QSpace Community: Queen's UniversityInformation
http://hdl.handle.net/1974/6
Queen's UniversityInformation2016-08-26T13:47:25ZEstimation of Sample Size and Power For Quantile Regression
http://hdl.handle.net/1974/14739
Title: Estimation of Sample Size and Power For Quantile Regression
Authors: Gong, Zhenxian
Abstract: Quantile regression (QR) was first introduced by Roger Koenker and Gilbert Bassett in 1978. It is robust to outliers which affect least squares estimator on a large scale in linear regression. Instead of modeling mean of the response, QR provides an alternative way to model the relationship between quantiles of the response and covariates. Therefore, QR can be widely used to solve problems in econometrics, environmental sciences and health sciences.
Sample size is an important factor in the planning stage of experimental design and observational studies. In ordinary linear regression, sample size may be determined based on either precision analysis or power analysis with closed form formulas. There are also methods that calculate sample size based on precision analysis for QR like C.Jennen-Steinmetz and S.Wellek (2005). A method to estimate sample size for QR based on power analysis was proposed by Shao and Wang (2009). In this paper, a new method is proposed to calculate sample size based on power analysis under hypothesis test of covariate effects.
Even though error distribution assumption is not necessary for QR analysis itself, researchers have to make assumptions of error distribution and covariate structure in the planning stage of a study to obtain a reasonable estimate of sample size. In this project, both parametric and nonparametric methods are provided to estimate error distribution. Since the method proposed can be implemented in R, user is able to choose either parametric distribution or nonparametric kernel density estimation for error distribution. User also needs to specify the covariate structure and effect size to carry out sample size and power calculation.
The performance of the method proposed is further evaluated using numerical simulation. The results suggest that the sample sizes obtained from our method provide empirical powers that are closed to the nominal power level, for example, 80%.2016-08-24T04:00:00ZCodomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation
http://hdl.handle.net/1974/14738
Title: Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation
Authors: Milne, TRISTAN
Abstract: We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2016-08-23 15:00:13.2832016-08-23T04:00:00ZCodomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation
http://hdl.handle.net/1974/14738
Title: Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation
Authors: Milne, TRISTAN
Abstract: We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2016-08-23 15:00:13.2832016-08-23T04:00:00ZCodomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation
http://hdl.handle.net/1974/14738
Title: Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation
Authors: Milne, TRISTAN
Abstract: We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2016-08-23 15:00:13.2832016-08-23T04:00:00Z