Shape analysis of curves and surfaces is a very important tool in many applications ranging from computer vision to bioinformatics and medical imaging. There are many difficulties when analyzing shapes of parameterized curves and surfaces. Firstly, it is important to develop representations and metrics such that the analysis is invariant to parameterization in addition to the standard transformations (rigid motion and scaling). Furthermore, under the chosen representations and metrics, the analysis must be performed on infinite-dimensional and sometimes non-linear spaces, which poses an additional difficulty. In this work, we develop and apply methods which address these issues. We begin by defining a framework for shape analysis of parameterized open curves and extend these ideas to shape analysis of surfaces. We utilize the presented frameworks in various classification experiments spanning multiple application areas. In the case of curves, we consider the problem of clustering DT-MRI brain fibers, classification of protein backbones, modeling and segmentation of signatures and statistical analysis of biosignals. In the case of surfaces, we perform disease classification using 3D anatomical structures in the brain, classification of handwritten digits by viewing images as quadrilateral surfaces, and finally classification of cropped facial surfaces. We provide two additional extensions of the general shape analysis frameworks that are the focus of this dissertation. The first one considers shape analysis of marked spherical surfaces where in addition to the surface information we are given a set of manually or automatically generated landmarks. This requires additional constraints on the definition of the re-parameterization group and is applicable in many domains, especially medical imaging and graphics. Second, we consider reflection symmetry analysis of planar closed curves and spherical surfaces. Here, we also provide an example of disease detection based on brain asymmetry measures. We close with a brief summary and a discussion of open problems, which we plan on exploring in the future. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester, 2012. / April 27, 2012. / curves, geodesics, medical imaging, statistical shape analysis, surfaces / Includes bibliographical references. / Anuj Srivastava, Professor Directing Thesis; Eric Klassen, University Representative; Wei Wu, Committee Member; Fred Huffer, Committee Member; Ian Dryden, Committee Member.

First, we present two novel semiparametric survival models with log-linear median regression functions for right censored survival data. These models are useful alternatives to the popular Cox (1972) model and linear transformation models (Cheng et al., 1995). Compared to existing semiparametric models, our models have many important practical advantages, including interpretation of the regression parameters via the median and the ability to address heteroscedasticity. We demonstrate that our modeling techniques facilitate the ease of prior elicitation and computation for both parametric and semiparametric Bayesian analysis of survival data. We illustrate the advantages of our modeling, as well as model diagnostics, via reanalysis of a small-cell lung cancer study. Results of our simulation study provide further guidance regarding appropriate modelling in practice. Our second goal is to develop the methods of analysis and associated theoretical properties for interval censored and current status survival data. These new regression models use log-linear regression function for the median. We present frequentist and Bayesian procedures for estimation of the regression parameters. Our model is a useful and practical alternative to the popular semiparametric models which focus on modeling the hazard function. We illustrate the advantages and properties of our proposed methods via reanalyzing a breast cancer study. Our other aim is to develop a model which is able to account for the heteroscedasticity of response, together with robust parameter estimation and outlier detection using sparsity penalization. Some preliminary simulation studies have been conducted to compare the performance of proposed model and existing median lasso regression model. Considering the estimation bias, mean squared error and other identication benchmark measures, our proposed model performs better than the competing frequentist estimator. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester, 2012. / March 26, 2012. / Bayesian Analysis, Median regression, Semiparametric, Survival Analysis, Transform-Both-Sides / Includes bibliographical references. / Debajyoti Sinha, Professor Directing Thesis; Yi Zhou, University Representative; Stuart Lipsitz, Committee Member; Dan McGee, Committee Member; Xu-Feng Niu, Committee Member; Yiyuan She, Committee Member.

We propose a Riemannian framework for shape analysis of annotated curves, curves that have certain attributes defined along them, in addition to their geometries.These attributes may be in form of vector-valued functions, discrete landmarks, or symbolic labels, and provide auxiliary information along the curves. The resulting shape analysis, that is comparing, matching, and deforming, is naturally influenced by the auxiliary functions. Our idea is to construct curves in higher dimensions using both geometric and auxiliary coordinates, and analyze shapes of these curves. The difficulty comes from the need for removing different groups from different components: the shape is invariant to rigid-motion, global scale and re-parameterization while the auxiliary component is usually invariant only to the re-parameterization. Thus, the removal of some transformations (rigid motion and global scale) is restricted only to the geometric coordinates, while the re-parameterization group is removed for all coordinates. We demonstrate this framework using a number of experiments. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester, 2011. / August 15, 2011. / Annotated curve, Riemannian, Shape analysis / Includes bibliographical references. / Anuj Srivastava, Professor Directing Dissertation; Jinfeng Zhang, Professor Co-Directing Dissertation; Eric P. Klassen, University Representative; Fred Huffer, Committee Member.

We propose a novel Riemannian framework for analyzing orientation distribution functions (ODFs), or their probability density functions (PDFs), in HARDI data sets for use in comparing, interpolating, averaging, and denoising PDFs. This is accomplished by separating shape and orientation features of PDFs, and then analyzing them separately under their own Riemannian metrics. We formulate the action of the rotation group on the space of PDFs, and define the shape space as the quotient space of PDFs modulo the rotations. In other words, any two PDFs are compared in: (1) shape by rotationally aligning one PDF to another, using the Fisher-Rao distance on the aligned PDFs, and (2) orientation by comparing their rotation matrices. This idea improves upon the results from using the Fisher-Rao metric in analyzing PDFs directly, a technique that is being used increasingly, and leads to geodesic interpolations that are biologically feasible. This framework leads to definitions and efficient computations for the Karcher mean that provide tools for improved interpolation and denoising. We demonstrate these ideas, using an experimental setup involving several PDFs. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester, 2011. / August 3, 2011. / Fisher-Rao, HARDI, Interpolation, ODF, Orientation, Riemannian Framework / Includes bibliographical references. / Anuj Srivastava, Professor Directing Dissertation; Eric Klassen, University Representative; Wei Wu, Committee Member; Xufeng Niu, Committee Member.

High blood pressure is a major determinant of risk for Coronary Heart Disease (CHD) and stroke, leading causes of death in the industrialized world. A myriad of pharmacological treatments for elevated blood pressure, defined as a blood pressure greater than 140/90mmHg, are available and have at least partially resulted in large reductions in the incidence of CHD and stroke in the U.S. over the last 50 years. The factors that may increase blood pressure levels are not well understood, but body mass is thought to be a major determinant of blood pressure level. Obesity is measured through various methods (skinfolds, waist-to-hip ratio, bioelectrical impedance analysis (BIA), etc.), but the most commonly used measure is body mass index,BMI= Weight(kg)/Height(m)2 / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester, 2012. / October 29, 2012. / Includes bibliographical references. / Daniel McGee, Professor Directing Dissertation; Daniel Lackland, Professor Co-Directing Dissertation; Myra Hurt, University Representative; Eric Chicken, Committee Member; Xufeng Niu, Committee Member.

Over the past twenty years, there has been a rapid development in Nonparametric Statistical Analysis on Manifolds applied to Medical Imaging problems. In this body of work, we focus on two different medical imaging problems. The first problem corresponds to analyzing the CT scan data. In this context, we perform nonparametric analysis on the 3D data retrieved from CT scans of healthy young adults, on the Size-and-Reflection Shape Space of k-ads in general position in 3D. This work is a part of larger project on planning reconstructive surgery in severe skull injuries which includes pre-processing and post-processing steps of CT images. The next problem corresponds to analyzing MR diffusion tensor imaging data. Here, we develop a two-sample procedure for testing the equality of the generalized Frobenius means of two independent populations on the space of symmetric positive matrices. These new methods, naturally lead to an analysis based on Cholesky decompositions of covariance matrices which helps to decrease computational time and does not increase dimensionality. The resulting nonparametric matrix valued statistics are used for testing if there is a difference on average between corresponding signals in Diffusion Tensor Images (DTI) in young children with dyslexia when compared to their clinically normal peers. The results presented here correspond to data that was previously used in the literature using parametric methods which also showed a significant difference. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester, 2012. / February 29, 2012. / Cholesky Decomposition, Nonparametric Bootstrap, Nonparametric Two-Sample Tests, Riemannian Homogeneous Spaces, Segmentation, Size-and-Reflection Shape Space / Includes bibliographical references. / Victor Patrangenaru, Professor Directing Dissertation; Xiuwen Liu, University Representative; Adrian Barbu, Committee Member; Eric Chicken, Committee Member.

This research is motivated by an analysis of reading research data. We are interested in modeling the test outcome of ability to fluently recode letters into sounds of kindergarten children aged between 5 and 7. The data showed excessive zero scores (more than 30% of children) on the test. In this dissertation, we carefully examine the models dealing with excessive zeros, which are based on the mixture of distributions, a distribution with zeros and a standard probability distribution with non negative values. In such cases, a log normal variable or a Poisson random variable is often observed with probability from semicontinuous data or count data. The previously proposed models, mixed-effects and mixed-distribution models (MEMD) by Tooze(2002) et al. for semicontinuous data and zero-inflated Poisson (ZIP) regression models by Lambert(1992) for count data are reviewed. We apply zero-inflated Poisson models to repeated measures data of zero-inflated data by introducing a pair of possibly correlated random effects to the zero-inflated Poisson model to accommodate within-subject correlation and between subject heterogeneity. The model describes the effect of predictor variables on the probability of nonzero responses (occurrence) and mean of nonzero responses (intensity) separately. The likelihood function is maximized using dual quasi-Newton optimization of an approximated by adaptive Gaussian quadrature. The maximum likelihood estimates are obtained through standard statistical software package. Using different model parameters, the number of subject, and the number of measurements per subject, the simulation study is conducted and the results are presented. The dissertation ends with the application of the model to reading research data and future research. We examine the number of correct letter sound counted of children collected over 2008 -2009 academic year. We find that age, gender and socioeconomic status are significantly related to the letter sound fluency of children in both parts of the model. The model provides better explanation of data structure and easier interpretations of parameter values, as they are the same as in standard logistic models and Poisson regression models. The model can be extended to accommodate serial correlation which can be observed in longitudinal data. Also, one may consider multilevel zero-inflated Poisson model. Although the multilevel model was proposed previously, parameter estimation by penalized quasi likelihood methods is questionable, and further examination is needed. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester, 2012. / March 27, 2012. / Includes bibliographical references. / Xufeng Niu, Professor Directing Dissertation; Shouping Hu, University Representative; Stephanie Dent Al Otaiba, Committee Member; Daniel McGee, Committee Member; Wei Wu, Committee Member.

Recent advancements in data collection allow scientists and researchers to obtain massive amounts of information in short periods of time. Often this data is functional and quite complex. Wavelet transforms are popular, particularly in the engineering and manufacturing fields, for handling these type of complicated signals. A common application of wavelets is in statistical process control (SPC), in which one tries to determine as quickly as possible if and when a sequence of profiles has gone out-of-control. However, few wavelet methods have been proposed that don't rely in some capacity on the assumption that the observational errors are normally distributed. This dissertation aims to fill this void by proposing a simple, nonparametric, distribution-free method of monitoring profiles and estimating changepoints. Using only the magnitudes and location maps of thresholded wavelet coefficients, our method uses the spatial adaptivity property of wavelets to accurately detect profile changes when the signal is obscured with a variety of non-Gaussian errors. Wavelets are also widely used for the purpose of dimension reduction. Applying a thresholding rule to a set of wavelet coefficients results in a "denoised" version of the original function. Once again, existing thresholding procedures generally assume independent, identically distributed normal errors. Thus, the second main focus of this dissertation is a nonparametric method of thresholding that does not assume Gaussian errors, or even that the form of the error distribution is known. We improve upon an existing even-odd cross-validation method by employing block thresholding and level dependence, and show that the proposed method works well on both skewed and heavy-tailed distributions. Such thresholding techniques are essential to the SPC procedure developed above. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester, 2013. / March 4, 2013. / Nonparametric Statistics, Profile Monitoring, Statistical Process Control, Thresholding, Wavelets / Includes bibliographical references. / Eric Chicken, Professor Directing Thesis; Peter Hoeflich, University Representative; Xufeng Niu, Committee Member; Jinfeng Zhang, Committee Member.

Spatiotemporal modeling is increasingly used in a diverse array of fields, such as ecology, epidemiology, health care research, transportation, economics, and other areas where data arise from a spatiotemporal process. Spatiotemporal models describe the relationship between observations collected from different spatiotemporal sites. The modeling of spatiotemporal interactions arising from spatiotemporal data is done by incorporating the space-time dependence into the covariance structure. A main goal of spatiotemporal modeling is the estimation and prediction of the underlying process that generates the observations under study and the parameters that govern the process. Furthermore, analysis of the spatiotemporal correlation of variables can be used for estimating values at sites where no measurements exist. In this work, we develop a framework for estimating quantities that are functions of complete spatiotemporal data when the spatiotemporal data is incomplete. We present two classes of conditional autoregressive (CAR) models (the homogeneous CAR (HCAR) model and the weighted CAR (WCAR) model) for the analysis of sparse spatiotemporal data (the log of monthly mean zooplankton biomass) collected on a spatiotemporal lattice by the California Cooperative Oceanic Fisheries Investigations (CalCOFI). These models allow for spatiotemporal dependencies between nearest neighbor sites on the spatiotemporal lattice. Typically, CAR model likelihood inference is quite complicated because of the intractability of the CAR model's normalizing constant. Sparse spatiotemporal data further complicates likelihood inference. We implement Monte Carlo likelihood (MCL) estimation methods for parameter estimation of our HCAR and WCAR models. Monte Carlo likelihood estimation provides an approximation for intractable likelihood functions. We demonstrate our framework by giving estimates for several different quantities that are functions of the complete CalCOFI time series data. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester, 2013. / December 13, 2012. / CalCOFI, Conditional Autoregressive Models, Missing Data, Monte Carlo Likelihood, Penalized Likelihood, Spatiotemporal Analysis / Includes bibliographical references. / Fred Huffer, Professor Directing Dissertation; Betsy Becker, University Representative; Xufeng Niu, Committee Member; Anuj Srivastava, Committee Member.

Nonstationary nonparametric densities occur naturally including applications such as monitoring the amount of toxins in the air and in monitoring internet streaming data. Progress has been made in estimating these densities, but there is little current work on monitoring them for changes. A new statistic is proposed which effectively monitors these nonstationary nonparametric densities through the use of transformed wavelet coefficients of the quantiles. This method is completely nonparametric, designed for no particular distributional assumptions; thus making it effective in a variety of conditions. Existing methods for monitoring sequential data typically focus on using a single value upper control limit (UCL) based on a specified in control average run length (ARL) to detect changes in these nonstationary statistics. However, such a UCL is not designed to take into consideration the false alarm rate, the power associated with the test or the underlying distribution of the ARL. Additionally, if the monitoring statistic is known to be monotonic over time (which is typical in methods using maxima in their statistics, for example) the flat UCL does not adjust to this property. We propose several methods for creating UCLs that provide improved power and simultaneously adjust the false alarm rate to user-specified values. Our methods are constructive in nature, making no use of assumed distribution properties of the underlying monitoring statistic. We evaluate the different proposed UCLs through simulations to illustrate the improvements over current UCLs. The proposed method is evaluated with respect to profile monitoring scenarios and the proposed density statistic. The method is applicable for monitoring any monotonically nondecreasing nonstationary statistics. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester, 2013. / February 27, 2013. / ARL, Change Points, Nonparametric Nonstationary Density, Power, Profile Monitoring, Type I Error / Includes bibliographical references. / Eric Chicken, Professor Directing Thesis; Guosheng Liu, University Representative; Debajyoti Sinha, Committee Member; Wei Wu, Committee Member.