Mean conservation for density estimation via diffusion using the finite element method

TL;DR

This paper introduces boundary conditions for the diffusion equation that preserve mean and mass in density estimation, supported by finite element method experiments and potential applications in one and two dimensions.

Contribution

It develops a novel boundary condition framework for diffusion-based density estimation that maintains key statistical properties of the data.

Findings

Boundary conditions preserve initial mean and total mass.

Finite element method effectively implements the proposed framework.

Potential for extending to two-dimensional diffusion problems.

Abstract

We propose boundary conditions for the diffusion equation that maintain the initial mean and the total mass of a discrete data sample in the density estimation process. A complete study of this framework with numerical experiments using the finite element method is presented for the one dimensional diffusion equation, some possible applications of this results are presented as well. We also comment on a similar methodology for the two-dimensional diffusion equation for future applications in two-dimensional domains.