- Mathematical Sciences - Doctoral Theses

# Mathematical Sciences - Doctoral Theses

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- ItemMetaheuristics 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 Ireland
Show more In 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.Show more - ItemImproved 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 Foundation
Show more Positron 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.Show more - ItemMathematical and computational approaches to contagion dynamics on networks(University College Cork, 2022-09-27) Humphries, Rory; Hoevel, Philipp; Mulchrone, Kieran F.
Show more 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.Show more - ItemThe role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model(University College Cork, 2022-07-04) Maulana, Heru; Kelly, Conall; Indonesia Endowment Fund for Education (LPDP)
Show more We demonstrate the effectiveness of an adaptive explicit Euler method for the approximate solution of the Cox-Ingersoll-Ross model. This relies on a class of path-bounded timestepping strategies which work by reducing the stepsize as solutions approach a neighbourhood of zero. The method is hybrid in the sense that a convergent backstop method is invoked if the timestep becomes too small, or to prevent solutions from overshooting zero and becoming negative. Under parameter constraints that imply Feller’s condition, we prove that such a scheme is strongly convergent, of order at least 1/2. Control of the strong error is important for multi-level Monte Carlo techniques. Under Feller’s condition we also prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. Numerically, we compare this adaptive method to fixed step implicit and explicit schemes, and a novel semi-implicit adaptive variant. We observe that the adaptive approach leads to methods that are competitive in a domain that extends beyond Feller’s condition, indicating suitability for the modelling of stochastic volatility in Heston-type asset models.Show more - ItemOption pricing and CVA calculations using the Monte Carlo-Tree (MC-Tree) method(University College Cork, 2022-07-18) Trinh, Yen Thuan; Hanzon, Bernard; Ma, Jingtang
Show more The thesis introduces a new method, the MC-Tree method, for pricing certain financial derivatives, especially options with high accuracy and efficiency. Our solution is to combine Monte Carlo (MC) method and Tree method by doing a mixing distribution on the tree, and the output is the compound distribution on the tree. The compound distribution in the tree output (after a logarithmic transformation of the asset prices) is not the ideal Gaussian distribution but has entropy values close to the maximum possible Gaussian entropy. We can get closer using entropy maximization. We introduce two correction techniques: distribution correction and bias correction to improve the accuracy and completeness of the model. The thesis presents an algorithm and numerical results for calculations of CVA on an American put option using the MC-Tree method. The MC-Tree method with the distribution correction technique significantly improves accuracy, resulting in practically exact solutions, compared to analytical solutions, at the tree depth $N=50$ or $100$ and MC-drawings $M=10^5$. The bias-correction technique makes the resulting tree model complete in the sense of financial mathematics and obtains the risk-neutral probability. Besides, we have obtained new formulae for the calculations of the entropy and the Kullback-Leibler divergence for rational densities and approximate entropy of finite Gaussian mixture.Show more