High-dimensional statistics is one of the most studied topics in the field of statistics. The most interesting problem to arise in the last 15 years is variable selection or subset selection. Variable selection is a strong statistical tool that can be explored in functional data analysis. In the first part of this thesis, we implement a Bayesian variable selection method for automatic knot selection. We propose a spike-and-slab prior on knots and formulate a conjugate stochastic search variable selection for significant knots. The computation is substantially faster than existing knot selection methods, as we use Metropolis-Hastings algorithms and a Gibbs sampler for estimation. This work focuses on a single nonlinear covariate, modeled as regression splines. In the next stage, we study Bayesian variable selection in additive models with high-dimensional predictors. The selection of nonlinear functions in models is highly important in recent research, and the Bayesian method of selection has more advantages than contemporary frequentist methods. Chapter 2 examines Bayesian sparse group lasso theory based on spike-and-slab priors to determine its applicability for variable selection and function estimation in nonparametric additive models.The primary objective of Chapter 3 is to build a classification method using longitudinal volumetric magnetic resonance imaging (MRI) data from five regions of interest (ROIs). A functional data analysis method is used to handle the longitudinal measurement of ROIs, and the functional coefficients are later used in the classification models. We propose a P\\'olya-gamma augmentation method to classify normal controls and diseased patients based on functional MRI measurements. We obtain fast-posterior sampling by avoiding the slow and complicated Metropolis-Hastings algorithm. Our main motivation is to determine the important ROIs that have the highest separating power to classify our dichotomous response. We compare the sensitivity, specificity, and accuracy of the classification based on single ROIs and with various combinations of them. We obtain a sensitivity of over 85% and a specificity of around 90% for most of the combinations.Next, we work with Bayesian classification and selection methodology. The main goal of Chapter 4 is to employ longitudinal trajectories in a significant number of sub-regional brain volumetric MRI data as statistical predictors for Alzheimer's disease (AD) classification. We use logistic regression in a Bayesian framework that includes many functional predictors. The direct sampling of regression coefficients from the Bayesian logistic model is difficult due to its complicated likelihood function. In high-dimensional scenarios, the selection of predictors is paramount with the introduction of either spike-and-slab priors, non-local priors, or Horseshoe priors. We seek to avoid the complicated Metropolis-Hastings approach and to develop an easily implementable Gibbs sampler. In addition, the Bayesian estimation provides proper estimates of the model parameters, which are also useful for building inference. Another advantage of working with logistic regression is that it calculates the log of odds of relative risk for AD compared to normal control based on the selected longitudinal predictors, rather than simply classifying patients based on cross-sectional estimates. Ultimately, however, we combine approaches and use a probability threshold to classify individual patients. We employ 49 functional predictors consisting of volumetric estimates of brain sub-regions, chosen for their established clinical significance. Moreover, the use of spike-and-slab priors ensures that many redundant predictors are dropped from the model.Finally, we present a new approach of Bayesian model-based clustering for spatiotemporal data in chapter 5 . A simple linear mixed model (LME) derived from a functional model is used to model spatiotemporal cerebral white matter data extracted from healthy aging individuals. LME provides us with prior information for spatial covariance structure and brain segmentation based on white matter intensity. This motivates us to build stochastic model-based clustering to group voxels considering their longitudinal and location information. The cluster-specific random effect causes correlation among repeated measures. The problem of finding partitions is dealt with by imposing prior structure on cluster partitions in order to derive a stochastic objective function.
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