Statistical Analysis of Functions on Surfaces, With an Application to Medical Imaging

In functional data analysis, data are commonly assumed to be smooth functions on a fixed interval of the real line. In this work, we introduce a comprehensive framework for the analysis of functional data, whose domain is a two-dimensional manifold and the domain itself is subject to variability from sample to sample. We formulate a statistical model for such data, here called functions on surfaces, which enables a joint representation of the geometric and functional aspects, and propose an associated estimation framework. We assess the validity of the framework by performing a simulation study and we finally apply it to the analysis of neuroimaging data of cortical thickness, acquired from the brains of different subjects, and thus lying on domains with different geometries. Supplementary materials for this article are available online

Geometric Functional Data Analysis

In this thesis, we introduce a comprehensive framework for the analysis of statistical samples that are functional data with non-trivial geometry. Geometry can interplay with functional data in different forms. The most general setting considered here is that of functional data supported on random non-linear smooth manifolds. This is a situation often encountered in neuroimaging, where modern imaging modalities are now able to produce structural brain representations coupled with functional information. Practitioners have commonly approached the analysis of such data with a two step approach. In the first step the manifolds are registered to a template and in the second step the functional information is analyzed on the template ignoring the registration step. The separation of the two steps precludes studies aimed at understanding how geometric variations relate to functional variations. On the other hand, functional data analysis has mostly developed tools for simplified settings, such as one-dimensional functional samples, limiting their applicability to real data. We formulate a model which is able to jointly represent geometric and functional variations. In this setting, modeling functional information requires the formulation of models able to incorporate structural information on the geometry of the underlying domains, with the aim of mitigating the curse of dimensionality. This is achieved by adopting regularized models involving differential operator penalties. Modeling random smooth manifolds requires the formulation of models constrained to produce `sensible' shapes, e.g. not self-intersecting. This is achieved by means of diffeomorphic flows. The proposed models have been applied to real data to perform studies able to relate structural changes to functional changes, and specifically, to study associations between brain shape and cerebral cortex thickness. We can also deal with more complex functional samples, themselves constrained to lie in a non-linear subspace. This is for instance the case of covariance operators, describing brain connectivity, which are symmetric and positive semi-definite operators. Thanks to the proposed models, we are able to model connectivity as an `object' and study its variations in time or across individuals. We also consider further extensions of this framework to the inverse problems setting, which is the setting where each sample is a latent object, and only indirect measurements are available.EPSRC Centre for Doctoral Training in Analysis (Cambridge Centre for Analysis) EP/L016516/

Representation and reconstruction of covariance operators in linear inverse problems

Abstract We introduce a framework for the reconstruction and representation of functions in a setting where these objects cannot be directly observed, but only indirect and noisy measurements are available, namely an inverse problem setting. The proposed methodology can be applied either to the analysis of indirectly observed functional images or to the associated covariance operators, representing second-order information, and thus lying on a non-Euclidean space. To deal with the ill-posedness of the inverse problem, we exploit the spatial structure of the sample data by introducing a flexible regularizing term embedded in the model. Thanks to its efficiency, the proposed model is applied to MEG data, leading to a novel approach to the investigation of functional connectivity.</jats:p

Smooth Principal Component Analysis over two-dimensional manifolds with an application to neuroimaging

Motivated by the analysis of high-dimensional neuroimaging signals located over the cortical surface, we introduce a novel Principal Component Analysis technique that can handle functional data located over a two-dimensional manifold. For this purpose a regularization approach is adopted, introducing a smoothing penalty coherent with the geodesic distance over the manifold. The model introduced can be applied to any manifold topology, can naturally handle missing data and functional samples evaluated in different grids of points. We approach the discretization task by means of finite element analysis and propose an efficient iterative algorithm for its resolution. We compare the performances of the proposed algorithm with other approaches classically adopted in literature. We finally apply the proposed method to resting state functional magnetic resonance imaging data from the Human Connectome Project, where the method shows substantial differential variations between brain regions that were not apparent with other approaches.Engineering and Physical Sciences Research Council (Grants EP/K021672/2, EP/N014588/1)This is the author accepted manuscript. The final version is available from the Institute of Mathematical Statistics via https://doi.org/10.1214/16-AOAS97

Statistical Analysis of Functions on Surfaces, With an Application to Medical Imaging

In functional data analysis, data are commonly assumed to be smooth functions on a fixed interval of the real line. In this work, we introduce a comprehensive framework for the analysis of functional data, whose domain is a two-dimensional manifold and the domain itself is subject to variability from sample to sample. We formulate a statistical model for such data, here called functions on surfaces, which enables a joint representation of the geometric and functional aspects, and propose an associated estimation framework. We assess the validity of the framework by performing a simulation study and we finally apply it to the analysis of neuroimaging data of cortical thickness, acquired from the brains of different subjects, and thus lying on domains with different geometries. Supplementary materials for this article are available online

Functional random effects modeling of brain shape and connectivity

We present a statistical framework that jointly models brain shape and functional connectivity, which are two complex aspects of the brain that have been classically studied independently. We adopt a Riemannian modeling approach to account for the non-Euclidean geometry of the space of shapes and the space of connectivity that constrains trajectories of co-variation to be valid statistical estimates. In order to disentangle genetic sources of variability from those driven by unique environmental factors, we embed a functional random effects model in the Riemannian framework. We apply the proposed model to the Human Connectome Project dataset to explore spontaneous co-variation between brain shape and connectivity in young healthy individuals

Eigen-Adjusted Functional Principal Component Analysis

Functional Principal Component Analysis (FPCA) has become a widely-used dimension reduction tool for functional data analysis. When additional covariates are available, existing FPCA models integrate them either in the mean function or in both the mean function and the covariance function. However, methods of the first kind are not suitable for data that display second-order variation, while those of the second kind are time-consuming and make it difficult to perform subsequent statistical analyses on the dimension-reduced representations. To tackle these issues, we introduce an eigen-adjusted FPCA model that integrates covariates in the covariance function only through its eigenvalues. In particular, different structures on the covariate-specific eigenvalues -- corresponding to different practical problems -- are discussed to illustrate the model's flexibility as well as utility. To handle functional observations under different sampling schemes, we employ local linear smoothers to estimate the mean function and the pooled covariance function, and a weighted least square approach to estimate the covariate-specific eigenvalues. The convergence rates of the proposed estimators are further investigated under the different sampling schemes. In addition to simulation studies, the proposed model is applied to functional Magnetic Resonance Imaging scans, collected within the Human Connectome Project, for functional connectivity investigation