- Thesis:
- Inauguraldissertation zur Erlangung der Doktorwürde der Naturwissenschaftlich-Mathematischen Gesamtfakultät der Ruprecht-Karls-Universität Heidelberg
- urn:nbn:de:bsz:16-heidok-296774
- Author:
- Christof Schötz (Heidelberg University)
- Title:
- The Fréchet mean and statistics in non-euclidean spaces
- Supervisors and examiners:
- Jan JOHANNES
- Enno Mammen (Heidelberg University)
- Second examiner:
- Hans Georg Müller (University of California, Davis)
- Abstract:
- In this thesis, we study statistical properties of the Fréchet mean and its generalizations in abstract settings. These settings include large classes of scenarios, which may be of great interest in practice when dealing with nonstandard data. Our main focus is on the convergence of sample Fréchet means of independent observations to their population counterpart. The results are exemplarily applied to some specific spaces. The expectation of a real-valued, square-integrable random variable is characterized by being the unique constant value that minimizes the expected squared difference to the random variable. One can use this property to generalize the notion of mean. A Fréchet mean of a metric space-valued random variable is any minimizer of the expected squared distance to that random variable. This definition achieves two important things: Firstly, it encompasses many commonly used types of mean – like the expectation, the median, or the geometric mean – allowing to state powerful, general, and far-reaching theorems about properties of means. Secondly, it defines a mean for non-Euclidean spaces – like the sphere, the space of phylogenetic trees, or Wasserstein spaces – opening up these spaces for profound applications of probability theory and statistics. We show strong laws of large numbers of Fréchet mean sets with two different notions of convergence of sets assuming only a first moment condition. After having established consistency of the sample Fréchet mean, we investigate the rate of this convergence. We demonstrate, using projected means, an instance of the Fréchet mean, that Fréchet means may exhibit very different rates depending on the geometry of the metric space and properties of the distribution of the data. Then we prove rates of convergence in a general setting under some conditions. One of these is the quadruple inequality – a generalization of the Cauchy-Schwarz inequality. This and some other conditions are fulfilled in Hadamard spaces – geodesic metric spaces of nonpositive curvature – which makes them particularly interesting to study in the context of Fréchet means. We show a quadruple inequality for certain powers of Hadamard metrics – a purely geometric result with an intriguingly complex proof. Lastly, we examine regression models where responses live in a metric space and the regression function is a conditional Fréchet mean. We compare two approaches to transform known estimators to this non-Euclidean setting. In doing so, we establish rates of convergence for four different estimation procedures, two of which are new methods. To illustrate these regression estimators, an R-package was developed that allows their application and comparison on the sphere.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |