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 da...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
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2020
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| Online Access: | https://eprints.nottingham.ac.uk/65862/ |
| _version_ | 1848800276643840000 |
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| author | Severn, Katie |
| author_facet | Severn, Katie |
| author_sort | Severn, Katie |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:48:59Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-65862 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:48:59Z |
| publishDate | 2020 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-658622021-07-19T13:26:23Z https://eprints.nottingham.ac.uk/65862/ Manifold-valued data analysis of networks and shapes Severn, Katie 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. 2020-07-24 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en cc_by https://eprints.nottingham.ac.uk/65862/1/KatieSevernApprovedThesis.pdf Severn, Katie (2020) Manifold-valued data analysis of networks and shapes. PhD thesis, University of Nottingham. manifold-valued data analysis multivariate data |
| spellingShingle | manifold-valued data analysis multivariate data Severn, Katie Manifold-valued data analysis of networks and shapes |
| title | Manifold-valued data analysis of networks and shapes |
| title_full | Manifold-valued data analysis of networks and shapes |
| title_fullStr | Manifold-valued data analysis of networks and shapes |
| title_full_unstemmed | Manifold-valued data analysis of networks and shapes |
| title_short | Manifold-valued data analysis of networks and shapes |
| title_sort | manifold-valued data analysis of networks and shapes |
| topic | manifold-valued data analysis multivariate data |
| url | https://eprints.nottingham.ac.uk/65862/ |