Nonparametric functional models have a wide range of applications in cancer and other biomedical research, such as biomarker data, hormone profiles, circadian patterns and EEG data. Most of the current literature focuses on developing flexible nonparametric models for the mean structure while the stochastic variation around the mean is treated as a nuisance component. In many biomedical applications, such as the data described in this proposal, the inferential focus is on the stochastic variation and on how it is related to the covariates and experimental treatment. Our first specific aim is to develop methods to nonparametrically estimate the unknown covariance structure in the functional data analysis setting. The estimate of the covariance structure can in turn be used to obtain more efficient estimates of the mean parameters.
Our second aim i s to extend the concept of functional data analysis to time series data, in which the basic unit of the data analysis is a time series and the focus on the analysis is not on the mean, but on how the covariates are related to the stochastic variation over time. A time series is uniquely defined by its spectrum and when we average across a group of spectra, we can obtain a group-average spectrum that uniquely defines a group-average"""""""" time series. Functional linear models and functional mixed effects models can also be applied to the spectra to make inference on covariates and treatment effects. We will focus our methods development on nonstationary time series data, which are most common in biomedical research. We will also develop computationally efficient estimation procedures through construction of equivalent state space models. Our third specific aim is to generalize the concept of functional data analysis to density data, in which the basic unit of the data analysis is a density. The proposed density models allow us to investigate covariates or treatment effects without making explicit assumptions on the underlying distributions. The proposed methods are motivated by and will be applied to the clinical studies that the principal investigator is directly or indirectly involved.
