Neuroimaging has become ubiquitous in the study of human brain anatomy, yet it continues
to pose multiple statistical challenges, namely high-dimensionality and spatial dependence.
Functional data analysis (FDA) can address this by modelling observations as
functions, applicable to images. In this thesis we consider neuroimaging datasets where a
series of 3D images is captured over time, creating a 4 dimensional dataset for which there
are no FDA methods. We model these data using FDA, proposing a novel model that
preserves the spatial relations between voxels and simultaneously modelling the temporal
correlation whilst maintaining computational efficiency.
Whilst all images are captured on a regular, high-dimensional grid, the time dimension
can vary in density. This thesis considers two types of temporal data, densely collected
images in the form of fMRI and sparse data collected longitudinally in a large-scale
study. Current methods that model multi-dimensional functional data are limited to two
dimensions and cannot be applied to imaging. We begin by introducing a non-parametric
functional principal component model for the representation of spatio-temporal images as
a product of time-invariant basis functions and subject specific score functions that can
vary over time. We propose an estimation method that avoids calculating the covariance
matrix, making our approach computationally efficient. The performance of the model
and its estimation are studied via simulation. This method is applied to a task fMRI
dataset. The obtained score functions are used to model the associations between brain
activity and risk behaviour. In low dimensions we design a large simulation study to
compare the performance of our model to state-of-the-art functional models. The simulation
provides insight on appropriate use cases for the proposed model as well as shows
that it has comparable performance in such cases.
The second type of functional data considered is on a sparse temporal grid. In this
case we adapt our methodology to consider a longitudinal dataset of MRI images with
missingness at several time points. In this application, the estimated score functions are
modelled with a random slope model which described a subject’s trajectory over time
associated with a PC. The random intercepts and slopes are used to associate with, and
later predict subject outcomes. Given that machine learning has become increasingly
prevalent in image analysis for outcome prediction, we propose a deep neural network for
disease state prediction. A framework for comparison between our proposed functional
principal component (FPC) model and the network is proposed.
In this thesis, we propose a novel model for dimensionality reduction of images over time
alongside an efficient estimation method. This method is investigated via simulation and
is used to analyse two imaging datasets over dense and sparse temporal grids. Data
analysis on fMRI and ADNI revealed that temporal variation plays an important role in
outcome association and prediction. Whilst machine learning methods, especially neural
networks, are frequently used in image analysis, FDA is particularly useful on small
and complex datasets. Our approach provides an efficient and interpretable approach of
modelling high dimensional data which can be used for association or prediction.
