Mathematics and Statistics, Department ofQueen's University Informationhttps://hdl.handle.net/1974/6https://qspace.library.queensu.ca/retrieve/4a39211c-e083-4c71-963c-9986a27e5cd4/2024-02-27T10:02:43Z2024-02-27T10:02:43Z2071A Framework for the Meta-Analysis of Survey DataFox, Karla Michelle Phillips Cooperhttps://hdl.handle.net/1974/69002020-02-03T20:19:56Z2011-12-06T00:00:00Zdc.title: A Framework for the Meta-Analysis of Survey Data
dc.contributor.author: Fox, Karla Michelle Phillips Cooper
dc.description.abstract: The research outlined in this thesis covers various different statistical issues relating to meta-analysis of survey data. These issues include the creation of an original comprehensive methodological framework for combining survey data, a comparison of this framework with the traditional one proposed by Cochran for the combination of experiments, a proposal for a new weighting method that takes into account the differences in variability due to the sampling plan, an examination of the convergence of meta-analytic estimators, and a discussion on the numerous implicit assumptions researchers make when they are using meta-analysis methods with survey data along with guidelines for completing and reporting reviews when the data come from surveys.
dc.description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-12-06 15:11:24.717
2011-12-06T00:00:00ZA Geometric Approach to Energy ShapingGharesifard, Bahmanhttps://hdl.handle.net/1974/51142020-02-03T20:09:02Z2009-09-02T22:51:42Zdc.title: A Geometric Approach to Energy Shaping
dc.contributor.author: Gharesifard, Bahman
dc.description.abstract: In this thesis is initiated a more systematic geometric exploration of energy shaping. Most of the previous results have been dealt wih particular cases and
neither the existence nor the space of solutions has been discussed with any degree of generality. The geometric theory of partial differential equations originated by Goldschmidt and Spencer in late 1960s is utilized to analyze the partial differential equations in energy shaping. The energy shaping partial differential equations
are described as a fibered submanifold of a $ k $-jet bundle of a fibered manifold. By revealing the nature of kinetic energy shaping, similarities are noticed between the problem of kinetic energy shaping and some well-known problems in Riemannian geometry. In particular, there is a strong similarity between kinetic energy shaping and the problem of finding a metric connection initiated by Eisenhart and Veblen. We notice that the necessary conditions for the set of so-called $ \lambda $-equation restricted to the control
distribution are related to the Ricci identity, similarly to the Eisenhart and Veblen metric connection problem. Finally, the set of $ \lambda $-equations for kinetic energy shaping are coupled with the integrability results of potential energy shaping. The procedure shows how a poor design of closed-loop metric can make it impossible to achieve any flexibility in the character of the possible closed-loop
potential function. The integrability results of this thesis have been used to answer some interesting questions about the energy shaping. In particular, a geometric proof is provided which shows that linear controllability is sufficient for energy shaping of linear simple mechanical systems. Furthermore, it is shown that all linearly controllable mechanical control systems with one degree of underactuation can be stabilized using energy shaping
feedback. The result is geometric and completely characterizes the
energy shaping problem for these systems. Using the geometric approach of this thesis, some new open problems in energy shaping
are formulated. In particular, we give ideas for relating the kinetic energy shaping problem to a problem on holonomy groups.
Moreover, we suggest that the so-called Fakras lemma might be used for investigating the stabilization condition of energy shaping.
dc.description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-09-02 12:12:55.051
2009-09-02T22:51:42ZA Mathematical Discussion of Corotational Finite Element ModelingCraighead, John Wesleyhttps://hdl.handle.net/1974/63462020-03-31T14:09:38Z2011-03-31T13:18:37Zdc.title: A Mathematical Discussion of Corotational Finite Element Modeling
dc.contributor.author: Craighead, John Wesley
dc.description.abstract: This thesis discusses the mathematics of the Element Independent Corotational (EICR) Method and the more general Unified Small-Strain Corotational Formulation. The former was developed by Rankin, Brogan and Nour-Omid [106]. The latter, created by Felippa and Haugen [49], provides a theoretical frame work for the EICR and similar methods and its own enhanced methods.
The EICR and similar corotational methods analyse non-linear deformation of a body by its discretization into finite elements, each with an orthogonal frame rotating (and translating) with the element. Such methods are well suited to deformations where non-linearity arises from rigid body deformation but local strains are small (1-4%) and so suited to linear analysis. This thesis focuses on such small-strain, non-linear deformations.
The key concept in small-strain corotational methods is the separation of deformation into its rigid body and elastic components. The elastic component then can be analyzed linearly. Assuming rigid translation is removed first, this separation can be viewed as a polar decomposition (F = vR) of the deformation gradient (F) into a rigid rotation (R) followed by a small, approximately linear, stretch (v). This stretch usually causes shear as well as pure stretch.
Using linear algebra, Chapter 3 explains the EICR Method and Unified Small-Strain Corotational Formulation initially without, and then with, the projector operator, reflecting their historical development. Projectors are orthogonal projections which simplify the isolation of elastic deformation and improve element strain invariance to rigid body deformation.
Turning to Lie theory, Chapter 4 summarizes and applies relevant Lie theory to explore rigid and elastic deformation, finite element methods in general, and the EICR Method in particular. Rigid body deformation from a Lie perspective is well represented in the literature which is summarized. A less developed but emerging area in differential geometry (notably, Marsden/Hughes [82]), elastic deformation is discussed thoroughly followed by various Lie aspects of finite element analysis. Finally, the EICR Method is explored using Lie theory. Given the available research, complexity of the area, and level of this thesis, this exploration is less developed than the earlier linear algebraic discussion, but offers a useful alternative perspective on corotational methods.
dc.description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-03-30 21:40:25.831
2011-03-31T13:18:37ZA Mathematical Model for Tick Transmission of Lyme Disease: Does Questing Style Impact the Competition Between Two Strains of Infection?Wasser, Alexandra J.https://hdl.handle.net/1974/326942024-01-05T09:26:07Zdc.title: A Mathematical Model for Tick Transmission of Lyme Disease: Does Questing Style Impact the Competition Between Two Strains of Infection?
dc.contributor.author: Wasser, Alexandra J.
dc.description.abstract: Borrelia burgdorferi is the bacteria responsible for Lyme disease and there
are multiple strains of B. burgdorferi that exist. There is a perception that all
ticks carry Lyme disease and all bacteria responsible for the disease are equal.
Each strain of B. burgdorferi can be characterised by varying persistence in
a host. By investigating the questing behaviours of ticks, an analysis is
conducted to explore the conditions conducive to different strains. A model
is constructed to reproduce the dynamics of the life cycle of the tick and the
relationship with hosts to replicate natural interactions between the tick and
a host species. The model construction captures the intra-seasonal dynamics
using differential equations and inter-seasonal dynamics using a set of discrete
recursions. Using an invasion analysis, the conditions that allow for mutant
invasion and environmental factors that favour invasion are investigated. The
questing styles, both synchronous and asynchronous, are replicated using
mathematical techniques. It is found that the relative fitness of competing
strains is not impacted by questing style but rather the absolute fitness of
a strain is impacted. A conclusion can be drawn that chronic infection is
predominantly found in asynchronous systems and with the worsening state
of climate change, the persistence of acute infection is less likely.
A Morse Index-Maslov Index Theorem for Discrete Lagrangian SystemsKavle, Henryhttps://hdl.handle.net/1974/304642022-10-12T07:03:21Zdc.title: A Morse Index-Maslov Index Theorem for Discrete Lagrangian Systems
dc.contributor.author: Kavle, Henry
dc.description.abstract: We develop a discrete analog to Morse theory for discrete Lagrangian systems, including a discrete Morse index theorem and a correspondence theorem between the Morse indices of critical trajectories in discrete Lagrangian systems and the Maslov index of symplectic paths arising in associated discrete linear Hamiltonian systems. We use this correspondence to adapt criteria for linear stability of periodic trajectories in Lagrangian dynamics to this discrete setting. Finally, we apply some of these results to dynamical billiards.
A New Approach in Survival Analysis with Longitudinal CovariatesPavlov, Andreyhttps://hdl.handle.net/1974/55852020-02-03T19:50:30Z2010-04-27T21:36:33Zdc.title: A New Approach in Survival Analysis with Longitudinal Covariates
dc.contributor.author: Pavlov, Andrey
dc.description.abstract: In this study we look at the problem of analysing survival data in the presence of
longitudinally collected covariates. New methodology for analysing such data has
been developed through the use of hidden Markov modeling. Special attention has
been given to the case of large information volume, where a preliminary data reduction
is necessary. Novel graphical diagnostics have been proposed to assess goodness of fit
and significance of covariates.
The methodology developed has been applied to the data collected on behaviors
of Mexican fruit flies, which were monitored throughout their lives. It has been found
that certain patterns in eating behavior may serve as an aging marker. In particular it
has been established that the frequency of eating is positively correlated with survival
times.
dc.description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2010-04-26 18:34:01.131
2010-04-27T21:36:33ZA Polya Urn Stochastic Model for the Analysis and Control of Epidemics on NetworksHayhoe, Mikhailhttps://hdl.handle.net/1974/220142020-02-03T21:23:11Zdc.title: A Polya Urn Stochastic Model for the Analysis and Control of Epidemics on Networks
dc.contributor.author: Hayhoe, Mikhail
dc.description.abstract: This thesis introduces a model for epidemics on networks based on the classical Polya process. Temporal contagion processes are generated on the network nodes using a modified Polya sampling scheme that accounts for spatial infection among neigh- bouring nodes. The stochastic properties and asymptotic behaviour of the resulting network Polya contagion process are analyzed. Given the complicated nature of this process, three classical Polya processes, one computational and two analytical, are proposed to statistically approximate the contagion process of each node, demon- strating a good fit for a range of system parameters. An optimal control problem is formulated for minimizing the average infection using a limited curing budget, and a number of different curing strategies are presented, including a proven conver- gent gradient descent algorithm. The feasibility of the problem is proven under high curing budgets by deriving conservative lower bounds that turn some processes into supermartingales. Extensive simulations run on large-scale networks demonstrate the effectiveness of our proposed strategies.
A Quantification of Long Transient DynamicsLiu, Ankaihttps://hdl.handle.net/1974/304102022-09-23T14:05:03Zdc.title: A Quantification of Long Transient Dynamics
dc.contributor.author: Liu, Ankai
dc.description.abstract: We present a systematic study of transient dynamics starting with a technical definition of transient points which are initial data of an autonomous system of ordinary differential equations that can lead to “long transient dynamics.” We then define transient centers, which are points in the state space that cause long transient behaviors, and reachable transient centers, which are transient centers that can be reached from initializations that do not need to be nearby. These points give rise to dynamics where a prescribed observable changes arbitrarily slowly for arbitrarily long time durations. We demonstrate the many interesting properties of transient centers, including how it easily translates from point to point: if an initial point is a transient center then so are all the points in its entire trajectory forward and backward in time, as well as any limit points. Finally, we also explore how these ideas are related to established concepts in dynamical systems such as slow-fast systems, turning points and Lyapunov regularity.
A Study of the Minimum p-value and Related Methods for the Identification of Treatment-Sensitive GroupsLi, Nahttps://hdl.handle.net/1974/303252022-08-30T07:07:55Zdc.title: A Study of the Minimum p-value and Related Methods for the Identification of Treatment-Sensitive Groups
dc.contributor.author: Li, Na
dc.description.abstract: In clinical practices, a fundamentally important problem is to identify a subgroup of patients who may benefit more in terms of a clinical outcome from a certain treatment based on a specific clinical variable or biomarker, which is referred as predictive classification in this thesis. The clinical variable or biomarker used for predictive classification is usually continuous and, therefore, a cutpoint needs to be determined for the definition of the subgroup. The commonly adopted methods, the minimum p-value method and the profile method, suffer from the type I error inflation and/or the identifiability issue. In this thesis research, we first propose bootstrap-based methods for the adjustment of the minimum p-value method for predictive classifications with respect to a continuous clinical outcome in both identifiable and non-identifiable cases under random designs and fixed designs, respectively. Since the minimum p-value test statistics diverge at a rate sqrt(n) (n is the sample size) under the framework of the generalized linear model with non-identity link function and the Cox model for respectively categorical and time to event clinical outcomes, bootstrap-based methods are proposed to adjust the profile methods for predictive classifications in both identifiable and non-identifiable cases. The theoretical properties of the proposed adjustments are investigated and simulation studies are conducted to evaluate their fixed sample size performance. In addition, the proposed methods are applied to analyze the data from a cancer clinical trial.
A Superlacunary Ito-Kawada Theorem, and Applications to the Equidistribution of Generalized Rudin-Shapiro PolynomialsCloutier, Danielhttps://hdl.handle.net/1974/304692022-10-15T07:03:20Zdc.title: A Superlacunary Ito-Kawada Theorem, and Applications to the Equidistribution of Generalized Rudin-Shapiro Polynomials
dc.contributor.author: Cloutier, Daniel
dc.description.abstract: The Ito-Kawada Theorem is a classical result in the theory of random walks on compact groups. It states that essentially the only obstruction to the equidistribution of a random walk with independent and identically distributed increment is if the support of the increments is confined to proper, closed subgroup of G, or a coset of a proper, closed, normal subgroup of G. We prove a version of the Ito-Kawada theorem for a class of weakly dependent random variables arising from superlacunary sequences. We then show that a conjecture of Doche about the even moments of generalized Rudin-Shapiro polynomials follows from a conjectured Ito-Kawada theorem for lacunary random variables.