An ever-increasing number of biomedical studies yield functional data sampled on a fine grid. These type of data are frequently high dimensional and complex with many irregular features like peaks and change points. There is currently a dearth of existing rigorous statistical methods for analyzing this type of data. The goal of this research program is to develop new Bayesian methodology that provides a unified framework for modeling and performing inference on samples of curves that is flexible enough to apply to a variety of applications, from various experimental designs, and can answer a broad range of research questions. 1. We will develop new methodology within the wavelet-based functional mixed model framework that accommodates outlying curves, a broader class of within- curve covariance structures, and higher dimensional functional data, making it applicable to a broad range of functional data. 2. We will develop methods to classify individuals based on their functional data, e.g. proteomic profiles, in a way that allows us to combine information across functional and scalar factors of multiple sources. We will develop methods to perform Bayesian functional hypothesis testing. 3. We will develop adaptive methods for relating functional predictors to functional responses. 4. We will develop methods for adaptive functional principal components analysis and for principal component-based functional mixed models, which represents a data-driven modeling framework that is extremely flexible in taking into account the complex structure that may be present in the functional data. 5. We will apply the methods to a number of cancer-related studies yielding functional data, including various types of proteomics and genomics data. 6. We will develop efficient, easy-to-use, freely available code to fit the methods described in this proposal.
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