Estimating multidimensional probability fields using the Field Estimator for Arbitrary Spaces (FiEstAS) with applications to astrophysics

TL;DR

FiEstAS is a versatile algorithm that estimates continuous probability density fields in multi-dimensional spaces, useful for astrophysical data analysis, by constructing a data-driven, metric-independent tessellation and applying kernel density estimation with bias correction.

Contribution

The paper introduces FiEstAS, a novel, metric-independent tessellation-based method for estimating probability density fields in arbitrary dimensions, with practical bias correction and application to astrophysics.

Findings

Effective in 2D and 6D astrophysical models

Provides a balance between resolution and variance

Offers guidelines for optimal parameter selection

Abstract

The Field Estimator for Arbitrary Spaces (FiEstAS) computes the continuous probability density field underlying a given discrete data sample in multiple, non-commensurate dimensions. The algorithm works by constructing a metric-independent tessellation of the data space based on a recursive binary splitting. Individual, data-driven bandwidths are assigned to each point, scaled so that a constant "mass" M0 is enclosed. Kernel density estimation may then be performed for different kernel shapes, and a combination of balloon and sample point estimators is proposed as a compromise between resolution and variance. A bias correction is evaluated for the particular (yet common) case where the density is computed exactly at the locations of the data points rather than at an uncorrelated set of locations. By default, the algorithm combines a top-hat kernel with M0=2.0 with the balloon estimator and applies the corresponding bias correction. These settings are shown to yield reasonable results for a simple test case, a two-dimensional ring, that illustrates the performance for oblique distributions, as well as for a six-dimensional Hernquist sphere, a fairly realistic model of the dynamical structure of stellar bulges in galaxies and dark matter haloes in cosmological N-body simulations. Results for different parameter settings are discussed in order to provide a guideline to select an optimal configuration in other cases. Source code is available upon request.