Statistical methods for high-dimensional data and continuous glucose monitoring

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.Ph. D.Includes bibliographical referencesIncludes vitaby Tingni Su