k-diagram: Technical Report — Derivations and Details

This technical report provides the detailed mathematical foundations for the k-diagram Python package, a novel toolkit for visualising and diagnosing probabilistic forecasts. This document serves as a citable supplement and technical appendix to the main JOSS paper, "k-diagram: Rethinking Forecasting Uncertainty via Polar-based Visualisation". It contains the formal derivations and equations for the package's core visualisation methods, including: Score calculation, normalisation, and polar coordinate mapping. Formulations for key plots such as coverage evaluation, error violins, velocity diagrams, polar heatmaps, and regression performance charts. The definition of the pinball loss for quantile forecast evaluation. This report is essential for researchers, developers, and practitioners who wish to gain a deeper theoretical understanding of the diagnostic tools provided by k-diagram. k-diagram GitHub Repository: https://github.com/earthai-tech/k-diagram k-diagram Documentation: https://k-diagram.readthedocs.io/