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.

Abstract: With the advance of high-throughput sequencing technologies, it has become feasible to investigate the influence of the entire spectrum of sequencing variations on complex human diseases. Although association studies utilizing the new sequencing technologies hold great promise to unravel novel genetic variants, especially rare genetic variants that contribute to human diseases, the statistical analysis of high-dimensional sequencing data remains a challenge. Advanced analytical methods are in great need to facilitate high-dimensional sequencing data analyses. In this talk, we will introduce a generalized genetic random filed (GGRF) method for association analyses of sequencing data in case-control studies. We will then further extend GGRF method to a family-based GGRF (FB-GGRF) method for family-based association studies. Both GGRF and FB-GGRF methods are compared with other existing methods through simulation studies and real data applications for investigating the genetic etiology of complex diseases/traits.

Abstract: Asymptotic behavior of the tuning parameter selection in the standard cross-validation methods is investigated for the high-dimensional variable selection problem. It is shown that the shrinkage effect of the Lasso penalty is not always the true reason for the over-selection phenomenon in the cross-validation based tuning parameter selection. After identifying the potential problems with the standard cross-validation methods, we propose a new procedure, Consistent Cross-Validation (CCV), for selecting the optimal tuning parameter. CCV is shown to enjoy the tuning parameter selection consistency property under certain technical conditions. Extensive simulations and real data analysis support the theoretical results and demonstrate that CCV also works well in terms of prediction.