Brown Analysis Seminar
Brown University Center for Statistical Sciences Seminar
Abstract: The accuracy (sensitivity and specificity) of a new screening test can be compared with that of a standard test by applying both tests to a group of subjects in which disease status can be determined by a gold standard (GS) test. However, it is not always feasible to administer a GS test to all study subjects. For example, a study is planned to determine if a new screening test for cervical cancer is better than the standard test, and in this setting it is not feasible (or ethical) to determine disease status by biopsy in order to identify women with and without disease for participation in a study. When determination of disease status is not possible for all study subjects, the ratio of the accuracy of two screening tests can still be estimated using a paired screen positive (PSP) design in which all subjects receive both screening tests, but only have the GS test if one of the screening tests is positive. Unfortunately in the cervical cancer example, the PSP design is also infeasible because it is not technically possible to administer both screening tests at the same time. In this talk a randomized paired screen positive (RPSP) design is described in which subjects are randomized to receive one of the two screening tests initially, and only receive the other screening test and gold standard if the first screening test is positive. We derive maximum likelihood estimators and confidence intervals for the relative accuracy of the two screening tests, and assess the small sample behavior of these estimators using simulation studies. Sample size formulae are derived and applied to the cervical cancer screening trial example, and the efficiency of the RPSP design is compared with other designs.
Lefschetz Center for Dynamical Systems Seminar
Abstract: Many stochastic partial differential equation models arising in applications generate complex time-evolving patterns which are hard to quantify due to the lack of any underlying regular structure. The influence of stochasticity leads to variations in the detail structure of the patterns and forces one to concentrate on rougher common geometric features. In many of these instances, such as for example in phase-field type models in materials science, one is interested in the geometry of sublevel sets of a function in terms of their topology, in particular, their homology. Recent computational advances make it possible to compute the homology of discrete structures efficiently and fast. Such methods can be applied to the above situation if the sublevel sets of interest are approximated using an underlying discretization of the considered evolution equation. Yet, this method immediately raises the question of the accuracy of the computed homology. In this talk, I will present a probabilistic approach which gives insight into the suitability of the above method in the context of random fields. We will obtain explicit probability estimates for the correctness of the homology computations, which in turn yield a-priori bounds for the suitability of certain grid sizes as well as information on the optimal location of sampling points.
Department of Mathematics Colloquium
Abstract:
This documentary film by Agnes Handwerk and Harrie Willems tells the moving
story of a young Jewish mathematician who is tragically caught in the difficult
times of World War II. During the winter of 1939-40, while serving in the
French army, he wrote a mathematics manuscript entitled "On Kolmogorov's
equation". He sealed and sent this to the Academy of Sciences in Paris. Later
that winter, when trapped by German soldiers, he committed suicide. The sealed
letter was not opened until May 2000; when deciphered, the manuscript showed
that Doeblin developed a formula to calculate the role of chance in continuous
random processes comparable to the formula that Kiyoshi Ito developed some
years later. The film explores the biography of Wolfgang Doeblin, the
intriguing history of his sealed letter with the manuscript, and the
mathematics in the manuscript.
This 55-minute film will be shown of interest to a general audience. It will
be followed by a more mathematical, 25-minute film having to do with stochastic
processes.
Center for Computational Molecular Biology Seminar
Abstract:
Theoretical population genetics offers us the ability to make inferences about
past evolutionary forces from genetic variation observed in natural populations.
However, the models we apply to data often make very simplistic assumptions
about both demographic processes and the strength of natural selection over
time. Such assumptions can lead to spurious conclusions about the
likelihood of past evolutionary events.
I will discuss two spatial aspects of human genetic variation: the geographic
distribution of genetic variation, and genetic differences within the genome
due to chromosomal location on either the X chromosome or the autosomes.
These studies highlight the role human demographic history has played in
shaping the distribution of human genetic variability, and lead to interesting
questions about the appropriate spatial and temporal scales at which we might
study the signatures of evolutionary forces in the genomic era.
Scientific Computing Seminar
Abstract: We propose a novel semi-Lagrangian method for the Vlasov equation, which combines Strang splitting in time with WENO reconstruction in space. A key insight is that the spatial interpolation matrices, used in the reconstruction process, can be factored into flux matrices, because of which WENO can be applied. The CFL time step restriction is removed in the semi-Lagrangian framework. The quality of the method is demonstrated by several classical problems in plasma physics.
PDE Seminar