Parameter clustering in Bayesian functional PCA of neuroscientific data

This thesis provides novel methodologies for functional Principal Component Analysis of dependent time series (curves) with particular emphasis on those arising from neuroscientific experiments. In this context, the extraordinary advances in neuroscientific technology for brain recordings over the last decades have led to increasingly complex spatio-temporal datasets. We propose new models that merge ideas from Functional Data Analysis and Bayesian nonparametrics to obtain a flexible exploration of spatiotemporal data. In the first part of the thesis, we developed a Dirichlet process Gaussian mixture model to cluster functional Principal Component scores within the standard Bayesian functional Principal Component Analysis framework. This approach allows us to capture the structure of spatial dependence among smoothed curves and its interaction with the time domain. Moreover, by moving the mixture from data to functional Principal Component scores, we obtain a more general clustering procedure, allowing a substantially finer curve classification and higher level of intricate insight and understanding of the data. We present results from a Monte Carlo simulation study showing improvements in curve and correlation reconstruction compared with different Bayesian and frequentist functional Principal Component Analysis models, Further, we apply our method to a resting-state fMRI data analysis providing a rich exploration of the spatio-temporal patterns in brain time series. In the second part of the thesis, we extend our model to the challenging case of multilevel functional data where multiple curves are nested within subjects, and subjects are divided in groups. We develop a method based on a parsimonious trade-off between group behaviours and individual deviations, returning a comprehensive exploration of intricate multilevel functional data. We obtained excellent classification and improved curve reconstruction with high-noise level due to the eigen-dimension-specific borrowing of strength among subjects’ functional scores in the same group. Finally, we discuss future extensions in the direction of a general and flexible modelling framework for complex spatio-temporal data