Mathematics

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    Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State
    (MDPI, 2021-05-27) Zhang, Xuze; Pyne, Saumyadipta; Kedem, Benjamin
    In disease modeling, a key statistical problem is the estimation of lower and upper tail probabilities of health events from given data sets of small size and limited range. Assuming such constraints, we describe a computational framework for the systematic fusion of observations from multiple sources to compute tail probabilities that could not be obtained otherwise due to a lack of lower or upper tail data. The estimation of multivariate lower and upper tail probabilities from a given small reference data set that lacks complete information about such tail data is addressed in terms of pertussis case count data. Fusion of data from multiple sources in conjunction with the density ratio model is used to give probability estimates that are non-obtainable from the empirical distribution. Based on a density ratio model with variable tilts, we first present a univariate fit and, subsequently, improve it with a multivariate extension. In the multivariate analysis, we selected the best model in terms of the Akaike Information Criterion (AIC). Regional prediction, in Washington state, of the number of pertussis cases is approached by providing joint probabilities using fused data from several relatively small samples following the selected density ratio model. The model is validated by a graphical goodness-of-fit plot comparing the estimated reference distribution obtained from the fused data with that of the empirical distribution obtained from the reference sample only.
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    Harmonic Analysis Inspired Data Fusion for Applications in Remote Sensing
    (2014) Doster, Timothy; Benedetto, John J; Czaja, Wojciech; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    This thesis will address the fusion of multiple data sources arising in remote sensing, such as hyperspectral and LIDAR. Fusing of multiple data sources provides better data representation and classification results than any of the independent data sources would alone. We begin our investigation with the well-studied Laplacian Eigenmap (LE) algorithm. This algorithm offers a rich template to which fusion concepts can be added. For each phase of the LE algorithm (graph, operator, and feature space) we develop and test different data fusion techniques. We also investigate how partially labeled data and approximate LE preimages can used to achieve data fusion. Lastly, we study several numerical acceleration techniques that can be used to augment the developed algorithms, namely the Nystrom extension, Random Projections, and Approximate Neighborhood constructions. The Nystrom extension is studied in detail and the application of Frame Theory and Sigma-Delta Quantization is proposed to enrich the Nystrom extension.
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    Data Representation for Learning and Information Fusion in Bioinformatics
    (2013) Rajapakse, Vinodh Nalin; Czaja, Wojciech; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    This thesis deals with the rigorous application of nonlinear dimension reduction and data organization techniques to biomedical data analysis. The Laplacian Eigenmaps algorithm is representative of these methods and has been widely applied in manifold learning and related areas. While their asymptotic manifold recovery behavior has been well-characterized, the clustering properties of Laplacian embeddings with finite data are largely motivated by heuristic arguments. We develop a precise bound, characterizing cluster structure preservation under Laplacian embeddings. From this foundation, we introduce flexible and mathematically well-founded approaches for information fusion and feature representation. These methods are applied to three substantial case studies in bioinformatics, illustrating their capacity to extract scientifically valuable information from complex data.