# Probabilistic multi-catalogue positional cross-match

**Authors:** F. -X. Pineau, S. Derriere, C. Motch, F. J. Carrera, F. Genova, L. Michel, B. Mingo, A. Mints, A. Nebot Gómez-Morán, S. R. Rosen, A. Ruiz Camuñas  
**arXiv:** [1609.00818](https://arxiv.org/abs/1609.00818)  
**Date:** 2016-09-03  
**Summary page:** https://www.summarizepaper.com/en/paper/1609.00818v1/

## Key points

- Statistical framework for multi-catalogue cross-correlation and cross-identification
- Emphasizes the importance of considering both astrometric and photometric data
- Proposes using Chi (or Chi-square) test with 2(n-1) degrees of freedom for matching candidates
- Merges astrometric probabilities with photometric likelihoods using Bayes' formula
- Estimates expected numbers of spurious associations by calculating volume of Chi-test domain of acceptance
- Derives priors for spurious associations using a frequentist approach based on geometrical considerations
- Validates theoretical results through generation and cross-matching of synthetic catalogues
- Results do not depend on the order used to cross-correlate the catalogues
- Applies formalism to construct multi-wavelength catalogues for ARCHES project
- Makes cross-matching engine publicly available through a multi-purpose web interface
- Plans to integrate into CDS XMatch Service in future developments

## Summary

This paper presents a statistical framework for multi-catalogue cross-correlation and cross-identification, focusing primarily on the astrometric part of the identification process. The authors emphasize the importance of considering both astrometric and photometric data in order to achieve proper identification. They propose using the Chi (or Chi-square) test with 2(n-1) degrees of freedom to select matching candidates in n catalogues, referring to this cross-match as a chi-match. To merge the astrometric probabilities with photometric likelihoods, they employ Bayes' formula and consider exhaustive sets of hypotheses based on combinatorial analysis. They estimate the expected numbers of spurious associations by calculating the volume of the Chi-test domain of acceptance, represented as a 2(n-1)-dimensional acceptance ellipsoid. Priors for these numbers are derived using a frequentist approach that relies on simple geometrical considerations. Likelihoods in this framework are based on normalized Rayleigh, Chi, and Poisson distributions over the Chi-test acceptance domain. The theoretical results are validated through the generation and cross-matching of synthetic catalogues. The authors highlight that their results do not depend on the order used to cross-correlate the catalogues. They apply their formalism to construct multi-wavelength catalogues for the ARCHES (Astronomical Resource Cross-matching for High Energy Studies) project and make their cross-matching engine publicly available through a multi-purpose web interface. Additionally, they plan to integrate it into CDS XMatch Service in future developments. Overall, this paper provides a comprehensive statistical framework for multi catalogue positional cross match which offers insights into astrometric probabilities and their integration with photometric likelihoods.