Brown University - Division of Biology and Medicine
Center for Statistical Sciences Seminar
Department of Biostatistics, Harvard School of Public Health Candidate for Assistant Professor (Tenure Track) in the Public Health Program/Department of Community Health | |
1st Floor Conference Room 106 (Coffee at 3:45 p.m.) |
Abstract: Recent technological advances in continuous biological monitoring and personal exposure assessment have led to collection of subject-specific functional data. Primary goal of these investigations was the study of the relationship between the functional predictors and the functional responses.
In order to model the dependence of the response on the history of the predictor values, we propose an extension of the historical functional linear model (HFLM), first introduced by Malfait and Ramsay (2003), by using a variety of regularization techniques in the estimation of regression coefficients. First, we approximate the regression surface relating the predictor to the outcome by a finite dimensional basis expansion, and then we penalize the coefficients of the neighboring basis functions by restricting the size of the coefficient differences to be small.
We study penalties based on the absolute values of the basis functions coefficient differences (corresponding to the LASSO) and the squares of the differences (corresponding to the P-spline methodology). We compare the fits using an extension of the Akaike Information Criterion which combines the error variance estimate, degrees of freedom of the fit and the norm of the basis functions coefficients. In the simulation settings considered, we find that the LASSO penalty on the linearly transformed coefficients gives sparser representation of the estimated regression surface, while the quadratic penalty provides solutions with the smallest L2-norm or the basis functions coefficients.
We apply our methodology to analyze the effects of occupational particulate matter (PM) exposure on the heart rate variability (HRV) in a cohort of boilermaker workers. Results suggest that the strongest association between PM exposure and HRV in these workers occurs as a result of point exposures to the increased levels of particulate matter.
Brown Analysis Seminar
Probability/Statistics Seminar
Brown University, Division of Applied Mathematics
Department of Mathematics, University of California at Berkeley Candidate for the Junior Probability/Statistics Position | |
From the threefold way to a continuous \beta future | |
Abstract: In classical probability, the Gaussian, Chi-square, and Beta are three of the most studied distributions, with wide applicability. In the last century, matrix equivalents to these three distributions have emerged from nuclear physics (Gaussian ensembles) and multivariate statistics (Wishart and MANOVA ensembles). Their eigenvalue statistics have been studied in deppth for three values of a parameter (\beta = 1, 2 and 4) which defines the "threefold way" and can be thought of as a counting tool for their real, complex, or quaternion entries.
The re-examination of the Selberg integral formula, in the late `80s, has brought the advent of general \beta-ensembles, which subsume the classical cases, and for which the Boltzmann parameter \beta acts as an inverse temperature. Their eigenvalue statistics interpolate between the isolated instances 1, 2, and 4, offering a "behind the scenes" perspective.
With the discovery of matrix models for the general \beta-ensembles in the early 00's, we have entered a new stage in the understanding of the complex phenomena that lie beneath the threefold way. While the \beta = 1,2 and 4 cases are and will always be special, we believe that the future of the classical ensembles is written in terms of a continuous \beta parameter.
The J. P. LaSalle Memorial Lecture
Abstract: A perspective will be presented on stochastic equations of fluid dynamics for turbulent velocities. These include: transport equations driven by the Kraichnan velocity, stochastic Navier-Stokes and Euler equations, etc. The stochastic Navier-Stokes equation will be derived from first principles.
Propagation of Wiener chaos by the transport and Navier-Stokes equations and its utility for the closure problem will be discussed. The "propagator", a deterministic parabolic system for the Fourier coefficients in a Wiener chaos expansion of the velocity field will be presented. It will be shown that the statistical moments of the velocity field (in particular the N-point correlators ) can be expressed straightforwardly via the solution of the propagator. Error analysis and some preliminary numerical results will be discussed.
Probability/Statistics Seminar
Brown University, Division of Applied Mathematics
Abstract: Thermal or stochastic effects are prevalent in physical, chemical, and biological systems. Particularly in small systems, noise can overpower the deterministic dynamics and lead to "rare events," events which would never be seen in the absence of noise. One example is the thermally-driven switching of the magnetization in small memory elements. Wentzell-Freidlin large deviation theory is a mathematical tool for studying rare events. It estimates their probability and also the "most likely switching pathway," which is the pathway in phase space by which rare events are most likely to occur. We explain how large deviation theory and concepts from stochastic resonance may be applied to analyze thermally-activated magnetization reversal in the context of the spatially uniform Landau-Lifschitz-Gilbert equations. The time-scales of theexperiment are critical. One surprising and physically relevant result is that in multiple-pulse experiments, nonconvential "short-time switching pathways" can dominate. The effect is dramatic: the usual pathway (connected with the Arrhenius-law) underestimates the probability of switching by an exponential factor.
An advantage of the method via large deviation theory is that it generalizes to systems with spatial variation. To discuss the complications and richnessthat emerge when spatial variation is taken into account, we consider the (simpler) Allen-Cahn equation. In this context, the rare event of interest is phase transformation from u = -1 to u = +1, and the most likely switching pathway is a pathway through function space. A natural reduced problem emerges in the "sharp-interface limit." We give a brief overview of some results (rigorous in d = 1, heuristic in d > 1).
The first part of the talk is joint work with Bob Kohn and Eric Vanden? Eijnden. The second part includes work that is also joint with Felix Otto and Yoshihiro Tonegawa.
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