Field Formulation of Parzen Data Analysis

TL;DR

This paper introduces a formal field-based framework for Parzen window density estimation, linking it to scalar fields and differential equations, enhancing data analysis and clustering techniques.

Contribution

It presents a novel formalism connecting Parzen density, potential fields, and differential equations, providing new insights for clustering and data exploration.

Findings

Loci of extrema serve as clustering focal points.

Potential and density fields are interconnected through differential equations.

The formalism extends to Schrödinger and diffusion equations in data spaces.

Abstract

The Parzen window density is a well-known technique, associating Gaussian kernels with data points. It is a very useful tool in data exploration, with particular importance for clustering schemes and image analysis. This method is presented here within a formalism containing scalar fields, such as the density function and its potential, and their corresponding gradients. The potential is derived from the density through the dependence of the latter on the common scale parameter of all Gaussian kernels. The loci of extrema of the density and potential scalar fields are points of interest which obey a variation condition on a novel indicator function. They serve as focal points of clustering methods depending on maximization of the density, or minimization of the potential, accordingly. The mixed inter-dependencies of the different fields in d-dim data-space and 1-d scale-space, are discussed. They lead to a Schr\H{o}dinger equation in d-dim, and to a diffusion equation in (d+1)-dim