Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds, appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

Monitoring and supporting daily routines in households with dementia is highly important. Here we analyse daily activity in 104 households with dementia across 24 months using passive motion sensors, door sensors and bed mat. We use Markov chain to extract the transitions around the house and we compare these transitions over time. We found that transitions around the house can be used to uniquely match the household as a distinctive fingerprint. We observed that the fingerprint of single-occupancy households is stronger than multiple occupancy and that predicting consecutive periods of time has a higher accuracy due to less change in routine. We focused the latter part of our study on the single occupancy households, specifically looking at the impact of the house floorplan. We observed that despite having similar or identical floorplans, the household fingerprint is still unique and related to the level of activity, and routine.

Functional connections between brain regions are supported by structural connectivity. Both functional and structural connectivity are estimated from in vivo magnetic resonance imaging and offer complementary information on brain organization and function. However, imaging only provides noisy measures, and we lack a good neuroscientific understanding of the links between structure and function. Therefore, inter-subject joint modeling of structural and functional connectivity, the key to multimodal biomarkers, is an open challenge. We present a probabilistic framework to learn across subjects a mapping from structural to functional brain connectivity. Expanding on our previous work [1], our approach is based on a predictive framework with multiple sparse linear regression. We rely on the randomized LASSO to identify relevant anatomo-functional links with some confidence interval. In addition, we describe resting-state functional magnetic resonance imaging in the setting of Gaussian graphical models, on the one hand imposing conditional independences from structural connectivity and on the other hand parameterizing the problem in terms of multivariate autoregressive models. We introduce an intrinsic measure of prediction error for functional connectivity that is independent of the parameterization chosen and provides the means for robust model selection. We demonstrate our methodology with regions within the default mode and the salience network as well as, atlas-based cortical parcellation.