Brown University Center for Statistical Sciences Seminar
Abstract:
In medical diagnostic research, biomarkers are used as the basis for detecting
or predicting disease. There has been an increased interest in using the
receiver operating characteristic (ROC) curve to assess the accuracy of biomarkers. In certain situations, a single
biomarker is not enough to achieve the desired level of accuracy; furthermore, newly discovered biomarkers can provide
additional information for a specific disease. Even though numerous methods have been developed for a single biomarker,
few statistical methods exist to accommodate multiple biomarkers simultaneously. In this paper, we propose a multivariate
binormal ROC model to assess multiple biomarkers. Our model assumes that biomarkers follow multivariate normal
distribution after unknown and marker-specific transformations. Random effects are introduced to account for
within-subject correlation among biomarkers. Nonparametric maximum likelihood estimation is used for inference and the
parameter estimators are shown to be asymptotically normal and efficient. Both simulation study and real data application
are used to illustrate the proposed method. (This is joint work with Professor Donglin Zeng)
Candidate for Assistant Professor in the Biostatistics Section of the Program in Public Health
Lefschetz Center for Dynamical Systems Seminar
Abstract: The one-dimensional nonlinear Schrodinger equation (1D NLS) emerges as a first order model in a variety of fields--from high intensity laser beam propagation to Bose-Einstein condensation to water waves theory. The 1D NLS is completely integrable, hence solvable, on the infinite line or with periodic boundary conditions. The realization that the integrable structure might not persist under small perturbations led to the study of the periodic 1D NLS perturbed by a slightly conservative periodic forcing. In this talk I will describe our studies in which we show co-existence of various types of perturbed solutions such as ordered, temporal chaotic and spatiotemporal chaotic solutions. The prediction of the initial profiles that evolve into the different types of perturbed solutions is performed by utilizing a novel geometrical phase space description of the integrable unperturbed equation. As a result, we identify three mechanisms of temporal chaos in the perturbed NLS: homoclinic chaos, hyperbolic resonance, and parabolic resonance. For the latter mechanism we show that it serves as a route from initial data near an unperturbed stable plane wave to a regime of spatiotemporal chaos. Statistical measures are employed to demonstrate that this spatiotemporal chaos is intermittent: there are windows in time for which the solution gains spatial coherence. This is a joint work with V. Rom-Kedar.
Stochastic Systems Seminar
Abstract: The theory of homogenization for equations with random coefficients is now quite well-developed. What is less studied is the theory for the correctors to homogenization, which asymptotically characterize the randomness in the solution of the equation and as such are important to quantify in many areas of applied sciences. I will present recent results in the theory of correctors for elliptic and parabolic problems and briefly mention how such correctors may be used to improve reconstructions in inverse problems. Homogenized (deterministic effective medium) solutions are not the only possible limits for solutions of equations with highly oscillatory random coefficients as the correlation length in the medium converges to zero. When fluctuations are sufficiently large, the limit may take the form of a stochastic equation and stochastic partial differential equations (SPDE) are routinely used to model small scale random forcing. In the very specific setting of a parabolic equation with large, Gaussian, random potential, I will show the following result: in low spatial dimensions, the solution to the parabolic equation indeed converges to the solution of a SPDE, which however needs to be written in a (somewhat unconventional) Stratonovich form; in high spatial dimension, the solution to the parabolic equation converges to a homogenized (hence deterministic) equation and randomness appears as a central limit-type corrector. One of the possible corollaries for this result is that SPDE models may indeed be appropriate in low spatial dimensions but not necessarily in higher spatial dimensions.
Stochastic Systems Seminar
Abstract: Motivated by numerical representations of robust utility functionals, due to Maccheroni et al(2006), we study the problem of partially hedging a European option $H$ when a hedging strategy is selected through a robust convex loss functional $L(\cdot)$ involving a penalization term $\gamma(\cdot)$ and a class of absolutely continuous probability measures $\mq$. We present three results. An optimization problem is defined in a space of stochastic integrals with value function $EH(\cdot)$. Extending the method of F\"ollmer and Leukert(2000), it is shown how to construct an optimal strategy. The optimization problem $EH(\cdot)$ as criterion to select a hedge, is of a ``minimax'' type. In the second, a dual-representation formula for this value is presented, which is of a ``maxmax'' type. This leads us to a dual optimization problem. In the third result, we apply some key arguments in the robust convex-duality theory developed by Schied(2007) to construct optimal solutions to the dual problem, if the loss functional $L(\cdot)$ has an associated convex risk measure $\rho^L(\cdot)$ which is continuous from below, and if the European option $H$ is unifomly bounded.
Scientific Computing Seminar
Abstract: In this talk, we discuss local discontinuous Galerkin (LDG) methods for solving the integrable nonlinear hyperbolic equations which contain nonlinear high order derivatives. The equations contains the Camassa-Holm equation which is an integrable model equation for shallow water waves, the Hunter-Saxton equation which is the high-frequency limit of the Camassa-Holm equation and the Degasperis-Procesi equation which has a similar form to the limiting case of the Camassa-Holm equation. The stabillty and the error estimates of the LDG methods will be discussed. Numerical simulation results for different types of solutions illustrate the accuracy and capability of the LDG methods.
***Faculty Search Candidate Talk***
Abstract: Conditional inference has proven useful for exploratory analysis of neurophysiological point process data. I will illustrate this approach and then focus on a specific sub-problem: random generation of binary matrices with margin constraints. Sequential importance sampling (SIS) is an effective technique for approximate uniform sampling of binary matrices with specified margins. I will describe how to simplify and improve existing SIS procedures using improved asymptotic enumeration and dynamic programming (DP). The DP approach is interesting because it facilitates generalizations.