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- Genre: Academic theses
Description
Solving partial differential equations on surfaces has many applications including modeling chemical diffusion, pattern formation, geophysics and texture mapping. This dissertation presents two techniques for solving time dependent partial differential equations on various surfaces using the partition of unity method. A novel spectral cubed sphere method that utilizes the windowed Fourier technique is presented and used for both approximating functions on spherical domains and solving partial differential equations. The spectral cubed sphere method is applied to solve the transport equation as well as the diffusion equation on the unit sphere. The second approach is a partition of unity method with local radial basis function approximations. This technique is also used to explore the effect of the node distribution as it is well known that node choice plays an important role in the accuracy and stability of an approximation. A greedy algorithm is implemented to generate good interpolation nodes using the column pivoting QR factorization. The partition of unity radial basis function method is applied to solve the diffusion equation on the sphere as well as a system of reaction-diffusion equations on multiple surfaces including the surface of a red blood cell, a torus, and the Stanford bunny. Accuracy and stability of both methods are investigated.
ContributorsIslas, Genesis Juneiva (Author) / Platte, Rodrigo (Thesis advisor) / Cochran, Douglas (Committee member) / Espanol, Malena (Committee member) / Kao, Ming-Hung (Committee member) / Renaut, Rosemary (Committee member) / Arizona State University (Publisher)
Created2021
Description
Functional magnetic resonance imaging (fMRI) is one of the popular tools to study human brain functions. High-quality experimental designs are crucial to the success of fMRI experiments as they allow the collection of informative data for making precise and valid inference with minimum cost. The primary goal of this study is on identifying the best sequence of mental stimuli (i.e. fMRI design) with respect to some statistically meaningful optimality criteria. This work focuses on two related topics in this research field. The first topic is on finding optimal designs for fMRI when the design matrix is uncertain. This challenging design issue occurs in many modern fMRI experiments, in which the design matrix of the statistical model depends on both the selected design and the experimental subject's uncertain behavior during the experiment. As a result, the design matrix cannot be fully determined at the design stage that makes it difficult to select a good design. For the commonly used linear model with autoregressive errors, this study proposes a very efficient approach for obtaining high-quality fMRI designs for such experiments. The proposed approach is built upon an analytical result, and an efficient computer algorithm. It is shown through case studies that our proposed approach can outperform the existing method in terms of computing time, and the quality of the obtained designs. The second topic of the research is to find optimal designs for fMRI when a wavelet-based technique is considered in the fMRI data analysis. An efficient computer algorithm to search for optimal fMRI designs for such cases is developed. This algorithm is inspired by simulated annealing and a recently proposed algorithm by Saleh et al. (2017). As demonstrated in the case studies, the proposed approach makes it possible to efficiently obtain high-quality designs for fMRI studies, and is practically useful.
ContributorsZhou, Lin (Author) / Kao, Ming-Hung (Thesis advisor) / Welfert, Bruno (Thesis advisor) / Jackiewicz, Zdzislaw (Committee member) / Reiser, Mark R. (Committee member) / Stufken, John (Committee member) / Taylor, Jesse Earl (Committee member) / Arizona State University (Publisher)
Created2017
Description
Spatial regression is one of the central topics in spatial statistics. Based on the goals, interpretation or prediction, spatial regression models can be classified into two categories, linear mixed regression models and nonlinear regression models. This dissertation explored these models and their real world applications. New methods and models were proposed to overcome the challenges in practice. There are three major parts in the dissertation.
In the first part, nonlinear regression models were embedded into a multistage workflow to predict the spatial abundance of reef fish species in the Gulf of Mexico. There were two challenges, zero-inflated data and out of sample prediction. The methods and models in the workflow could effectively handle the zero-inflated sampling data without strong assumptions. Three strategies were proposed to solve the out of sample prediction problem. The results and discussions showed that the nonlinear prediction had the advantages of high accuracy, low bias and well-performed in multi-resolution.
In the second part, a two-stage spatial regression model was proposed for analyzing soil carbon stock (SOC) data. In the first stage, there was a spatial linear mixed model that captured the linear and stationary effects. In the second stage, a generalized additive model was used to explain the nonlinear and nonstationary effects. The results illustrated that the two-stage model had good interpretability in understanding the effect of covariates, meanwhile, it kept high prediction accuracy which is competitive to the popular machine learning models, like, random forest, xgboost and support vector machine.
A new nonlinear regression model, Gaussian process BART (Bayesian additive regression tree), was proposed in the third part. Combining advantages in both BART and Gaussian process, the model could capture the nonlinear effects of both observed and latent covariates. To develop the model, first, the traditional BART was generalized to accommodate correlated errors. Then, the failure of likelihood based Markov chain Monte Carlo (MCMC) in parameter estimating was discussed. Based on the idea of analysis of variation, back comparing and tuning range, were proposed to tackle this failure. Finally, effectiveness of the new model was examined by experiments on both simulation and real data.
In the first part, nonlinear regression models were embedded into a multistage workflow to predict the spatial abundance of reef fish species in the Gulf of Mexico. There were two challenges, zero-inflated data and out of sample prediction. The methods and models in the workflow could effectively handle the zero-inflated sampling data without strong assumptions. Three strategies were proposed to solve the out of sample prediction problem. The results and discussions showed that the nonlinear prediction had the advantages of high accuracy, low bias and well-performed in multi-resolution.
In the second part, a two-stage spatial regression model was proposed for analyzing soil carbon stock (SOC) data. In the first stage, there was a spatial linear mixed model that captured the linear and stationary effects. In the second stage, a generalized additive model was used to explain the nonlinear and nonstationary effects. The results illustrated that the two-stage model had good interpretability in understanding the effect of covariates, meanwhile, it kept high prediction accuracy which is competitive to the popular machine learning models, like, random forest, xgboost and support vector machine.
A new nonlinear regression model, Gaussian process BART (Bayesian additive regression tree), was proposed in the third part. Combining advantages in both BART and Gaussian process, the model could capture the nonlinear effects of both observed and latent covariates. To develop the model, first, the traditional BART was generalized to accommodate correlated errors. Then, the failure of likelihood based Markov chain Monte Carlo (MCMC) in parameter estimating was discussed. Based on the idea of analysis of variation, back comparing and tuning range, were proposed to tackle this failure. Finally, effectiveness of the new model was examined by experiments on both simulation and real data.
ContributorsLu, Xuetao (Author) / McCulloch, Robert (Thesis advisor) / Hahn, Paul (Committee member) / Lan, Shiwei (Committee member) / Zhou, Shuang (Committee member) / Saul, Steven (Committee member) / Arizona State University (Publisher)
Created2020