Joint Estimation of Multiple Graphical Models from High Dimensional Time Series

In this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions. We assume that the data are collected from n subjects, each of which consists of T possibly dependent observations. The graphical models of subjects vary, but are assumed to change smoothly corresponding to a measure of closeness between subjects. We propose a kernel based method for jointly estimating all graphical models. Theoretically, under a double asymptotic framework, where both (T,n) and the dimension d can increase, we provide the explicit rate of convergence in parameter estimation. It characterizes the strength one can borrow across different individuals and impact of data dependence on parameter estimation. Empirically, experiments on both synthetic and real resting state functional magnetic resonance imaging (rs-fMRI) data illustrate the effectiveness of the proposed method.Comment: 40 page

Phantom of the Hartle-Hawking instanton: connecting inflation with dark energy

If the Hartle-Hawking wave function is the correct boundary condition of our universe, the history of our universe will be well approximated by an instanton. Although this instanton should be classicalized at infinity, as long as we are observing a process of each history, we may detect a non-classicalized part of field combinations. When we apply it to a dark energy model, this non-classicalized part of fields can be well embedded to a quintessence and a phantom model, i.e., a quintom model. Because of the property of complexified instantons, the phantomness will be naturally free from a big rip singularity. This phantomness does not cause perturbative instabilities, as it is an effect emergent from the entire wave function. Our work may thus provide a theoretical basis for the quintom models, whose equation of state (EoS) can cross the cosmological constant boundary (CCB) phenomenologically.Comment: 20 pages, 7 figure

Complex Dynamics in Fed-Batch Systems: Modeling, Analysis and Control of Alcoholic Fermentations

Modeling and control of fed-batch fermentation processes has been a subject of great interest to realize high productivity and yields from the fermentation technique. The goal of this dissertation was to gain insights into how the complex dynamic behaviors exhibited in fed-batch fermentation systems affect the stability of standard single-loop as well as non-standard feedback control structures. Novel PID stability theorems were established to help construct the controller stabilizing regions