Department of Mathematics and StatisticsNo Descriptionhttps://hdl.handle.net/1828/1322024-10-16T05:20:56Z2024-10-16T05:20:56Z6721Invariant conic optimization with basis-dependent cones: Scaled diagonally dominant matrices and real *-algebra decompositionNeshat Taherzadeh, Khashayarhttps://hdl.handle.net/1828/203792024-09-05T22:03:41Z2024-01-01T00:00:00Zdc.title: Invariant conic optimization with basis-dependent cones: Scaled diagonally dominant matrices and real *-algebra decomposition
dc.contributor.author: Neshat Taherzadeh, Khashayar
dc.description.abstract: Symmetry reduction for a semidefinite program (SDP) with symmetries makes computational solution of the SDP easier by decomposing the semidefiniteness constraint into multiple smaller semidefiniteness constraints. This decomposition requires changing to a symmetry-adapted basis that block diagonalizes the matrix variable, but this does not change the optimum value of the SDP because the semidefinite cone is basis-independent. For other cones that are basis-dependent, if optimization problems over those cones have symmetries one can still change to a symmetry-adapted basis that block diagonalizes the matrix. However, this change of basis generally changes the constraint cone and can change the optimum. In this thesis, we develop a framework for determining when symmetry reduction for basis-dependent conic optimization makes the optimum increase, decrease, or stay the same. The aim is to determine this using general features such as the symmetry group of the optimization problem, without having to solve the problem computationally. We then use our framework to prove various results of this type for scaled diagonally dominant programs (SDDPs), which are convex optimization problems over the cone of scaled diagonally dominant matrices. These results depend on the orbital structure of the underlying representation of invariant SDDPs. Using the regular representation, we demonstrate that analysis of SDDPs of any size can be confined to a smaller SDDP that is invariant under a particular representation. Our approach uses real *-algebra decomposition of equivariant maps, which is not needed for existing symmetry reduction of SDPs. Because polynomial optimization problems with sum-of-squares and sum-of-binomial-squares can be represented as SDPs and SDDPs, respectively, our results on SDDPs have implications for polynomial optimization. Using several polynomial optimization problems as examples, we give computational results that illustrate our theorems. For polynomial optimization subject to sum-of-binomial-squares, our examples include cases in which symmetry reduction causes the optimum to increase, decrease, or stay the same.
2024-01-01T00:00:00ZAdaptVarLM: A linear regression model for covariate-dependent non-constant error varianceWang, Wanmenghttps://hdl.handle.net/1828/203702024-09-04T22:03:19Z2024-01-01T00:00:00Zdc.title: AdaptVarLM: A linear regression model for covariate-dependent non-constant error variance
dc.contributor.author: Wang, Wanmeng
dc.description.abstract: In biological research, traditional multiple regression models assume homoscedasticity — constant variance of error terms — an assumption that is difficult to maintain in complex biological data. This thesis introduces AdaptVarLM, a novel linear regression model specialized in dealing with non-constant error variance dependent on one covariate. AdaptVarLM integrates an auxiliary linear relationship between the logarithmic variance of the error term and a specific explanatory variable, and uses maximum likelihood estimation (MLE) in the iterative updating process to improve the parameter estimation accuracy. By modelling non-constant error variance, AdaptVarLM outperforms the traditional regression model in capturing the complex variability inherent in biological data. Applying to the study of Alzheimer's disease, AdaptVarLM detects genetically linked genes associated with the disease and error variance. The results of analyzing both bulk and single-cell data validate the effectiveness of AdaptVarLM in detecting significant genes.
2024-01-01T00:00:00ZA non-local reaction advection-diffusion model for self-interacting speciesYue, Zongzhihttps://hdl.handle.net/1828/203072024-08-26T22:01:06Z2024-01-01T00:00:00Zdc.title: A non-local reaction advection-diffusion model for self-interacting species
dc.contributor.author: Yue, Zongzhi
dc.description.abstract: In biological models, advection is inherently a non-local process. In this thesis, we proposed a natural extension of the non-local advection-diffusion model in [7] to include the reaction term (birth and death process). This thesis begins with an investigation of the well-posedness and existence of travelling wave solutions for this non-local reaction-advection-diffusion (RAD) equation. We prove the local-in-time existence and positivity of solutions under H³(R) initial conditions and provide a continuation criterion of the equation. Subsequently, we explore the existence of travelling wave solutions of this non-local RAD using a combination of perturbation methods, Fredholm operator theory, and Banach's fixed point theorem. Our analysis reveals that such solutions exist when the non-local advection term is small. Finally, we simulate the travelling wave solution to verify our theoretical findings and speculate the solution with a large advection term.
2024-01-01T00:00:00ZDevelopment of a disease analytic model for estimating the hidden population using the stratified-Petersen estimatorMa, Siyinghttps://hdl.handle.net/1828/202932024-08-18T00:54:19Z2024-01-01T00:00:00Zdc.title: Development of a disease analytic model for estimating the hidden population using the stratified-Petersen estimator
dc.contributor.author: Ma, Siying
dc.description.abstract: The COVID-19 pandemic brought the need for novel disease analytic models capable of estimating the true number of infections, including those that evaded detection. Statistical methods, such as the stratified-Petersen estimator, provide effective ways in wildlife population modelling to estimate hard-to-reach population size. We developed a novel disease analytic model to estimate the levels of underreported COVID-19 cases and the true population size based on the idea of developing a Bayesian version of the stratified-Petersen estimator under a state-space formulation using individual-level capture-recapture data. We obtained the capture events from individuals’ electronic health records and treated the occurrence of positive SARS-CoV-2 diagnostic test results and 2020 COVID-19-related hospitalizations as the tagging and recapture processes. Applying this model to the data from the Northern Health Authority region in British Columbia, Canada in 2020 by using a Bayesian Markov chain Monte Carlo (MCMC) approach, we found that the estimate of the size of the COVID-19 population (Nˆ = 2, 967) is 1.58 (95% CI: (1.53, 1.63)) times greater than the observed cases (nobs = 1, 880), which is a comparable result to those reported in other studies.
2024-01-01T00:00:00ZOptimization problems with variational inequality constraintsYe, Xiang Yanghttps://hdl.handle.net/1828/202872024-08-16T04:50:00Z1995-01-01T00:00:00Zdc.title: Optimization problems with variational inequality constraints
dc.contributor.author: Ye, Xiang Yang
1995-01-01T00:00:00ZMega-dose vitamins and minerals for the treatment of breast cancer : a comparison study of treated nonmetastatic patients versus two control groups.Zhao, Yangrhttps://hdl.handle.net/1828/202762024-08-16T04:52:49Z2000-01-01T00:00:00Zdc.title: Mega-dose vitamins and minerals for the treatment of breast cancer : a comparison study of treated nonmetastatic patients versus two control groups.
dc.contributor.author: Zhao, Yangr
2000-01-01T00:00:00ZConvexity of minimal total dominating functions of graphsYu, Bohttps://hdl.handle.net/1828/202472024-08-16T04:49:27Z1992-01-01T00:00:00Zdc.title: Convexity of minimal total dominating functions of graphs
dc.contributor.author: Yu, Bo
1992-01-01T00:00:00ZApplication of Bayesian networks to testing theoryYamada, Shinichihttps://hdl.handle.net/1828/202252024-08-16T04:36:16Z2001-01-01T00:00:00Zdc.title: Application of Bayesian networks to testing theory
dc.contributor.author: Yamada, Shinichi
2001-01-01T00:00:00ZSubdifferentials and their applicationsWu, Zilihttps://hdl.handle.net/1828/202082024-08-16T04:37:37Z1997-01-01T00:00:00Zdc.title: Subdifferentials and their applications
dc.contributor.author: Wu, Zili
1997-01-01T00:00:00ZModelling and analysis of center of pressure dataWebber, Adam Matthewhttps://hdl.handle.net/1828/200642024-08-16T04:10:38Z2003-01-01T00:00:00Zdc.title: Modelling and analysis of center of pressure data
dc.contributor.author: Webber, Adam Matthew
2003-01-01T00:00:00Z