A "Rosetta Stone" for Studies of Spatial Variation in Astrophysical Data: Power Spectra, Semivariograms, Structure Functions, and More

TL;DR

This paper unifies various mathematical tools used in astrophysical spatial data analysis, such as power spectra and semivariograms, to facilitate cross-disciplinary understanding and comparison of results.

Contribution

It provides a comprehensive comparison and translation guide for different spatial correlation measures used across astrophysics subfields.

Findings

Clarifies the relationships between key spatial data products.

Provides conditions for the usefulness of each method.

Enables translation of results between different analysis techniques.

Abstract

From the turbulent interstellar medium to the cosmic web, astronomers in many different fields have needed to make sense of spatial data describing our Universe. Through different historical choices for mathematical conventions, many different subfields of spatial data analysis have evolved their own language for analysing structures and quantifying correlation in spatial data. Because of this history, terminology from a myriad of different fields is used, often to describe two data products that are mathematically identical. In this Note, we define and describe the differences and similarities between the power spectrum, the two-point correlation function, the covariance function, the semivariogram, and the structure functions, in an effort to unify the languages used to study spatial correlation. We also highlight under which conditions these data products are useful and describe how the results found using one method can be translated to those found using another, allowing for easier comparison between different subfields' native methods. We hope for this document to be a ``Rosetta Stone" for translating between different statistical approaches, allowing results to be shared between researchers from different backgrounds, facilitating more cross-disciplinary approaches to data analysis.