A note on averaging prediction accuracy, Green's functions and other kernels
This work provides a better mathematical understanding of the prediction accuracy index, which is used for identifying hot spots in predictive security and other applications, but it is incremental in nature.
The paper tackles the mathematical underpinnings of the prediction accuracy index by introducing an integral average transform and relating it to two-variable kernels, resulting in a novel integral representation for solutions of the Poisson equation.
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.