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Topics in descriptive Principal Component Analysis.

Cadima, Jorge Filipe Campinos Landerset (1992) Topics in descriptive Principal Component Analysis. Doctor of Philosophy (PhD) thesis, University of Kent. (doi:10.22024/UniKent/01.02.94253) (KAR id:94253)

Abstract

Principal Component Analysis (PCA) is viewed as a descriptive multivariate method for a set of n observations on p variables.

Geometric considerations in the inner product spaces associated with such nxp data sets play a central role throughout this thesis and provide the motivation for the main results. Among these spaces are spaces of matrices, whose geometry is meaningfully discussed in terms of PCA’s kej' concepts.

It is argued that the conventional interpretation of Principal Components, which is based on the magnitude of each variable's loading for that PC. can be misleading. Alternative approaches based on multiple regression are discussed.

The effects on PCA of linear transformations of the data are discussed in general terms, for non-singular and projective transformations. Specific applications are analyzed and a new solution to the problem of removing isometric size from morphometric data is suggested.

Indicators measuring the degree of similarity between the PCA of a data matrix and the PCA of some transformation of that matrix are provided, for various concepts of ‘similarity’. Methods for joint multiple comparisons of several such transformations are also suggested, discussed and exemplified.

Finally, a truljr scale-invariant alternative to PCA is suggested. At the core of Most Correlated Component Analysis (MCCA) lies a result by Hotelling in his 1933 pioneering paper on PCA. The new method and its performance relative to PCA are discussed in detail. This discussion provides new insights into the information provided by covariance and correlation matrices, as well as a new optimal criterion for PCA.

Item Type: Thesis (Doctor of Philosophy (PhD))
DOI/Identification number: 10.22024/UniKent/01.02.94253
Additional information: This thesis has been digitised by EThOS, the British Library digitisation service, for purposes of preservation and dissemination. It was uploaded to KAR on 25 April 2022 in order to hold its content and record within University of Kent systems. It is available Open Access using a Creative Commons Attribution, Non-commercial, No Derivatives (https://creativecommons.org/licenses/by-nc-nd/4.0/) licence so that the thesis and its author, can benefit from opportunities for increased readership and citation. This was done in line with University of Kent policies (https://www.kent.ac.uk/is/strategy/docs/Kent%20Open%20Access%20policy.pdf). If you feel that your rights are compromised by open access to this thesis, or if you would like more information about its availability, please contact us at ResearchSupport@kent.ac.uk and we will seriously consider your claim under the terms of our Take-Down Policy (https://www.kent.ac.uk/is/regulations/library/kar-take-down-policy.html).
Uncontrolled keywords: Matrix analysis
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
SWORD Depositor: SWORD Copy
Depositing User: SWORD Copy
Date Deposited: 19 May 2023 14:02 UTC
Last Modified: 19 May 2023 14:02 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/94253 (The current URI for this page, for reference purposes)

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