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On probability of union of events and applications in neuroimaging
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Lee, Joonhee. On probability of union of events and applications in neuroimaging. Retrieved from https://doi.org/doi:10.7282/t3-kdg4-8f60

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TitleOn probability of union of events and applications in neuroimaging
NameLee, Joonhee (author); Boros, Endre (chair); Katehakis, Michael (member); Baykal-Gürsoy, Melike (member); Lidbetter, Thomas (member); Kang, Hakmook (member); Rutgers University; School of Graduate Studies
Date Created2023
Other Date2023-05 (degree)
SubjectOperations research, Linear programming, Neuroimaging, Probability, Quadratic programming
Extent76 pages : illustrations
DescriptionGeorge Boole proposed the union bounding problem, which is a class of probabilistic satisfiability/optimization problems. In its most frequently studied case, we are given the probabilities of $n$ events and also the probabilities of their pairwise intersections, and want to know how small/large can the probability of their union be. Hailperin (1965) provided a linear programming model for this problem which involves $2^n-1$ variables. The model generates tight bounds (both the minimum and the maximum are realizable by events) but it may be hard to compute. In Chapter 2, we provide a proof of NP-Hardness for the feasibility of Hailperin's Linear Programming model. Large number of papers were published in the last 100 years that provide upper and/or lower bounds for this problem. Many are based on Hailperin's model and considers a relaxation that either has polynomial dimension in $n$ or at least is computable in time polynomial in $n$.
We study methods that aggregate subsets of variables in Hailperin's model in various ways. These aggregation methods can be viewed as monotone linear mappings $mathcal{L}$ to a lower $N$ dimensional space such that $mathcal{L}(mathbb{R}_+^{2^n-1}) subseteq mathbb{R}_+^N$. While several publications in the literature considered this type of relaxations of Hailperin's model, only few noted that we may have strict containment in the above relation. In Chapter 3, we provide a complete characterization of the cone $mathcal{L}(mathbb{R}_+^{2^n-1})$ for three different aggregation models from the literature -- the so called binomial moment problem (see e.g., Pr'ekopa (1988)), the relaxation by Pr'ekopa and Gao (2005), and the model by Yang, Alajaji, and Takahara (2018). These characterizations lead to strictly improved bounds in the last two of these cases. We also obtain an $O(n^2)$ dimensional polyhedral formulation that is equivalent with Hailperin's model, simultaneously for minimization and maximization, unlike the ones obtained by linear programming duality. An interesting side result is that there exists a new connection between Hailperin's model and the Boolean Quadric polytope (see e.g., Padberg (1985)). In Chapter 4, we generalize the previous result to an arbitrary degree $d$. As a result, we provide a complete polyhedral characterization of $L_{cE^d}(RR_+^gO)$. That is, we obtain an $O(n^d)$ dimensional linear program that is equivalent with Hailperin's model.
A branch in Neuroimaging focuses on the functional connectivity in a brain. Regions of Interest (ROIs) are clusters of voxels or brain regions based on the location or the function in a brain. Chapter 5 presents a possible application of the union bounding problem in resting state fMRI data.
NotePh.D.
NoteIncludes bibliographical references
Genretheses
Persistent URLhttps://doi.org/doi:10.7282/t3-kdg4-8f60
LanguageEnglish
CollectionSchool of Graduate Studies Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.
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