Covariance estimation and graphical models for infinite collections of random variables [electronic resource]
- David Montague.
- Physical description
- 1 online resource.
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- Estimating correlations for a large collection of random variables is an important part of an enormous number of modern applications of data science, statistics, and mathematics. In particular, it is often necessary to work with either an infinite collection of variables (such as in a spatial-temporal field) or to approximate a large high-dimensional finite stochastic system with an infinite-dimensional system. Two of the most common techniques used for high dimensional covariance estimation are Markov random fields, also known as graphical models, and positive definite functions, also known as covariance functions, but neither of these approaches has been adequately explored for use with infinite collections of random variables. In particular, the theory of Markov random fields is highly dependent on the finiteness of the graphs used to represent the random variables, and while covariance functions can be used for high-dimensional covariance estimation in certain contexts, they generalize poorly to non-spatiotemporal collections of random variables and have other drawbacks (e.g., they may induce poorly conditioned covariance matrices). This dissertation attempts to expand the mathematical theory of Markov random fields to enable infinite dimensional applications, and explores the relationship between graphical approaches to covariance estimation and positive definite functions.
- Publication date
- Submitted to the Department of Mathematics.
- Thesis (Ph.D.)--Stanford University, 2016.
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