Michael Grabchak

Abstract: Human neurodevelopment is a highly regulated biological process, and recent technological advances allow scientists to study the dynamic changes of neurodevelopment at the molecular level through the analysis of gene expression data from human brains. In this talk, we focus on the analysis of data sampled from 16 brain regions in 15 time periods of neurodevelopment. We will introduce a two-step statistical inferential procedure to identify expressed and unexpressed genes and to detect differentially expressed genes between adjacent time periods. Markov Random Field (MRF) models are used to efficiently utilize the information embedded in brain region similarity and temporal dependency in our approach. We develop and implement a Monte Carlo expectation-maximization (MCEM) algorithm to estimate the model parameters. Simulation studies suggest that our approach achieves lower misclassification error and potential gain in power compared with models not incorporating spatial similarity and temporal dependency. We will also describe our methods to infer dynamic co-expression networks from these data. This is joint work with Zhixiang Lin, Stephan Sanders, Mingfeng Li, Nenad Sestan, and Matthew State.

Abstract: In many epidemiological and medical studies, subjects are examined at regular or irregular follow-up visits. Panel count data arise when the response of interest is the count of some repeated events between consecutive examination times, while interval-censored data arise when the response of interest is the time to some particular event and only the status of the event is known at each examination time. Poisson process has been popular to model the panel count data in the literature. In this talk, we propose a gamma-frailty nonhomogeneous Poisson process model for analyzing panel count data to account for the within-subject correlation and develop an easy estimation method using EM algorithm. We also propose a computationally efficient method for analyzing general interval-censored data under the PH model using an EM algorithm. We developed a novel data augmentation by introducing a latent nonhomogeneous Poisson process to expand the observed likelihood. Both approaches have shown excellent performance in terms of estimation accuracy and computational advantages, such as being robust to initial values, converging fast, and providing variance estimates in closed-form. A joint modeling of panel count responses and the interval-censored failure time of a terminal event is discussed as a generalization of the proposed approaches.

Abstract: Dynamic treatment regimes (DTRs) are sequential decision rules tailored at each stage by potentially time-varying patient features and intermediate outcomes observed in previous stages. The complexity, patient heterogeneity and chronicity of many dis- eases and disorders calls for learning optimal DTRs which best dynamically tailor treatment to each individual’s response over time. Proliferation of personalized data (e.g., genetic and imaging data) provides opportunities for deep tailoring as well as new challenges for statistical methodology. In this work, we propose a robust and hybrid learning method, namely Augmented Multistage Outcome-Weighted Learning (AMOL), to identify optimal DTRs from the Sequential Multiple Assignment Randomization Trials (SMARTs). For multiple-stage SMART studies, we develop a sequentially backward learning method to infer DTRs, making use of the robustness of single-stage outcome weighted learning and the imputation ability of regression model-based Q- learning at each stage. The proposed AMOL remains valid even if the imputation model assumed in the Q-learning is misspecified. We establish theoretical properties of AMOL, including double robustness and efficiency of the imputation step, as well as consistency of estimated rules and rates of convergence to the optimal value function. The comparative advantage of AMOL over existing methods is demonstrated in extensive simulation studies and applications to two SMART data sets: a two-stage trial for attention deficit and hyperactive disorder (ADHD) and the STAR*D trial for major depressive disorder (MDD).

Abstract: With the abundance of high dimensional data in various disciplines, sparse regularized techniques are very popular these days. In this talk, we use the structure information among predictors to improve sparse regression models. Typically, such structure information can be modeled by the connectivity of an undirected graph. Most existing methods use this graph edge-by-edge to encourage the regression coefficients of corresponding connected predictors to be similar. However, such methods may require expensive computation when the predictor graph has many edges. Furthermore, they do not directly utilize the neighborhood information. In this work, we incorporate the graph information node-by-node instead of edge-by-edge. Our proposed method is quite general and it includes adaptive Lasso, group Lasso and ridge regression as special cases. Both theoretical study and numerical study demonstrate the effectiveness of the proposed method for simultaneous estimation, prediction and model selection. Applications to Alzheimer’s disease data and cancer data will be discussed as well.

Abstract: In 1921, Steklov made a conjecture that the polynomials orthonormal on a segment with respect to a weight bounded away from zero are uniformly bounded for every interior point of that segment. This conjecture was disproved by Rahmanov in 1979 but the sharp estimates on the polynomials from the Steklov class were still missing. We will discuss some recent results (joint with Aptekarev and Tulyakov) in which the full solution to the problem by Steklov was obtained.

Abstract: In this talk, I will present convergence results of singularly perturbed problems in the sense of PDEs, which is related to the vanishing viscosity limit. I also provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a P1 classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh. In the second part of my talk, I will present a finite volume scheme to solve the two dimensional inviscid primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically relevant boundary conditions. In that respect, a version of a projection method is introduced to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions.

Abstract: The problem of homogenization of random structures will be reviewed and some open mathematical questions will be outlined.

Abstract: A sign pattern (matrix) is a matrix whose entries are from the set {+, -, 0}. The minimum rank of a sign pattern matrix A is the smallest possible rank of a real matrix whose entries have signs indicated by A. A direct connection between an m by n sign pattern with minimum rank r>=2 and an m point–n hyperplane configuration in R^{r-1} is established. A possibly smallest example of a sign pattern (with minimum rank 3) whose minimum rank cannot be realized rationally is given. It is shown that for every sign pattern with at most 2 zero entries in each column, the minimum rank can be realized rationally. Using a new approach involving sign vectors of subspaces and oriented matroid duality, it is shown that for every m by n sign pattern with minimum rank >= n-2, rational realization of the minimum rank is possible. It is also shown that for every integer n>=9, there is a positive integer m, such that there exists an m by n sign pattern with minimum rank n-3 for which rational realization is not possible. A characterization of m by n sign patterns A with minimum rank n-1 is given, along with a more general description of sign patterns with minimum rank r, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of k-dimensional subspaces of R^n are discussed; this maximum number is equal to the total number of cells of a generic central hyperplane arrangement in R^k. In particular, it is shown that the maximum number of sign vectors of a 2-dimensional subspace of R^n is 4n+1 and the maximum number of sign vectors of a 3-dimensional subspace of R^n is 4n(n – 1) + 3. Related results and open problems are stated along the way.

Abstract: I will consider deterministic and stochastic perturbations of dynamical systems and stochastic processes. The perturbed system has a slow and fast components and the slow component is the most important characteristic of the long time behavior of the perturbed system. The slow component lives on the simplex  of normalized invariant measures of the non-perturbed system. In an appropriate time scale, the limiting slow motion is defined by a modified averaging principle or by large deviations. I will demonstrate how this general approach works for Landau-Lifshitz magnitization equation and for some PDEs with a small parameter.

Abstract: Many disease diagnoses involve subjective judgments. For example, through the inspection of a mammogram, MRI, radiograph, ultrasound image, etc., the clinician himself becomes part of the measuring instrument. Variability among raters examining the same item injects variability into the entire diagnostic process and thus adversely affect the utility of the diagnostic process itself. To reduce diagnostic errors and improve the quality of diagnosis, it is very important to quantify inter-rater variability, to investigate factors affecting the diagnostic accuracy, an to reduce the inter-rater variability over time. This paper focuses on a subjective binary decision process. A hierarchical model linking data on rater opinions with patient disease-development outcomes is proposed. The model allows for the quantification of patient-specific disease severity and rater-specific bias and diagnostic ability. The model can be used in an ongoing setting in a variety of ways, including calibration of rater opinions (estimation of the probability of disease development given opinions) and quantification of rater-specific sensitivities and specificities. Bayesian computational algorithm is developed. An extensive simulation study is conducted to evaluate the proposed method, and the proposed method is illustrated by a mammogram data set.