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

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 ( y , 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.

A discussion of current crime forecasting indices and an improvement to the prediction efficiency index for applications

“…Relevant statistical metrics are summarized in White and Hunt (2022). Crime analysts should avoid solely using PAI, as it is not a reliable metric (Joshi et al 2021;Galvis, Hernández-Romero, and Gómez 2022). Instead, we suggest analysts use PEI and/or PEI* as outlined here and look for better metrics as they are developed.…”

“…To address various study areas, one study developed the penalized predictive accuracy index (PPAI), where the area ratio is penalized (Joshi et al 2021). However, mathematicians have shown that PAI and its adaptions are still susceptible to artificial manipulation since PAI increases as subspace size (e.g., grid cell height, street length) decreases (Galvis, Hernández-Romero, and Gómez 2022). Hunt 2016 demonstrates this fact by explicitly showing that from a meso-to macro-place perspective, PAI is sensitive to grid cell size.…”