Plotting correlated data

TL;DR

This paper discusses the challenges of interpreting correlated uncertainties in data visualization plots and proposes methods to improve the assessment of model fit by visualizing principal components and conditional uncertainties.

Contribution

It introduces techniques to enhance data plots by explicitly showing principal components and conditional uncertainties, aiding better interpretation of correlated errors.

Findings

Correlated uncertainties complicate model fit assessment in plots.

Adding principal component contributions improves understanding of data-model agreement.

Displaying conditional uncertainties helps identify where models may be deficient.

Abstract

A very common task in data visualization is to plot many data points with some measured y-value as a function of fixed x-values. Uncertainties on the y-values are typically presented as vertical error bars that represent either a Frequentist confidence interval or Bayesian credible interval for each data point. Most of the time, these error bars represent a 68\% confidence/credibility level, which leads to the intuition that a model fits the data reasonably well if its prediction lies within the error bars of roughly two thirds of the data points. Unfortunately, this and other intuitions no longer work when the uncertainties of the data points are correlated. If the error bars only show the square root of diagonal elements of some covariance matrix with non-negligible off-diagonal elements, we simply do not have enough information in the plot to judge whether a drawn model line agrees well with the data or not. In this paper we will demonstrate this problem and discuss ways to add more information to the plots to make it easier to judge the agreement between the data and some model prediction in the plot, as well as glean some insight where the model might be deficient. This is done by explicitly showing the contribution of the first principal component of the uncertainties, and by displaying the conditional uncertainties of all data points.