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Modeling Functional Data with Spatially Heterogeneous Shape Characteristics


  • Ana-Maria Staicu,

    Corresponding author
    1. Department of Statistics, North Carolina State University, 2311 Stinson Drive, Campus Box 8203, Raleigh, North Carolina 27695-8203, U.S.A.
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  • Ciprian M. Crainiceanu,

    1. Department of Biostatistics, Johns Hopkins University, 615 North Wolfe Street, E3636 Baltimore, Maryland 21205, U.S.A.
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  • Daniel S. Reich,

    1. Translational Neuroradiology Unit, Neuroimmunology Branch, National Institute of Neurological Disorders and Stroke, National Institutes of Health, 10 Center Drive MSC 1400, Building 10 Room 5C103, Bethesda, Maryland 20892, U.S.A.
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  • David Ruppert

    1. Andrew Schultz, Jr., Professor of Engineering, School of Operational Research and Information Engineering, 1170 Comstock Hall, Cornell University, New York 14853-3801, U.S.A.
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Summary We propose a novel class of models for functional data exhibiting skewness or other shape characteristics that vary with spatial or temporal location. We use copulas so that the marginal distributions and the dependence structure can be modeled independently. Dependence is modeled with a Gaussian or t-copula, so that there is an underlying latent Gaussian process. We model the marginal distributions using the skew t family. The mean, variance, and shape parameters are modeled nonparametrically as functions of location. A computationally tractable inferential framework for estimating heterogeneous asymmetric or heavy-tailed marginal distributions is introduced. This framework provides a new set of tools for increasingly complex data collected in medical and public health studies. Our methods were motivated by and are illustrated with a state-of-the-art study of neuronal tracts in multiple sclerosis patients and healthy controls. Using the tools we have developed, we were able to find those locations along the tract most affected by the disease. However, our methods are general and highly relevant to many functional data sets. In addition to the application to one-dimensional tract profiles illustrated here, higher-dimensional extensions of the methodology could have direct applications to other biological data including functional and structural magnetic resonance imaging (MRI).