Electronic Dissertations and Theses - Mathematics http://hdl.handle.net/2376/667 This collection contains theses and dissertations by students in the Department of Mathematics and Statistics at WSU. 2021-06-16T07:14:54Z TAIL MUTUAL INFORMATION OF VINE COPULAS http://hdl.handle.net/2376/16764 TAIL MUTUAL INFORMATION OF VINE COPULAS Multivariate mutual information describes the amount of uncertainty among several random variables, whose scale-invariant dependence is captured by the copula of the joint distribution. In this dissertation, we first show that multivariate mutual information and conditional mutual information of a distribution are the same as these of its underlying copula. The invariance properties allows us to separate one-dimensional marginal distributions and multivariate dependence in the context of analyzing mutual information. Second, we derive the recursive formulas of multivariate mutual information for vine copulas in terms of conditional mutual informations of bivariate liking copulas, based on the tree structures. The important part of this dissertation consists of introducing for the first time in the literature and analyzing the tail mutual information of multivariate extremes. Specifically, we introduce upper and lower tail mutual information of a multivariate distribution and prove that the rates at which upper or lower conditional mutual information converge to zero can be expressed in terms of conditional mutual information of tail densities. We also establish the scaling properties of upper or lower tail mutual information and show that the scaling tail indexes are the same as that of the underlying multivariate regularly varying distribution or the underlying copula. Our results, such as explicit expressions on converging rates and scaling tail indexes, lay the foundations for multivariate extreme value analysis, based on mutual information. 2019-01-01T00:00:00Z A Modified Chang-Wilson-Wolff Inequality Via the Bellman Function http://hdl.handle.net/2376/16730 A Modified Chang-Wilson-Wolff Inequality Via the Bellman Function We produce the optimal constant in an inequality bounding the exponential integral of a function by the exponential integral of its dyadic square function. This work is motivated by a well known result due to Chang, Wilson, and Wolff which controls the exponential integral of a function in terms of the essential supremum of its dyadic square function. Perhaps more interesting than the result itself is the method of proof. We establish our inequality and find the optimal constant using a Bellman function argument. This type of argument was pioneered in the 1980s by Donald Burkholder in his work on martingale transforms. Along the way, we trace the origin of the Bellman function technique back to Richard Bellman’s development of dynamic programming in the 1950s. Doing so provides some historical context and lends insight into the genesis of Burkholder’s ideas. 2019-01-01T00:00:00Z EXPERIMENTS IN MEDICAL IMAGE SEGMENTATIONS http://hdl.handle.net/2376/16741 EXPERIMENTS IN MEDICAL IMAGE SEGMENTATIONS Non-invasive Radiology Imaging (e.g. CT, MRI, and PET) have been utilized tremendously in medical study for disease diagnosis, prognostication, and monitoring therapeutic response. And segmenting medical image for regions of interest is an essential step in computer assisted clinical interventions. Tumor detection in biomedical imaging is a time-consuming process for medical professionals and with nonneglectable human variation in recent decades, researchers have developed algorithmic techniques for image processing using a wide variety of mathematical methods, such as statistical modeling, variational techniques, and machine learning. Graph theory is the framework for the study explained in this thesis. We focus on both theoretical and practical aspects, with an emphasis on the experimental, practical parts. On the theoretical part, we propose a graph cut based semi-automatic method for liver segmentation of 2D CT scans into three labels denoting healthy, vessel, or tumor tissue. In chapter one, some definitions and algorithms for networks are introduced. Medical data and the idea of convolution are also introduced. In chapter two, we propose a new model to segment using the image sequences from dynamic CT scan. Also a particular image can be picked from the images sequence and then vectorized using a sequence of convolutions or neighborhoods. We also introduce a method of minimization that we call the α method. In chapter three, a moving mean method is developed and tested, using a more traditional train-test partition of the data. In chapter four, we explore the potential of deep neural networks. We decided to begin with U-Net, which consists of a contracting path to capture context and a symmetric expanding path that enables precise localization and we developed for medical data. 2019-01-01T00:00:00Z Spectrally Arbitrary Patterns Over Various Rings http://hdl.handle.net/2376/16801 Spectrally Arbitrary Patterns Over Various Rings A pattern $\mathcal{A}$ is a matrix where the location, but not the magnitude, of the nonzero entries are known. A subpattern of $\mathcal{A}$, say $\mathcal{B}$, is pattern where a nonzero entry from $\mathcal{A}$ may be zero in $\mathcal{B}$. Pattern $\mathcal{A}$ is spectrally arbitrary over $\mathscr{R}$, a commutative ring with unity, if for each $n$-th degree monic polynomial $f(x)\in\mathscr{R}[x]$, there exists a matrix $A$ over $\mathscr{R}$ with pattern $\mathcal{A}$ where the characteristic polynomial $p_A(x)=f(x)$. Similarly, a pattern $\mathcal{A}$ is relaxed spectrally arbitrary over $\mathscr{R}$ if for each $n$-th degree monic polynomial $f(x)\in\mathscr{R}[x]$, there exists a matrix $A$ over $\mathscr{R}$ with either pattern $\mathcal{A}$ or a subpattern of $\mathcal{A}$ where the characteristic polynomial $p_A(x)=f(x)$. We evaluate how the structure of rings, compared to the structure of fields, affects how we determine if a pattern is spectrally arbitrary. Using these results, we consider whether a pattern $\mathcal{A}$ that is spectrally arbitrary over $\mathscr{R}$ is spectrally arbitrary or relaxed spectrally arbitrary over another commutative ring $\mathscr{S}$ with unity, establishing some results using ring homomorphisms. Our results imply that a spectrally arbitrary pattern over the integers, $\mathbb{Z}$, is relaxed spectrally arbitrary over the integers modulo $m$, $\mathbb{Z}/m\mathbb{Z}$, and spectrally arbitrary over $\mathbb{Q}$. Similarly, a spectrally arbitrary pattern over the $p$-adic integers, $\mathbb{Z}_p$ for some prime $p$, is spectrally arbitrary over the $p$-adic numbers, $\mathbb{Q}_p$, and the finite field of order $p$, $\mathbb{F}_p$. We establish the minimum number of nonzero entries necessary for $n\times n$ pattern to be relaxed spectrally arbitrary over a ring when $n\leq4$. We also obtain some necessary conditions on the digraph of a relaxed spectrally arbitrary pattern over $\mathbb{F}_2$ and $\mathbb{Z}$. The combination of all our results provide us with some insight on how mappings preserve spectrally arbitrariness. Additionally, they provide a library of $3\times 3$ irreducible spectrally arbitrary and relaxed spectrally arbitrary patterns over various rings, along with a list of $2\times 2$ and $3\times 3$ patterns that are not relaxed spectrally arbitrary over $\mathbb{Z}$. Finally, we consider numerous open questions related to finite rings, relaxed spectrally arbitrary patterns, and potential techniques for finding spectrally arbitrary patterns over $\mathbb{Q}_p$. Thesis (Ph.D.), Mathematics, Washington State University 2019-01-01T00:00:00Z