have univariate luminosity functions at some wavelengths for a survey, but the joint distribution is unknown. Main limitation of the copula method is that it is not easy to extend a joint function to higher dimensions ($d > 2$), except some special cases like multidimensional Gaussian. Even if we find such a multivariate analytic function in some fortunate case, it would often be inflexible and impractical. In this work, we show a systematic method to extend the copula method to unlimitedly higher dimensions by a vine copula. This is based on the pair-copula decomposition of a general multivariate distribution. We show how the vine copula construction is flexible and extendable. We also present an example of the construction of an stellar mass--atomic gas--molecular gas 3-dimensional mass function. We demonstrate the maximum likelihood estimation of the best functional form for this function, as well as a proper model selection via vine copula.