A new goodness-of-fit statistic and its application to 21-cm cosmology

TL;DR

The paper introduces a new goodness-of-fit statistic called the reduced psi-squared, which is more sensitive than chi-squared in detecting low-level features in noisy data, demonstrated through 21-cm cosmology applications.

Contribution

A novel goodness-of-fit measure, the reduced psi-squared, that improves detection of subtle features in noisy data and is demonstrated with 21-cm cosmology data analysis.

Findings

The reduced psi-squared detects low-level features better than chi-squared.

It maintains sensitivity to outliers while identifying wide, low-level signals.

A Python tool, psipy, is released for calculating this statistic efficiently.

Abstract

The reduced chi-squared statistic is a commonly used goodness-of-fit measure, but it cannot easily detect features near the noise level, even when a large amount of data is available. In this paper, we introduce a new goodness-of-fit measure that we name the reduced psi-squared statistic. It probes the two-point correlations in the residuals of a fit, whereas chi-squared accounts for only the absolute values of each residual point, not considering the relationship between these points. The new statistic maintains sensitivity to individual outliers, but is superior to chi-squared in detecting wide, low level features in the presence of a large number of noisy data points. After presenting this new statistic, we show an instance of its use in the context of analyzing radio spectroscopic data for 21-cm cosmology experiments. We perform fits to simulated data with four components: foreground emission, the global 21-cm signal, an idealized instrument systematic, and noise. This example is particularly timely given the ongoing efforts to confirm the first observational result for this signal, where this work found its original motivation. In addition, we release a Python script dubbed $\texttt{psipy}$ which allows for quick, efficient calculation of the reduced psi-squared statistic on arbitrary data arrays, to be applied in any field of study.