A note on averaging prediction accuracy, Green's functions and other kernels

TL;DR

This paper explores the mathematical foundations of the prediction accuracy index, introducing an integral average transform and relating it to kernels, with applications to solving the Poisson equation and identifying hot spots.

Contribution

It defines the integral average transform, relates it to kernel functions, and provides a new integral representation for solutions to the Poisson equation.

Findings

Integral average transform relates to kernel functions.

Re-interpretation of Poisson equation solutions via averages.

Enhanced understanding of the prediction accuracy index.

Abstract

We present the mathematical context of the predictive accuracy index and then introduce the definition of integral average transform. We establish the relation of our definition with two variables kernels $K({\bf y},{\bf x})$. As an example of an application we show that integrating against the fundamental solution of the Laplace operator, that is, solving the Poisson equation, can be re-interpreted as an integral of averages of the forcing term over balls. As a result, we obtained a novel integral representation of the solution of the Poisson equation. Our motivation comes from the need for a better mathematical understanding of the prediction accuracy index. This index is used to identify hot spots in predictive security and other applications.