Joint Materials/Solid Mechanics Seminar Series
Brown University Center for Statistical Sciences Seminar
Abstract: The diagnostic criteria for defining mental disorders in the Diagnostic and Statistical Manuals (DSM) and the International Classification of Diseases (ICD) have come about by general consensus of experts in the field. While this is a valid means for creating diagnostic criteria, it is not driven by empirical evidence. As a result, it is important to compare the classifications based on these criteria with classifications based purely on empirical methods. One method for achieving this is to use a latent class model (LCM) for defining disease classes. The LCM takes discrete indicators of symptoms and, assuming a fixed number of classes k, uses the symptoms to define k distinct diseases classes via a mixture model approach. The general idea is that each individual in the population belongs to one of the k classes and that the symptom patterns provide evidence as to the disorder class. The LCM estimation finds the definitions of classes most consistent with the distribution of symptom patterns in the population.
There are several approaches for estimation of the LCM. An EM algorithm is often used for estimating the maximum likelihood (ML) estimates of the parameters of the LCM. We instead use a Bayesian approach, where we estimate the model via a Markov chain Monte Carlo (MCMC) algorithm using relatively weak prior information. The benefit of the MCMC approach over the ML approach is that the posterior distributions of any function of the parameters can be estimated. This is crucial to our inferences about the operating characteristics of the diagnostic criteria: sensitivity, specificity, positive predictive value, and negative predictive value. Using the MCMC approach we can get estimates of these quantities in addition to estimates of their standard errors.These methods are demonstrated on the third wave of Baltimore's Epidemiologic Area Catchment Area (ECA) study, which is an epidemiologic sample of data collected between 1993 and 1996. Data were collected on mental disorder symptoms including depression. We compare the LCM definition of depression in the sample of 1920 individuals who completed the depression symptom questions to the diagnostic definitions from the DSM-III, DSM-IV, and the mild, moderate, and severe definitions of depression in the ICD-10. We consider the latent class categorization to be our gold standard and can compare diagnoses from the ICD and the DSM to this gold standard and estimate sensitivity, specificity, positive predictive value, and negative predictive value for each diagnostic criteria. This allows us to formally evaluate the agreement between empirical evidence and more theoretical definitions of disorders.
Stochastic Systems Seminar
Abstract: We are concerned with learning to act optimally in a dynamic, noisy, and competitive environment. Our model deviates from classical game theory as we assume that the other agents in the environment may act irrationally and arbitrarily. We look for strategies that take advantage of possible deviations of the other agents from an adversarial behavior.
We present two solution concepts for such environments. The first concept is based on the empirical measurement of statistics related to the non-stationary elements in the environment. We take a worst-case estimate over unknown information and re-define the problem as a vector-valued stochastic game. We define a concrete goal, the convex Bayes envelope, and show that it is attainable. This envelope is shown to be safe (above the worst-case performance guarantees) and adaptive (strictly above the worst-case performance guarantees if the non-stationary elements appear non-hostile). The second concept is based on viewing the stochastic game as a repeated super-game with partial monitoring. We show that this approach leads to different performance guarantees which are also safe and adaptive.
Brown Analysis Seminar
Scientific Computing Seminar
Abstract: It is well known that solutions of elliptic PDEs usually are singular in corners of the spatial domain. These 'corner singularities' have observable consequences. In buildings, cracks tend to radiate from the corners of walls, windows, and doors. Airplanes have rounded windows to minimize cracks that can cause catastrophic failures, and small nested vortices called 'Moffatt eddies' appear in the corners of fluid-filled regions. What has been much less noticed is that the solutions to virtually all time-dependent PDEs will exhibit singularities in the corners of the time-space domain, i.e. where initial conditions and boundary conditions meet. The consequences for numerical calculations can be severe. Without special corrections, the accuracy of Chebyshev and other spectral methods will get reduced to that of a low order finite difference scheme (either for brief initial moments only, or for all times, dependent on the character of the PDE). Time-space singularities are multiple-scale phenomena. Analytical and numerical tools to analyze and remedy them will be discussed in the cases of convective-diffusive and dispersive PDEs.
The topic was brought to my attention by Dr. Natasha Flyer, and the seminar describes our recent collaborative work on the subject.
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