Supervised and unsupervised classification algorithms separate a data set into classes or clusters respectively by making use of prior knowledge or some measure of similarity. Vectors in the same class represent objects that have some characteristic in common. In this thesis the application of classification algorithms to inverse problem data is investigated. Frequently, the ill-posedness of the inverse problem Ax = y is due to the compactness of the linear operator A, which maps the state space X onto the measurement space Y. Theory from the field of inverse problems allows the reconstruction of state space data from measurement data via regularization techniques. Often, in applications, discrimination algorithms are applied to measurement data without taking care of the ill-posed nature inherent in the data. The comparison of classification in measurement space with classification of reconstructed data in the state space is the primary focus of this thesis. Two classes are linearly separable if a hyperplane exists that separates the classes. In this thesis, two classes are referred to as being stably separable if there exist two separating hyperplanes with a positive distance between them. For linear classifications it is shown that the instability of the inverse problem is inherited by the classification problem when it is applied to the data in the measurement space. For elliptic classes the stability of the separation is preserved, providing the range of the adjoint of the compact linear operator A is dense in the state space. The illposedness of the classification problem is reflected by a corresponding ill-posedness of standard classification methods, which - as the inverse problem - need regularization. We investigate Fisher's Linear Discriminant, Principal Component Analysis and the K-Means Algorithm. We apply the theory to magnetic tomography for fuel cells, demonstrating the validity of the results in a practically relevant application.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:610939 |
| Date | January 2013 |
| Creators | Lowery, Natalie L. H. |
| Publisher | University of Reading |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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