Statistics and Applied Mathematics Seminar
|Friday, April 21, 2017 at 3:30 p.m. in ILB 360
Note the different day and the different room!
|Livia Corsi, Georgia Institute of Technology||
On the Persistence of Invariant Tori for Dynamical Systems
Abstract: Given a dynamical system (in finite or infinite dimension) it is very natural to look for finite dimensional invariant subspaces on which the dynamics is very simple. Of particular interest are the invariant tori on which the dynamics is conjugated to a linear one. The problem of persistence under perturbations of such objects has been widely studied starting form the 50's, and this gives rise to the celebrated KAM theory. The aim of this talk is to give an overview of the main difficulties and strategies, having in mind the application to PDE's.
|Wednesday, October 12, 2016||Stefan Siegmund, Dresden University of Technology, Germany||
Dynamical Systems Approach to Bone Remodeling
Abstract: Bone remodeling is a lifelong process where mature bone tissue is removed from the skeleton and new bone tissue is formed. These processes also control the reshaping or replacement of bone following injuries like fractures but also micro-damage, which occurs during normal activity. We discuss adequate general classes of two- and three-dimensional population models for the cell types involved in this process and show that two populations are not enough to explain what biologists observe. A three-dimensional model class not only explains the observations but also explains what is called Paget's disease of oscillating or disorganized bone remodeling. The mathematical tools are developed step by step and describe the basics of a 'bifurcation theory for an infinity number of ordinary differential equations with the same coupling structure'. This is joint work with Thilo Gross, Martin Zumsande and Dirk Stiefs.
|Thursday, March 24, 2016||Susmita Sadhu, Georgia College & State University||
Canards, Mixed-Mode Oscillations and Chaos in a Two-Trophic Ecological Model: Sensitivity to Parameters and Environmental Fluctuations
Abstract: We consider a two-trophic ecological model comprising of two predators competing for the same prey. Under the assumption that the growth rate of the prey is much larger than that of the predators, the problem is viewed as a singular perturbed system in one fast and two slow variables. We assume that one of the predators (territorial) exhibits a density dependent mortality rate. In the absence of the non-territorial predator, the subsystem exhibits a canard explosion, which refers to a change from an outbreak dynamics to small oscillations around the two species equilibrium state over an extremely narrow interval. As the mortality rate of the territorial predator is varied, the full-system exhibits a variety of rich dynamics, including but not limited to relaxation oscillations (which represent periodic outbreaks interspersed with collapses of the populations), mixed-mode oscillations (which are concatenations of small amplitude and large amplitude oscillations) that indicate the adaptability of the species to prolong the cycles of boom and bust, and chaos. Numerical simulations are carried out to demonstrate the sensitivity of the system to initial conditions and parameters. Finally, if time permits then we will briefly discuss the effect of "noise" on the system.
|Monday, November 16, 2015||Rajarshi Dey, University of South Alabama||
Hypothesis Tests for K-Dimensional ROC Manifolds
Abstract: Consider any classification procedure for K groups based on a certain marker X. Let; Xi ~ Fi; i = 1, 2, ..., K where Fi is continuous for i = 1, 2, ...,K. The most common index to measure the performance of the marker is HUM (Hyper-volume Under Manifold) of the ROC (Receiver Operating Characteristic) manifold obtained by this marker. A random marker obtains an HUM of 1/K!. So, a natural setup for testing the performance of the marker is H0: HUM = 1/K! vs. HA: HUM > 1/K!. However, for even moderately large K; this test is useless as 1/K! converges to 0. I will discuss two new tests for testing the performance of a marker on a K (> 2) group classification problem.
|Monday, November 9, 2015||Frazier Bindele, University of South Alabama||
Rank-Based Inference with Non-Ignorable Missing Responses
Abstract: Missing observations always occur whenever data are collected. The mechanism that causes the missingness is characterized by its degree of randomness. In this talk, the focus is placed on observations that have missing responses in the context of regression modeling. Such missingness is assumed not to be at random- also known as non-ignorable. Existing methods for estimating parameters in regression models when data contain outliers or in presence of errors distributions that are heavy tailed, provide estimates that are inefficient and/or non robust. We propose a rank-based approach to estimating the true regression parameters with the missing responses being imputed via either a simple imputation or an inverse marginal probability imputation. The imputation methods are then incorporated in the regression model leading to a robust rank-based estimate of the parameters via the considered objective function. Large sample properties of the proposed estimator are established under mild regularity conditions. Monte Carlo simulation experiments are carried out and show that the rank-based estimator is more efficient than the least squares estimator whenever the error distribution is heavy tailed or contaminated and under non-ignorable responses. Finally, an illustrative example is discussed.