Journal Article

A geometric approach to non-linear correlations with intrinsic scatter

Pauli Pihajoki

in Monthly Notices of the Royal Astronomical Society

Published on behalf of Royal Astronomical Society

Volume 472, issue 3, pages 3407-3424
ISSN: 0035-8711
Published online August 2017 | e-ISSN: 1365-2966 | DOI: http://dx.doi.org/10.1093/mnras/stx2179
A geometric approach to non-linear correlations with intrinsic scatter

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Abstract

We propose a new mathematical model for n − k-dimensional non-linear correlations with intrinsic scatter in n-dimensional data. The model is based on Riemannian geometry and is naturally symmetric with respect to the measured variables and invariant under coordinate transformations. We combine the model with a Bayesian approach for estimating the parameters of the correlation relation and the intrinsic scatter. A side benefit of the approach is that censored and truncated data sets and independent, arbitrary measurement errors can be incorporated. We also derive analytic likelihoods for the typical astrophysical use case of linear relations in n-dimensional Euclidean space. We pay particular attention to the case of linear regression in two dimensions and compare our results to existing methods. Finally, we apply our methodology to the well-known MBH–σ correlation between the mass of a supermassive black hole in the centre of a galactic bulge and the corresponding bulge velocity dispersion. The main result of our analysis is that the most likely slope of this correlation is ∼6 for the data sets used, rather than the values in the range of ∼4–5 typically quoted in the literature for these data.

Keywords: methods: data analysis; methods: statistical; galaxies: fundamental parameters

Journal Article.  16292 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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