Kriging prediction for manifold-valued random fields

The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly important in many applications, such as shape analysis, diffusion tensor imaging and the analysis of covariance matrices. In many cases, data are spatially distributed but it is not trivial to take into account spatial dependence in the analysis because of the non linear geometry of the manifold. This work proposes a solution to the problem of spatial prediction for manifold valued data, with a particular focus on the case of positive definite symmetric matrices. Under the hypothesis that the dispersion of the observations on the manifold is not too large, data can be projected on a suitably chosen tangent space, where an additive model can be used to describe the relationship between response variable and covariates. Thus, we generalize classical kriging prediction, dealing with the spatial dependence in this tangent space, where well established Euclidean methods can be used. The proposed kriging prediction is applied to the matrix field of covariances between temperature and precipitation in Quebec, Canada.This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jmva.2015.12.00

Non‐parametric regression for networks

Network data are becoming increasingly available, and so there is a need to develop suitable methodology for statistical analysis. Networks can be represented as graph Laplacian matrices, which are a type of manifold-valued data. Our main objective is to estimate a regression curve from a sample of graph Laplacian matrices conditional on a set of Euclidean covariates, for example in dynamic networks where the covariate is time. We develop an adapted Nadaraya-Watson estimator which has uniform weak consistency for estimation using Euclidean and power Euclidean metrics. We apply the methodology to the Enron email corpus to model smooth trends in monthly networks and highlight anomalous networks. Another motivating application is given in corpus linguistics, which explores trends in an author's writing style over time based on word co-occurrence networks

Manifold-valued data analysis of networks and shapes

This thesis is concerned with the study of manifold-valued data analysis. Manifold-valued data is a type of multivariate data that lies on a manifold as opposed to a Euclidean space. We seek to develop analogue classical multivariate analysis methods, which are appropriate for Euclidean data, for data that lie on particular manifolds. A manifold we particularly focus on is the manifold of graph Laplacians. Graph Laplacians can represent networks and for the majority of this thesis we focus on the statistical analysis of samples of networks by identifying networks with their graph Laplacian matrices. We develop a general framework for extrinsic statistical analysis of samples of networks by this representation. For the graph Laplacians we define metrics, embeddings, tangent spaces, and a projection from Euclidean space to the space of graph Laplacians. This framework provides a way of computing means, performing principal component analysis and regression, carrying out hypothesis tests, such as for testing for equality of means between two samples of networks, and classifying networks. We will demonstrate these methods on many different network datasets, including networks derived from text and neuroimaging data. We also briefly consider another well studied type of manifold-valued data, namely shape data, comparing three commonly used tangent coordinates used in shape analysis and explaining the difference between them and why they may not all be suitable to always use

Manifold-valued data analysis of networks and shapes

This thesis is concerned with the study of manifold-valued data analysis. Manifold-valued data is a type of multivariate data that lies on a manifold as opposed to a Euclidean space. We seek to develop analogue classical multivariate analysis methods, which are appropriate for Euclidean data, for data that lie on particular manifolds. A manifold we particularly focus on is the manifold of graph Laplacians. Graph Laplacians can represent networks and for the majority of this thesis we focus on the statistical analysis of samples of networks by identifying networks with their graph Laplacian matrices. We develop a general framework for extrinsic statistical analysis of samples of networks by this representation. For the graph Laplacians we define metrics, embeddings, tangent spaces, and a projection from Euclidean space to the space of graph Laplacians. This framework provides a way of computing means, performing principal component analysis and regression, carrying out hypothesis tests, such as for testing for equality of means between two samples of networks, and classifying networks. We will demonstrate these methods on many different network datasets, including networks derived from text and neuroimaging data. We also briefly consider another well studied type of manifold-valued data, namely shape data, comparing three commonly used tangent coordinates used in shape analysis and explaining the difference between them and why they may not all be suitable to always use