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        "event_title":"Statistical Methods for Interpretable Representation in Complex Data",
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        "combined_title":"Statistical Methods for Interpretable Representation in Complex Data: Katherine Ahn",
        "event_subtitle":"Katherine Ahn",
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        "description":"This dissertation develops novel statistical methodologies that provide interpretable representations of complex forms of data --- such as sparsely observed functional data, panel data, and notably multivariate data. While flexible methods aiming to capture complex intrinsic structures with minimal assumptions have been previously developed, they often lack explicit and interpretable representations. Here, we develop methods that address this gap. \n\nThe first part of the dissertation develops a method that allows sparsely observed functional responses to be used in a sufficient dimension reduction (SDR) regression. An inverse SDR procedure requires conditional means of the predictors given the functional responses. To accommodate the sparsely observed functional responses, we use local averaging of the predictors with weights inferred by an auxiliary Gaussian process model. As the procedure entails a nested moment estimation, it becomes computationally expensive with a kernel approach that depends on pairwise distances between responses. We view the moment estimation as a numerical integration problem and achieve scalability via a coarsening approach using anchor points on the response space. \n\nThe second part of the dissertation provides a method for constructing an interpretable approximation to the joint distribution of two random vectors. The method first represents each marginal distribution in finitely quantized form. Next, we use a transport-based optimization to probabilistically map jointly observed data to the Cartesian grid of marginal quantizations. Finally, we marginalize over the data to attain a representation on the quantizations. The method ultimately yields an approximation that preserves the marginal distribution structures and provides an interpretable representation of the dependence structure. \n\nThe third part of the dissertation revolves around the analysis of multi-level and panel data via distance-based statistics. First, we develop a framework for using energy distance on spaces of probability distributions to analyze multi-level data. We then extend this idea to the setting of multivariate dyadic panel data, where vector-valued observations are serially observed within paired units. In such data, we decompose the dependence structure into between- and within-pair components. Second, we propose an infinitesimal perturbation framework with case- and variable-weights to elucidate the findings of complex statistical analyses that lack a directly interpretable representation. We use this framework to identify the cases and variables that most strongly contribute to the structures captured by  distance-based statistics.",
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