Manifold Learning on Empirical Probability Distributions

This thesis presents dimension reduction methods for empirical probability distributions to explore the underlying structure of high-dimensional data, and these methods are applied to detect households with anomalous electricity usage patterns. The first main chapter deal with the non-Euclidean distance estimation when the data objects are probability distributions and the associated computational efficiency. The second main chapter shows how to correct the distortion from dimension reduction when estimating the density of the underlying structure. And the third main chapter explores the change in the distribution of electricity consumption brought by COVID-19 in Melbourne.