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Humphries, Melissa Tuke, Jono Glonek, Gary Ryan, Matthew James School of Mathematical Sciences 2023 https://hdl.handle.net/2440/138532 Over the past 30 years functional magnetic resonance imaging (fMRI) has become a fundamental tool in cognitive neuroimaging studies. In particular, the emergence of restingstate fMRI has gained popularity in determining biomarkers of mental health disorders (Woodward & Cascio, 2015). Resting-state fMRI can be analysed using the functional connectivity matrix, an object that encodes the temporal correlation of blood activity within the brain. Functional connectivity matrices are symmetric positive definite (SPD) matrices, but common analysis methods either reduce the functional connectivity matrices to summary statistics or fail to account for the positive definite criteria. However, through the lens of Riemannian geometry functional connectivity matrices have an intrinsic non-linear shape that respects the positive definite criteria (the affine-invariant geometry (Pennec, Fillard, & Ayache, 2006)). With methods from Riemannian geometric statistics, we can begin to explore the shape of the functional brain to understand this non-linear structure and reduce data-loss in our analyses. This thesis o↵ers two novel methodological developments to the field of Riemannian geometric statistics inspired by methods used in fMRI research. First we propose geometric- MDMR, a generalisation of multivariate distance matrix regression (MDMR) (McArdle & Anderson, 2001) to Riemannian manifolds. Our second development is Riemannian partial least squares (R-PLS), the generalisation of the predictive modelling technique partial least squares (PLS) (H. Wold, 1975) to Riemannian manifolds. R-PLS extends geodesic regression (Fletcher, 2013) to manifold-valued response and predictor variables, similar to how PLS extends multiple linear regression. We also generalise the NIPALS algorithm to Riemannian manifolds and suggest a tangent space approximation as a proposed method to fit R-PLS. In addition to our methodological developments, this thesis o↵ers three more contributions to the literature. Firstly, we develop a novel simulation procedure to simulate realistic functional connectivity matrices through a combination of bootstrapping and the Wishart distribution. Second, we propose the R2S statistic for measuring subspace similarity using the theory of principal angles between subspaces. Finally, we propose an extension of the VIP statistic from PLS (S. Wold, Johansson, & Cocchi, 1993) to describe the relationship between individual predictors and response variables when predicting a multivariate response with PLS. All methods in this thesis are applied to two fMRI datasets: the COBRE dataset relating to schizophrenia, and the ABIDE dataset relating to Autism Spectrum Disorder (ASD). We show that geometric-MDMR can detect group-based di↵erences between ASD and neurotypical controls (NTC), unlike its Euclidean counterparts. We also demonstrate the efficacy of R-PLS through the detection of functional connections related to schizophrenia and ASD. These results are encouraging for the role of Riemannian geometric statistics in the future of neuroscientific research. Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2023 en Riemannian manifolds, statistics, fMRI, functional connectivity, PLS, MDMR Riemannian statistical techniques with applications in fMRI Thesis This electronic version is made publicly available by the University of Adelaide in accordance with its open access policy for student theses. Copyright in this thesis remains with the author. This thesis may incorporate third party material which has been used by the author pursuant to Fair Dealing exceptions. If you are the owner of any included third party copyright material you wish to be removed from this electronic version, please complete the take down form located at: http://www.adelaide.edu.au/legalsopen.access