Type

Text

Type

Dissertation

Advisor

Frey, Robert J. | Mullhaupt, Andrew P | Douady, Raphael | Djuric, Petar.

Date

2017-08-01

Keywords

Covariance | Applied mathematics | Dependance | Information Geometry | Multivariate Distributions

Department

Department of Applied Mathematics and Statistics.

Language

en_US

Source

This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree

Identifier

http://hdl.handle.net/11401/78204

Publisher

The Graduate School, Stony Brook University: Stony Brook, NY.

Format

application/pdf

Abstract

Current multivariate distributions have a static covariance structure. This implies that the levels of covariance between components of a system are the same during “normal” times and tail events. Many real world systems do not exhibit constant covariance between components across time. There are currently two primary ways to deal with this issue. The first is to take a heterogeneous approach by fitting a multivariate distribution to “normal” data and use extreme value theory to analyze tail data. The second approach is to fit marginal distributions to the components and use them to construct a copula. We show that there are drawbacks to each of these techniques. We suggest a more homogenous approach through the development of a new class of multivariate distributions called dynamic elliptical distributions. Dynamic elliptical distributions have a covariance matrix whose entries are functions. This dynamic covariance matrix acts as a local metric on the sample space, which allows the degree of covariance to change as one moves from the center of the distribution to its tails. We develop sampling and fitting methods for dynamic elliptical distributions and show the role they play in information geometry. More specifically we derive the general information geometry of elliptical distributions and show conditions under which these smooth manifolds become Einstein manifolds. Finally we show that any dynamic elliptical distribution can be regarded as a submanifold on a manifold of elliptical distributions. | 101 pages

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