Nonparametric, data-based kernel interpolation for particle-tracking simulations and kernel density estimation

TL;DR

This paper introduces a nonparametric, data-driven kernel interpolation method for particle-tracking simulations that adaptively determines the kernel shape based on the particles' distribution, improving density estimation especially for skewed or heavy-tailed data.

Contribution

It proposes an iterative, nonparametric kernel construction method that aligns the kernel with the particle cloud, enhancing the accuracy of Green's function and density estimations.

Findings

Effective kernel density estimation for skewed data

Applicable to heavy-tailed distributions and breakthrough curves

Demonstrated improved interpolation accuracy

Abstract

Traditional interpolation techniques for particle tracking include binning and convolutional formulas that use pre-determined (i.e., closed-form, parameteric) kernels. In many instances, the particles are introduced as point sources in time and space, so the cloud of particles (either in space or time) is a discrete representation of the Green's function of an underlying PDE. As such, each particle is a sample from the Green's function; therefore, each particle should be distributed according to the Green's function. In short, the kernel of a convolutional interpolation of the particle sample "cloud" should be a replica of the cloud itself. This idea gives rise to an iterative method by which the form of the kernel may be discerned in the process of interpolating the Green's function. When the Green's function is a density, this method is broadly applicable to interpolating a kernel density estimate based on random data drawn from a single distribution. We formulate and construct the algorithm and demonstrate its ability to perform kernel density estimation of skewed and/or heavy-tailed data including breakthrough curves.