An ever-increasing number of biomedical studies yield functional data, in which the ideal units of observation are curves. The goal of this research program is to develop new Bayesian methodology that provides a unifying framework for performing nonparametric estimation and inference for samples of curves. These methods will be flexible enough to model functions obtained from a variety of experimental designs, provide answers to a broad range of research questions, and will be sufficiently adaptive to apply to functional data from a wide range of applications. We will apply these methods to model functional data from a series of cancer-related biomedical studies that have motivated our methodological thinking. The methods we propose are appropriate for functional data characterized by numerous local features like peaks since we employ adaptive regularization procedures which denoise the functions with minimal attenuation of the dominant local features.
The specific aims of this research are: 1. Introduce a unified functional mixed model framework for modeling samples of curves. Develop a wavelet-based method to fit this model and obtain adaptively regularized nonparametric estimates and Bayesian inference for fixed and random effect functions as well as covariance parameters. 2. Develop methodology to perform formal Bayesian inference and model selection in functional mixed models. Applications of this method include testing functional hypotheses on fixed/random effects, comparing models with different covariance structures, determining the number of basis functions, testing for correlation among different functional responses, and identifying appropriate piecewise constant compartment models. 3. Develop methods to perform wavelet-regularized functional principal component analysis. Extend our wavelet-based functional mixed model methods developed in Specific Aims 1 and 2 to other basis functions, including wavelet-regularized eigen functions and splines. 4. Apply the methods we develop in Specific Aims 1, 2, and 3 to a series of biomedical applications involving functional data, including colon carcinogenesis studies, a Planet Health children's activity study, an animal study investigating acute renal failure, and medical studies involving proteomics. 5. Produce publicly available statistical software for implementing the methods developed in this proposal.
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