This thesis contains two parts. The first part concerns three connected problems with high-dimensional data in Chapters 2-4. The second part, Chapter 5, provides dynamic Bayes models to improve the continuous glucose monitoring. In the first part, we propose a unified scale invariant method for the estimation of parameters in linear regression, precision matrix and partial correlation. In Chapter 2, scaled Lasso is introduced to jointly estimate regression coefficients and noise level with a gradient descent algorithm. Under mild regularity conditions, we derive oracle inequalities for the prediction and estimation of the noise level and regression coefficients. These oracle inequalities provide sufficient conditions for the consistency and asymptotic normality of the noise level estimator, including certain cases where the number of variables is of greater order than the sample size. Chapter 3 considers the estimation of precision matrix, which is closely related to linear regression. The proposed estimator is constructed via the scaled Lasso, and guarantees the fastest convergence rate under the spectrum norm. Besides the estimation of high-dimensional objects, the estimation of low-dimensional functionals of high-dimensional objects is also of great interest. A rate minimax estimator of a high-dimensional parameter does not automatically yield rate minimax estimates of its low-dimensional functionals. We consider efficient estimation of partial correlation between individual pairs of variables in Chapter 4. Numerical results demonstrate the superior performance of the proposed methods. In the second part, we develop statistical methods to produce more accurate and precise estimates for continuous glucose monitoring. The continuous glucose monitor measures the glucose level via an electrochemical glucose biosensor, inserted into subcutaneous fat tissue, called interstitial space. We use dynamic Bayes models to incorporate the linear relationship between the blood glucose level and interstitial signal, the time series aspects of the data, and the variability depending on sensor age. The Bayes method has been tested and evaluated with an important large dataset, called ``Star I'', from Medtronic, Inc., composed of continuous monitoring of glucose and other measurements. The results show that the Bayesian blood glucose prediction outperforms the output of the continuous glucose monitor in the STAR 1 trial.
Subject (authority = RUETD)
Topic
Statistics and Biostatistics
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Rutgers University. Graduate School - New Brunswick
AssociatedObject
Type
License
Name
Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.