A General Approach to Visualizing Uncertainty in Statistical Graphics

TL;DR

This paper introduces a general framework for visualizing uncertainty in static 2-D graphics by treating a base visualization as a function of an uncertain quantity, which induces a distribution over images. By sampling from the base distribution and aggregating the resulting base-visualization outputs, it yields a single uncertainty-encoded image with formal coverage guarantees, without explicitly computing uncertainty. A reference implementation, bootplot, demonstrates IID-based applicability and a broad design space across common and novel visualization channels. The work bridges traditional sample-based overlays with formal inference, offering a practical tool for teaching, rapid prototyping, and uncovering flaws in base visualizations while enabling visual inference tasks.

Abstract

We present a general approach to visualizing uncertainty in static 2-D statistical graphics. If we treat a visualization as a function of its underlying quantities, uncertainty in those quantities induces a distribution over images. We show how to aggregate these images into a single visualization that represents the uncertainty. The approach can be viewed as a generalization of sample-based approaches that use overlay. Notably, standard representations, such as confidence intervals and bands, emerge with their usual coverage guarantees without being explicitly quantified or visualized. As a proof of concept, we implement our approach in the IID setting using resampling, provided as an open-source Python library. Because the approach operates directly on images, the user needs only to supply the data and the code for visualizing the quantities of interest without uncertainty. Through several examples, we show how both familiar and novel forms of uncertainty visualization can be created. The implementation is not only a practical validation of the underlying theory but also an immediately usable tool that can complement existing uncertainty-visualization libraries.

Figures (11)

• Figure 1: Illustrative examples of uncertainty visualizations in four common scenarios.
• Figure 2: Two examples of our approach to uncertainty visualization.
• Figure 3: This code snippet illustrates bootplot's use. The dataset is just an array of numbers, and the quantity of interest is the mean. The user provides code for visualizing the mean (the base visualization), following a prescribed signature, and passes it to a bootplot call. Both the left-hand and right-hand image are the result of the proposed method of generating images of multiple samples from the target distribution and then combining them. The only difference is that for the right-hand the axes were not set (we remove line 4). Instead, the axes are set automatically and vary across samples, resulting in unwanted variability.
• Figure 4: An illustrative example of the transformation of the intensities in the case of two unique values.
• Figure 5: The top-left image is the unaltered aggregate image ($\tau = 0$ and $k = 1$ is the identity transform). The bottom-middle image results in a solid color ($\tau = \frac{1}{2}$ is a constant function). A solid color also appears when the slope parameter $k$ is taken to either extreme, as shown in the bottom-left and bottom-right images. Intermediate values of $k$ control the gradient, as seen in the top-middle and top-right images.