Mathematical Sciences - Doctoral Theses
Permanent URI for this collection
Browse
Recent Submissions
Item Metaheuristics and machine learning for joint stratification and sample allocation in survey design(University College Cork, 2022-01) O'Luing, Mervyn; Prestwich, Steve; Tarim, Armagan; European Regional Development Fund; Science Foundation IrelandIn this thesis, we propose a number of metaheuristics and machine learning techniques to solve the joint stratification and sample allocation problem. Finding the optimal solution to this problem is hard when the sampling frame is large, and the evaluation algorithm is computationally burdensome. To advance the research in this area, we explore and evaluate different algorithmic methods of modelling and solving this problem. Firstly, we propose a new genetic algorithm approach using "grouping" genetic operators instead of traditional operators. Experiments show a significant improvement in solution quality for similar computational effort. Next, we combine the capability of a simulated annealing algorithm to escape from local minima with delta evaluation to exploit the similarity between consecutive solutions and thereby reduce evaluation time. Comparisons with two recent algorithms show the simulated annealing algorithm attaining comparable solution qualities in less computation time. Then, we consider the combination of the k-means and clustering algorithms with a hill climbing algorithm in stages and report the solution costs, evaluation times and training times. The multi-stage combinations generally compare well with recent algorithms, and provide the survey designer with a greater choice of algorithms to choose from. Finally, we combine the explorative properties of an estimation of distribution algorithm (EDA) to model the probabilities of an atomic stratum belonging to different strata with the exploitative search properties of a simulated annealing algorithm to create a hybrid estimation of distribution algorithm (HEDA). Results of comparisons with the best solution qualities from our earlier experiments show that the HEDA finds better solution qualities, but requires a longer total execution time than alternative approaches we considered.Item Improved statistical quantitation of dynamic PET scans(University College Cork, 2023-03-15) Gu, Fengyun; O'Sullivan, Finbarr; Huang, Jian; University College Cork; Science Foundation Ireland; National Cancer Institute USA; Swedish Cancer Foundation; IEEE FoundationPositron emission tomography (PET) scanning is an important diagnostic imaging technique used in the management of cancer patients and in medical research. It plays a key role in a variety of tasks related to diagnosis, therapy planning, prognosis and treatment monitoring by injecting a radiotracer to characterize the specific biologic process (e.g., tumor metabolism, proliferation or blood flow). The standardized uptake value (SUV) obtained at a single time point is widely employed in clinical practice. However, well beyond this simple static uptake measure, more detailed metabolic information may be recovered from dynamic PET scanning with multiple time frames. Assuming a tracer’s interaction with the tissue is linear and time-invariant, the tissue time course can be expressed as a convolution between arterial input function (AIF) and the tissue impulse response/residue function. Kinetic analysis is concerned with the estimation of residue and associated physiological parameters such as flow, flux and volume of distribution. Some traditional methods including Patlak and compartmental modeling are well-established with a given form of residue function (constant or mixture of exponentials), but they are not flexible to represent data in which in-vivo biochemistry is not clear, especially for the whole-body imaging on the long axial field of view (LAFOV) PET systems. The main goals of this thesis are to develop novel statistical approaches for improving and evaluating parametric imaging extracted from dynamic PET scans. The non-parametric residue mapping (NPRM) procedure has been constructed by a fully automatic process incorporating data-adaptive segmentation, non-parametric residue analysis of segment data and voxel-level kinetic mapping scheme. Based on this approach, the benefits of pooling data in multiple injection PET scans are investigated. Spatial and temporal patterns of residuals recovered by model diagnostics exhibit a non-Gaussian structure, which defines a bootstrap data generation process (DGP) in the image domain. The proposed bootstrap method has been used to assess the uncertainty (standard errors) in kinetic information and more complex regional summaries. We also examine its potential to improve the mean square error (MSE) characteristics of kinetic maps generated from either compartmental modeling or NPRM approach by averaging results from individual bootstrap samples. Dynamic breast cancer studies on the early, recent and latest LAFOV PET scanners are presented to illustrate these techniques. The performance of above models and schemes has been evaluated in a series of one and two-dimensional numerical simulations. Both direct filtered backprojection (FBP) and iterative maximum likelihood (ML) reconstructions are considered. The proposed NPRM approach has some important features like the flexibility for diverse tissue environments and consideration of delays for different parts, which make it promising to be applied to the emerging total-body PET imaging. The developed image-domain bootstrap provides a practical way to quantify the uncertainties of biomarkers. This mechanism has the potential to further support clinical decision-making and enhance personalized medicine.Item Mathematical and computational approaches to contagion dynamics on networks(University College Cork, 2022-09-27) Humphries, Rory; Hoevel, Philipp; Mulchrone, Kieran F.In this thesis, we firstly introduce the basic terminology and concepts needed for the the following chapters. In particular we introduce the basics of graph/network theory, epidemiological models (both well mixed and on networks), and mobility models (the gravity and radiation models). After the introduction of these topics, we propose a general framework for epidemiological network models from which the known individual-based and pair-based models can be derived. We then introduce a more exact pair-based model by showing previous iterations are a linearised version of it, and then we extend it further to the temporal setting. Next, we present a meta-population model for the spread of COVID-19 in Ireland which makes use of temporal commuting patters generated from the radiation model. Finally, we analyse a year worth of Irish cattle trade data. We then fit a number of mobility models and show that an altered version of the radiation model, which we call the generalised radiation model, is able to accurately reproduce the distance distribution of cattle trades in the country.Item Algebraic central limit theorems in noncommutative probability(University College Cork, 2022-01-24) Alahmade, Ayman; Koestler, Claus; Taibah UniversityDistributional symmetries and invariance principles in noncommutative probability theory provide sufficient conditions for the existence of central limit laws. In contrast to classical probability theory, there exist many different central limit laws for exchangeable sequences of noncommutative random variables and still little is known about their concrete form. This thesis goes one step further and investigates central limit laws for non-exchangeable spreadable sequences in the context of *-algebraic probability spaces. This provides first results on a new type of combinatorics underlying multivariate central limit theorems (CLTs). The starting point of the thesis has been a quite simple family of spreadable sequences, which is parametrized by a unimodular complex parameter ω. Each sequence of this family is spreadable, but not exchangeable for ω different from ±1. Moreover, the sequences from this family provide CLTs, which interpolate between the normal distribution (ω = 1) and the symmetric Bernoulli distribution (ω = −1), but differ from q-Gaussian distributions (−1 < q < 1). An algebraic structure, which underlies the considered family, is identified and used to define so-called ‘ω-sequences of partial isometries’. These ω-sequences encode all information, as it is relevant for computations of *-algebraic CLTs. Explicit combinatorial formulas are established for CLTs associated to such ω-sequences, which involve the counting of oriented crossings of directed ordered pair partitions. The limiting distributions of certain multivariate CLTs associated to ω-sequences show some features as they are defining for ‘z-circular systems’ in the work of Mingo and Nica. This similarity, as well as the well-known relation between q-circular systems and q-semicircular systems (for −1 ≤ q ≤ 1), guides the introduction of ‘z-semicircular systems’ in this thesis. Finally, it is shown that the class of z-semicircular systems is stable under certain multivariate central limits. In other words, the moment formulas of z-semicircular systems are reproduced in large N-limit formulas of central limit type.Item The inverse limit of GIT quotients of Grassmannians by the maximal torus(University College Cork, 2015) Yazdanpanah, Vahid; Mustata, Andrei; Mustata, Anca; Science Foundation IrelandThere are finitely many GIT quotients of 𝐺(3 𝑛) by maximal torus and between each two there is a birational map. These GIT quotients and the flips between them form an inverse system. This thesis describes this inverse system first and then, describes the inverse limit of this inverse system as a moduli space. An open set in this scheme represents the functor of arrangements of lines in planes. We show how to enrich this functor such that it is represented by the above inverse limit.