Lefschetz Center for Dynamical Systems Seminar
Abstract: We study slow modulation of patterns, or quasi-steady solutions, in forced dispersive systems as arise in models of optical parametric processes. The linearization about a pattern yields a family of non-self adjoint operators. The localization of the point spectrum of these operators requires new techniques. Moreover we obtain a rigorous decomposition of the flow near the pattern from a renormalization group method, which permits us to overcome not only the weak smoothing properties of the linear semi-groups, but also their possible secular behavior. We obtain a family of ODEs which describe the evolution of the pattern and estimates on the error.
Brown University Center for Statistical Sciences Seminar
Abstract: We propose a new method that is intermediate between the fully parametric and nonparametric maximum likelihood estimation approaches for fitting frailty models to clustered survival data. a parametric form is assumed on the baseline hazard for the purpose of imputing the unobserved frailties. We estimate the regression coefficients, however, by solving the average partial likelihood score based on the imputed frailties. We prove consistency and asymptotic normality of the resulting estimators and give associated closed-form varience estimators. The algorithm is easy to implement and reduces to the ordinary Cox partial likelihood approach when the frailties have a degenerate distribution. Simulations indicate high effucuebct abd ribystbess of the resulting estimates. We expect that this proposed imputed PLS methodology shall have broad applications in the analyses of clustered interval-censored survival data, survival data with measurement error in covariates and spatial survival data, where the unobserved random quantities, such as true survival times, true covariates and region-specific random effects, can be imputed simultaneously from their posterior distributions to construct unbiased average partial likelihood estimating equations.
*Reception following seminar at 167 Angell Street, 2nd floor conference room
Department of Mathematics Special Seminar
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: Most cost function based clustering or partitioning methods measure the compactness of groups of data in feature space, e.g., k-means clustering or pairwise clustering. This concept is appropriate for data sources which can be viewed as point sources in feature space. Data which are modified by a continuous transformation are better characterized by their distance to a low-dimensional manifold in feature space than by a prototype with a high density of data points in its neighborhood. To capture this property of data sets I describe a new clustering algorithm which stresses connectedness in feature space rather than compactness. Objects are similar if there exists a mediating path on the underlying object graph without an edge with large weight (dissimilarity). The cost function is optimized in a multi-scale fashion. The path based clustering concept is applied to segment textured images with strong texture gradients on the basis of pairwise dissimalarities between image patches.
In the second part of the talk I will discuss the relation of optimization principles for clustering to issues of model selection and robustness of solutions. The maximum entropy framework for inference plays a special role in clustering and provides a mechanism to determine the number of clusters from large deviation arguments.
(This talk summarizes joint work with Bernd Fischer and Thomas Zoeller.)
Brown Analysis Seminar
Brown University Graduate School
Dissertation Defense Information
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