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/