Semiparametric Bayesian local functional models for diffusion tensor tract statistics

We propose a semiparametric Bayesian local functional model (BFM) for the analysis of multiple diffusion properties (e.g., fractional anisotropy) along white matter fiber bundles with a set of covariates of interest, such as age and gender. BFM accounts for heterogeneity in the shape of the fiber bundle diffusion properties among subjects, while allowing the impact of the covariates to vary across subjects. A nonparametric Bayesian LPP2 prior facilitates global and local borrowings of information among subjects, while an infinite factor model flexibly represents low-dimensional structure. Local hypothesis testing and credible bands are developed to identify fiber segments, along which multiple diffusion properties are significantly associated with covariates of interest, while controlling for multiple comparisons. Moreover, BFM naturally group subjects into more homogeneous clusters. Posterior computation proceeds via an efficient Markov chain Monte Carlo algorithm. A simulation study is performed to evaluate the finite sample performance of BFM. We apply BFM to investigate the development of white matter diffusivities along the splenium of the corpus callosum tract and the right internal capsule tract in a clinical study of neurodevelopment in new born infants

Semiparametric Bayesian models for human brain mapping

Functional magnetic resonance imaging (fMRI) has led to enormous progress in human brain mapping. Adequate analysis of the massive spatiotemporal data sets generated by this imaging technique, combining parametric and non-parametric components, imposes challenging problems in statistical modelling. Complex hierarchical Bayesian models in combination with computer-intensive Markov chain Monte Carlo inference are promising tools.The purpose of this paper is twofold. First, it provides a review of general semiparametric Bayesian models for the analysis of fMRI data. Most approaches focus on important but separate temporal or spatial aspects of the overall problem, or they proceed by stepwise procedures. Therefore, as a second aim, we suggest a complete spatiotemporal model for analysing fMRI data within a unified semiparametric Bayesian framework. An application to data from a visual stimulation experiment illustrates our approach and demonstrates its computational feasibility

Bayesian Analysis of Varying Coefficient Models and Applications

The varying coefficient models have been very important analytic tools to study the dynamic pattern in biomedicine fields. Since nonparametric varying coefficient models make few assumptions on the specification of the model, the 'curse of dimensionality' is an very important issue. Nonparametric Bayesian methods combat the curse of dimensionality through specifying a sparseness-favoring structure. This is accomplished through the Bayesian penalty for model complexity (Jeffreys and Berger, 1992) and is aided through centering on a base Bayesian parametric model. This dissertation presents three novel semiparametric Bayesian methods for the analysis of longitudinal data, diffusion tensor imaging data, and longitudinal circumplex data. In longitudinal data analysis, we propose a semiparametric Bayes approach to allow the impact of the predictors to vary across subjects, which allows flexibly local borrowing of information across subjects. Local hypothesis testing and confidence bands are developed for the identification of time windows for significant predictor impact, adjusting for multiple comparisons. The methods are assessed using simulation studies and applied to a yeast cell-cycle gene expression data set. In analyzing diffusion tensor imaging data, we propose a semiparametric Bayesian local functional model to connect multiple diffusion properties along white matter fiber bundles with a set of covariates of interest. An LPP2 prior facilitates global and local borrowing of information among subjects, while an infinite factor model flexibly represents low-dimensional structure. Local hypothesis testing and confidence bands are developed to identify fiber segments for significant association of covariates with multiple diffusion properties, controlling for multiple comparisons. The method is assessed by a simulation study and illustrated via two fiber tract data sets for neurodevelopment. In analyzing longitudinal circumplex data, we propose a semiparametric Bayesian infinite state-space circumplex model to capture the dynamic transition pattern of affective experience, where affects are characterized as an ordering on the circumference of a circle. A sticky infinite state hidden Markov model via hierarchical Dirichlet proces is used to address the time related state-switching structure and the self-transition feature. The method is assessed by a simulation study and an emotion data set for the dynamics of emotion regulation

Functional Regression

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field

Functional generalized additive models

We introduce the functional generalized additive model (FGAM), a novel regression model for association studies between a scalar response and a functional predictor. We model the link-transformed mean response as the integral with respect to t of F{X(t), t} where F(·, ·) is an unknown regression function and X(t) is a functional covariate. Rather than having an additive model in a finite number of principal components as by Müller and Yao (2008), our model incorporates the functional predictor directly and thus our model can be viewed as the natural functional extension of generalized additive models. We estimate F(·, ·) using tensor-product B-splines with roughness penalties. A pointwise quantile transformation of the functional predictor is also considered to ensure each tensor-product B-spline has observed data on its support. The methods are evaluated using simulated data and their predictive performance is compared with other competing scalar-on-function regression alternatives. We illustrate the usefulness of our approach through an application to brain tractography, where X(t) is a signal from diffusion tensor imaging at position, t, along a tract in the brain. In one example, the response is disease-status (case or control) and in a second example, it is the score on a cognitive test. The FGAM is implemented in R in the refund package. There are additional supplementary materials available online. © 2013 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America

PENALIZED FUNCTION-ON-FUNCTION REGRESSION

We propose a general framework for smooth regression of a functional response on one or multiple functional predictors. Using the mixed model representation of penalized regression expands the scope of function on function regression to many realistic scenarios. In particular, the approach can accommodate a densely or sparsely sampled functional response as well as multiple functional predictors that are observed: 1) on the same or different domains than the functional response; 2) on a dense or sparse grid; and 3) with or without noise. It also allows for seamless integration of continuous or categorical covariates and provides approximate confidence intervals as a by-product of the mixed model inference. The proposed methods are accompanied by easy to use and robust software implemented in the pffr function of the R package refund. Methodological developments are general, but were inspired by and applied to a Diffusion Tensor Imaging (DTI) brain tractography dataset