The dynamical modelling of dwarf spheroidal galaxies using Gaussian-process emulation

Abstract

This thesis concerns the use of Gaussian-process mulation (Sacks et al., 1989) to build metamodels of computationally expensive dynamical models of dwarf spheriodal galaxies. These metamodels are computationally cheaper to evaluate than the models that they emulate, and hence have the potential to render tractable previously intractable problems in galactic dynamics. The first part of the thesis deals with the theoretical foundations of Gaussian-process emulation (GPE) while the second part deals with the application of GPE to the modelling of dwarf spheroidal galaxies. I give a description of the general principles of modelling and metamodelling, formally defining a physical model, and showing that the parameter spaces of such models may be made metric or pseudometric spaces. I give a formal treatment of the foundations of GPE and, building on the work of Parzen (1959), give a novel derivation of the GPE predictor and mean-squared error. I also set right some confusion and errors in the literature. In particular, I show that the GPE predictor presented by Rasmussen and Williams (2006) is biased. I quantify this bias, and discuss the circumstances under which it will be significant. In modelling dwarf spheroidal galaxies, I adopt the distribution-function approach, and use GPE to construct a metamodel of the log-likelihood. First, I construct a toy model of a dwarf spheroidal galaxy which I fit using synthetic data drawn from the same toy model. I maximize the log-likelihood using the method of efficient global optimization (Jones et al., 1998), finding that I am able to recover robust confidence regions for the parameter vector, galactic density, and velocity anisotropy with fewer than 100 model evaluations. Second, I construct a more general model. Although the resulting predictions are accurate, the metamodel fails validation, indicating that we may not trust the confidence regions associated with these predictions. We conclude that the usual simplifications made in implementing GPE render it inadequate for the task of predicting the log-likelihood in this case, and that we must consider more general methods.