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Biplots based on principal surfaces
Principal surfaces are smooth two-dimensional surfaces that pass through the middle of a p-dimensional data set. They minimise the distance from the data points, and provide a nonlinear summary of the data. The surfaces are nonparametric and their shape is suggested by the data. The formation of a s...
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| Format: | Thesis |
| Language: | English |
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Department of Statistical Sciences
2020
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| _version_ | 1867613216815710208 |
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| access_status_str | Open Access |
| author | Ganey, Raeesa |
| author2 | Er, Sebnem |
| author_browse | Er, Sebnem Ganey, Raeesa |
| author_facet | Er, Sebnem Ganey, Raeesa |
| author_sort | Ganey, Raeesa |
| collection | Thesis |
| description | Principal surfaces are smooth two-dimensional surfaces that pass through the middle of a p-dimensional data set. They minimise the distance from the data points, and provide a nonlinear summary of the data. The surfaces are nonparametric and their shape is suggested by the data. The formation of a surface is found using an iterative procedure which starts with a linear summary, typically with a principal component plane. Each successive iteration is a local average of the p-dimensional points, where an average is based on a projection of a point onto the nonlinear surface of the previous iteration. Biplots are considered as extensions of the ordinary scatterplot by providing for more than three variables. When the difference between data points are measured using a Euclidean embeddable dissimilarity function, observations and the associated variables can be displayed on a nonlinear biplot. A nonlinear biplot is predictive if information on variables is added in such a way that it allows the values of the variables to be estimated for points in the biplot. Prediction trajectories, which tend to be nonlinear are created on the biplot to allow information about variables to be estimated. The goal is to extend the idea of nonlinear biplot methodology onto principal surfaces. The ultimate emphasis is on high dimensional data where the nonlinear biplot based on a principal surface allows for visualisation of samples, variable trajectories and predictive sets of contour lines. The proposed biplot provides more accurate predictions, with an additional feature of visualising the extent of nonlinearity that exists in the data. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/31695 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:32:37.404Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2020 |
| publishDateRange | 2020 |
| publishDateSort | 2020 |
| publisher | Department of Statistical Sciences |
| publisherStr | Department of Statistical Sciences |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/31695 Biplots based on principal surfaces Ganey, Raeesa Er, Sebnem Lubbe, Sugnet Biplots Principal surfaces Nonparametric principal components Multidimensional scaling Principal surfaces are smooth two-dimensional surfaces that pass through the middle of a p-dimensional data set. They minimise the distance from the data points, and provide a nonlinear summary of the data. The surfaces are nonparametric and their shape is suggested by the data. The formation of a surface is found using an iterative procedure which starts with a linear summary, typically with a principal component plane. Each successive iteration is a local average of the p-dimensional points, where an average is based on a projection of a point onto the nonlinear surface of the previous iteration. Biplots are considered as extensions of the ordinary scatterplot by providing for more than three variables. When the difference between data points are measured using a Euclidean embeddable dissimilarity function, observations and the associated variables can be displayed on a nonlinear biplot. A nonlinear biplot is predictive if information on variables is added in such a way that it allows the values of the variables to be estimated for points in the biplot. Prediction trajectories, which tend to be nonlinear are created on the biplot to allow information about variables to be estimated. The goal is to extend the idea of nonlinear biplot methodology onto principal surfaces. The ultimate emphasis is on high dimensional data where the nonlinear biplot based on a principal surface allows for visualisation of samples, variable trajectories and predictive sets of contour lines. The proposed biplot provides more accurate predictions, with an additional feature of visualising the extent of nonlinearity that exists in the data. 2020-04-28T11:05:40Z 2020-04-28T11:05:40Z 2019 2020-04-28T10:26:42Z Doctoral Thesis Doctoral PhD https://hdl.handle.net/11427/31695 eng application/pdf Department of Statistical Sciences Faculty of Science |
| spellingShingle | Biplots Principal surfaces Nonparametric principal components Multidimensional scaling Ganey, Raeesa Biplots based on principal surfaces |
| thesis_degree_str | Doctoral |
| title | Biplots based on principal surfaces |
| title_full | Biplots based on principal surfaces |
| title_fullStr | Biplots based on principal surfaces |
| title_full_unstemmed | Biplots based on principal surfaces |
| title_short | Biplots based on principal surfaces |
| title_sort | biplots based on principal surfaces |
| topic | Biplots Principal surfaces Nonparametric principal components Multidimensional scaling |
| url | https://hdl.handle.net/11427/31695 |
| work_keys_str_mv | AT ganeyraeesa biplotsbasedonprincipalsurfaces |