Novel methods for functional data analysis with applications to neuroimaging studies

In recent years, there has been explosive growth in different neuroimaging studies such as functional magnetic resonance imaging (fMRI) and diffusion tensor imaging (DTI). The data generated from such studies are often complex structured which are collected for different individuals, via various time-points and across various modalities, thus paving the way for interesting problems in statistical methodology for analysis of such data. In this dissertation, some efficient methodologies are proposed with considerable development which have nice statistical properties and can be useful not only in neuroimaging but also in other scientific domains.A brief overview of the dissertation is provided in Chapter 1 and in particular, different kinds of data structures that are commonly used in consecutive chapters are described. Some useful mathematical results frequently used in the theoretical derivations in various chapters are also provided. Moreover, we raise some fundamental questions that arise due to some specific data structures with applications in neuroimaging and answer these questions in subsequent chapters.In Chapter 2, we consider the problem of estimation of coefficients in constant linear effect models for semi-parametric functional regression with functional response, where each response curve is decomposed into the overall mean function indexed by a covariate function with constant regression parameters and random error process. We provide an alternative semi-parametric solution to estimate the parameters using quadratic inference approach by estimating bases functions non-parametrically. Therefore, the proposed method can be easily implemented without assuming any working correlation structure. Moreover, we achieve a parametric ⁸́₍ \uD835\uDC5B-convergence rate of the proposed estimator under the proper choice of bandwidth and establish its asymptotic normality. A multi-step estimation procedure to simultaneously estimate the varying-coefficient functions using a local linear generalized method of moments (GMM) based on continuous moment conditions is developed in Chapter 3 under heteroskedasticity of unknown form. To incorporate spatial dependence, the continuous moment conditions are first projected onto eigen-functions and then combined by weighted eigen-values. This approach solves the challenges of using an inverse covariance operator directly. We propose an optimal instrumental variable that minimizes the asymptotic variance function among the class of all local linear GMM estimators, and it is found to outperform the initial estimates that do not incorporate spatial dependence.Neuroimaging data are increasingly being combined with other non-imaging modalities, such as behavioral and genetic data. The data structure of many of these modalities can be expressed as time-varying multidimensional arrays (tensors), collected at different time-points on multiple subjects. In Chapter 4, we consider a new approach to study neural correlates in the presence of tensor-valued brain images and tensor-valued predictors, where both data types are collected over the same set of time-points. We propose a time-varying tensor regression model with an inherent structural composition of responses and covariates. This development is a non-trivial extension of function-on-function concurrent linear models for complex and large structural data where the inherent structures are preserved. Through extensive simulation studies and real data analyses, we demonstrate the opportunities and advantages of the proposed methods.