School of Mathematics and Statistics  Theses
http://hdl.handle.net/11343/295
20180223T11:34:26Z

Variable selection confidence sets for high dimensional data
http://hdl.handle.net/11343/194522
Variable selection confidence sets for high dimensional data
Zheng, Chao
The traditional activity of model selection aims at discovering a single model superior to other candidate models. In the presence of noise in data, however, it is typically difficult to declare a single model significantly superior to all possible competitors, due to the prevailing effect of the model selection uncertainty. In this situation, multiple or even a large number of models may be equally supported by data. To resolve this model selection ambiguity, we extend the general approach of variable selection confidence sets (VSCSs) to general parametric family and ultrahigh dimensional scenario. A VSCS is defined as a list of models statistically indistinguishable from the true model at a userspecified level of confidence, which extends the familiar notion of confidence intervals to the model selection framework. Our approach guarantees asymptotically correct coverage probability of the true model when both sample size and model dimension increase. We derive conditions under which the VSCS contains all the relevant information about the true model structure. In addition, we propose an important subset of models (LBMs) and natural statistics based on the VSCS to measure importance of variables in a principled way that accounts for the overall model uncertainty. When the space of feasible models is large, VSCS is implemented by an adaptive stochastic search algorithm which samples VSCS models with high probability. The VSCS methodology is illustrated through numerical experiments on synthetic data and real data examples.
© 2017 Dr Chao Zheng
20170101T00:00:00Z

Triangulating CappellShaneson homotopy 4spheres
http://hdl.handle.net/11343/194367
Triangulating CappellShaneson homotopy 4spheres
Issa, Ahmad
The smooth 4dimensional Poincare conjecture states that if a smooth 4manifold is homeomorphic to the 4sphere then it is diffeomorphic to the standard 4sphere. Historically, one of the most promising families of potential counterexamples to this conjecture is the family of CappellShaneson homotopy 4spheres. Over time, with difficult Kirby calculus computations, an infinite subfamily of CappellShaneson homotopy 4spheres was shown to be standard, that is, diffeomorphic tothe 4sphere and hence are not counterexamples. More recently, Gompf showed that a strictly larger subfamily is standard using the fishtail surgery trick. In another direction, BudneyBurtonHillman discovered a fascinating ideal triangulation of a smooth 4manifold which they show is homeomorphic to a certain CappellShaneson 2knot complement X^4. They pose as a problem to show that it is in fact diffeomorphic to X. We solve this problem by directly triangulating X and observing that the triangulation we obtain is combinatorially isomorphic to theirs. In fact, we show that this triangulation is a layered triangulation. We then generalise our construction to show how to construct layered triangulations of oncepunctured 3torus bundles, generalising the wellknown FloydHatcher triangulations of oncepunctured torus bundles. Next, we describe how we can use these triangulations of oncepunctured 3torus bundles to construct triangulations of CappellShaneson homotopy 4spheres. This involves understanding the Gluck twisting operation via triangulations. For some particular examples, we simplify the triangulations using Pachner moves to a standard triangulation of the 4sphere. For these examples, this provides a new computational proof that the corresponding CappellShaneson homotopy 4spheres are diffeomorphic to the 4sphere.
© 2017 Ahmad Issa
20170101T00:00:00Z

Influenza viral dynamics models to explore the roles of innate and adaptive immunity
http://hdl.handle.net/11343/192580
Influenza viral dynamics models to explore the roles of innate and adaptive immunity
Yan, Ada W. C.
A mathematical model which captures how the immune response controls influenza infection is essential for predicting the effects of pharmaceutical interventions, and alleviating the public health burden of the disease. However, current models do not agree on how immune response components work together to control infection. Hence, the predicted effects of existing treatments differ between models, implying that predictions of the effects of novel treatments may be unreliable. The discrepancies between models arise because many models are only fit to viral load data from a single infection, from which it is difficult to distinguish between competing models.
This study focuses on the construction of a viral dynamics model which reproduces experimental observations of the protection conferred by a primary infection against a subsequent infection. Incorporating observations from multiple experimental conditions enables more accurate extraction of the timing and strength of crossimmunity, and thus quantification of the roles of each component of the immune response.
Following a literature review and mathematical preliminaries (Chapters 14), Chapter 5 details the analysis of data from experiments where ferrets are sequentially infected with two different influenza strains. The analysis shows that the protection conferred by a primary infection against a subsequent infection depends on the time between exposures as well as the strains used. Chapters 6 and 7 then present the construction of a viral dynamics model to show that the innate immune response can explain the delay of a secondary infection by a primary infection, and that both crossreactivity and memory in the cellular adaptive immune response are required to explain the shortening of a secondary infection by a primary infection. The model also reproduces qualitative observations from a range of knockout experiments.
To quantify the roles of each immune component, the model must be fitted to experimental data. However, because of the large number of model parameters, a simulation estimation study is first conducted in Chapters 8 and 9. The study shows that rather than examining marginal posterior distributions, using a fitted model to make predictions more clearly elucidates the role of each immune component in controlling infection. Moreover, a model fitted to sequential infection data accurately recovers the timing and extent of crossprotection between strains, whereas insufficient information is available from single infection data to enable inference of these quantities. This represents significant progress in ensuring that results of data fitting are interpreted appropriately, such that the fitted model provides a sound foundation upon which to explore the effects of interventions.
Lastly, Chapter 10 addresses the observation that under identical experimental conditions, some ferrets become infected when exposed to a second virus and some do not. Equations are derived for the extinction probability of the second virus for a stochastic version of one of the models in the study. The numerical solutions to these equations show that the dependence of the extinction probability on the interexposure interval is consistent with experimental observations. Thus, stochasticity in viral dynamics alone is a viable hypothesis for the difference in observed infection outcomes.
© 2017 Dr. Ada Wing Chi Yan
20170101T00:00:00Z

Skipstop schedules for public transportation
http://hdl.handle.net/11343/191748
Skipstop schedules for public transportation
Chew, Joanne Suk Chun
Public transit skipstop operations were first introduced to cope with the traffic congestion. Skipstop operations take place when each vehicle services a subset of stations along a route, which together covers all stations. Most skipstop scheduling problems in the literature were case based reallife problems. As a complementary contribution to the literature, we considered the generic fundamental skipstop scheduling problems, and designed heuristics to solve them.
We investigated variants of this problem. Five types of problems were considered: (i) minimizing passenger invehicle traveling time; (ii) minimizing passenger waiting time with equal passenger size problem; (iii) minimizing passenger waiting time with unequal passenger size and overtaking scheduling problem; (iv) minimizing passenger waiting time with unequal passenger size and nonovertaking scheduling problem; (v) minimizing passenger waiting time, invehicle traveling time and passenger capacity surplus time. A case study on Melbourne train network was also considered based on the problem (v).
Optimal heuristic algorithms were developed for problem (i). For problems (iii), (iv) and (v), various methods were designed for each problem. In particular, the heuristic algorithm of TBHA for the problem (iii) obtained competitive solutions with negligible running times for all the instances tested and had the worst case ratio of the total passenger waiting time found to the minimum total passenger waiting time found of 1.30. In problem (iv), push and ρpush heuristic algorithms produced highquality solutions and also take a negligible time to run the programs of all sizes. In problem (v), column generation with NOVHA as the subproblem outperformed the other methods used in terms of the computational times and solutions. When solving the case study, CG with NOVHA as the subproblem produced a solution that improved passenger waiting and invehicle traveling time from the current train system by 12.7%.
© 2017 Dr. Joanne Suk Chun Chew
20170101T00:00:00Z