8 - Multivariate analysis

Summary

The astronomical context

Whenever an astronomer is faced with a dataset that can be presented as a table — rows representing celestial objects and columns representing measured or inferred properties — then the many tools of multivariate statistics come into play. Multivariate datasets also arise in other situations. Astronomical images can be viewed as tables of three variables: right ascension, declination and brightness. Here the spatial variables are in a fixed lattice while the brightness is a random variable. An astronomical datacube has a fourth variable that may be wavelength (for spectro-imaging) or time (for multi-epoch imaging). High-energy (X-ray, gamma-ray, neutrino) detectors give tables where each row is a photon or event with columns representing properties such as arrival direction and energy. Calculations arising from astrophysical models also produce outputs that can be formulated as multivariate datasets, such as N-body simulations of star or galaxy interactions, or hydrodynamical simulations of gas densities and motion.

For multivariate datasets, we designate n for the number of objects in the dataset and p for the number of variables, the dimensionality of the problem. In traditional multivariate analysis, n is large compared to p; statistical methods for high-dimensional problems with p > n are now under development. The variables can have a variety of forms: real numbers representing measurements in any physical unit; integer values representing counts of some variable; ordinal values representing a sequence; binary variables representing “Yes/No” categories; or nonsequential categorical indicators.

We address multivariate issues in several chapters of this volume. The present chapter on multivariate analysis considers datasets that are commonly displayed in a table of objects and properties.