Abstract
(a) The concept of correlation. Covariance and the correlation coefficient; (b) Testing for correlation. Rank correlation tests; (c) Correlation test pitfalls. Least squares fitting. Fitting to a straight line; (d) Testing the fit. Curvilinear line fitting; (e) Arbitrary function fitting.
A large part of science concerns looking for, and then trying to understand, causal relations between observed quantities. This is much harder where random variables are concerned. For example, take a look at Fig. 9.1. Each data point represents a galaxy where both the mass of the “bulge” component of a galaxy, and the mass of the central supermassive black hole it contains, have been estimated. It looks like these things are connected, which could be important. But the points are rather scattered, and have large error bars. Are we just being fooled by a chance distribution of points drawn from some random distribution? Note also that the authors have drawn a line going through the data, hopefully representing the true relationship between black hole mass and bulge mass. But is the slope of the line right? How do we decide what the “best” slope is? And what if a straight line isn’t the right mathematical form? Can I test the prediction for my favourite theory?
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Notes
- 1.
The simulations were all generated from bivariate Gaussan PDFs, but in the bottom row, we have made the ring shape by subtracting one Gaussian from another.
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Websites (all accessed March 2019):
Blog post by Rasmus Bååth on Bayesian correlation testing. http://www.sumsar.net/blog/2013/08/bayesian-estimation-of-correlation/
Amusing list of spurious correlations by Tyler Vyglen. http://tylervigen.com/spurious-correlations
Wikipedia page on multi-variate Gaussians. https://en.wikipedia.org/wiki/Multivariate_normal_distribution
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Lawrence, A. (2019). Inference with Two Variables: Correlation Testing and Line Fitting. In: Probability in Physics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-04544-9_9
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