Statistics and Applied Mathematics Seminar

Semiparametric Methods for Survival Data with Measurement Error under Additive Hazards Cure Rate Models

Abstract: It is well established that measurement error has drastically negative impact on data analysis. It can not only bias parameter estimates but may also cause loss of power for testing relationship between variables. Although survival analysis of error-contaminated data has attracted extensive interest, relatively little attention has been paid to dealing with survival data with error-contaminated covariates when the underlying population is characterized by a cured fraction. In this paper, we consider this problem for which lifetimes of the non-cured individuals are featured by the additive hazards model and the measurement error process is described by an additive model. Unlike estimating the relative risk in the proportional hazards model, the additive hazards model allows us to estimate the absolute risk difference associated with the covariates. To allow the model flexibility, we incorporate time-dependent covariates in the model. We develop estimation methods for the two scenarios, without or with measurement error. The proposed methods are evaluated from both the theoretical view point and the numerical perspectives.

From Statistical Process Control (SPC) to Machine Learning and Fundamental Assisted Analytics (FAA): Transformation of the Steel Industry

Abstract: With over a billion ton plus global production, Steel is often considered as one of the most important manufacturing industries which involves many industrial processes and is capitally and operationally cost intensive. SPC has been an integral part of the quality control aspects for various steel plants, and therefore, steel plants have invested heavily to use SPC methods to improve their processes. With this in mind, the steel industry progressed steadily into the 1990s when steel-makers got interested in dynamic process control and Artificial Neural Network based dynamic models, based on plant data. In 2010, Big Data became the next buzzword for the steel mills. As much as it sounds good, steel plant data is nowhere close to ‘big data’ when compared to social networking and marketing data volumes. Since 2016 onwards, the buzz of artificial intelligence and machine learning became popular, and many equipment suppliers and consultants started developing intelligent machines. They claimed that such efforts would enable process control through real time analytics and the machines, and the steel plant as a whole would ‘self-learn’. On reviewing the reality of these claims, it was often observed that static models developed in the 1980s were being recycled with a fancier graphical user interfaces and named ‘intelligent’. The use of AI and ML based systems was not readily appreciated or adopted by steel makers, and instead people still believed in the fundamentals of metallurgy and steelmaking. Currently, steel makers are waiting for the next generation of SPC based on fundamentals assisted analytics, which would dominate the future of steel making, globally.

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.

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.

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.

Hypothesis Tests for K-Dimensional ROC Manifolds

Abstract: Consider any classification procedure for K groups based on a certain marker X. Let; X_i ~ F_i; i = 1, 2, ..., K where F_i 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. H_A: 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.

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.