A Mathematical Framework and Software Implementation for Uncertainty Visualisation

2026-06-23

AI Summary

Existing visualization methods lack systematic support for random variables, making it challenging to balance statistical rigor and graphical flexibility in exploratory data analysis. This work proposes a mathematical framework that treats visualizations as continuous functions, leveraging the Continuous Mapping Theorem to handle random-variable inputs. By decomposing visualization functions, the approach identifies and reconstructs components incompatible with uncertainty-aware data. As a core contribution, the study introduces the first complete integration of uncertainty into the grammar of graphics, ensuring that resulting visualizations inherit the convergence properties of their underlying random variables. To operationalize this framework, the authors develop ggdibbler, an R package extending ggplot2 that enables direct plotting of random variables, thereby enhancing analytical flexibility while preserving statistical validity.

Abstract

Random variables are the bread and butter of statistics, and visualisations are one of the most versatile tools in the field, so it is a wonder why we do not have a methodology for visualising random variables. This gap is particularly evident for exploratory data analysis (EDA). We address this gap by designing a mathematical framework for visualisation, which argues that we should consider visualisations to be continuous functions. In the case of random variable inputs, this means the visualisations should obey the continuous mapping theorem (CMT). By breaking the visual function down into its components, we are able to identify which parts of the mapping are ill-defined for random variable inputs and redefine them in a way that guarantees both the flexibility required for EDA and the statistical sensibility of CMT. This formalisation represents a complete integration of uncertainty into the grammar of graphics, which we show by implementing the theory in the R package, "ggdibbler". The ggdibbler software is a "ggplot2" extension that allows users to replace the data of any plotting function with a random variable, with the guarantee that the visualisation will have the same convergence properties as its underlying data.

Problem

• uncertainty visualisation
• random variables
• exploratory data analysis
• grammar of graphics

Innovation

• uncertainty visualisation
• continuous mapping theorem
• random variables
• grammar of graphics
• exploratory data analysis