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Improving modern dimension reduction methods through transformations

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posted on 2023-01-18, 15:40 authored by Alexandra Louise Garnham
A prominent difficulty facing researchers is the visualization of high dimensional data. Several dimension reduction techniques have been developed to facilitate this process; however they have limitations. The purpose of this thesis is to demonstrate how estimates produced from dimension reduction methodologies can be significantly improved using simple yet effective response transformations. In order to compare and assess the effectiveness of the applied response transformations, influence functions are utilized to evaluate the success of a method. Within this thesis, primary focus is on the dimension reduction methods Principal Hessian Directions (pHd) and Ordinary Least Squares (OLS). Furthermore, consideration is also given to Sliced Inverse Regression (SIR) and Sliced Average Variance Estimation (SAVE). When designing response transformations for pHd and OLS, the limitations of these methods were studied with a view toward creating transformations that overcame these issues. The selected transformations not only remove elements detrimental to estimation, they can also introduce beneficial components. To identify the most appropriate transformation, a minimal influence approach is implemented. With respect to OLS, an influence measure is also given which quantifies its effectiveness as a dimension reduction method. Additionally, several weighted estimators are provided that combine results following various transformations. These weighted techniques have allowed the creation of a method which enables OLS, traditionally a single-index model method, to be applied in the multi-index model setting. Several dimension reduction methodologies, including OLS and SIR, are severely affected by an impediment known as symmetric dependency. To overcome this problem, response and predictor transformations are provided which utilize estimates from methods unaffected by symmetric dependency to significantly improve results of those techniques that are. It is established that OLS and SIR can be improved greatly with this approach when faced with symmetric dependency.

Submission note: A thesis submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Engineering and Mathematical Sciences, Faculty of Science, Technology and Engineering, La Trobe University, Bundoora.

History

Center or Department

Faculty of Science, Technology and Engineering. School of Engineering and Mathematical Sciences.

Thesis type

  • Ph. D.

Awarding institution

La Trobe University

Year Awarded

2014

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The thesis author retains all proprietary rights (such as copyright and patent rights) over the content of this thesis, and has granted La Trobe University permission to reproduce and communicate this version of the thesis. The author has declared that any third party copyright material contained within the thesis made available here is reproduced and communicated with permission. If you believe that any material has been made available without permission of the copyright owner please contact us with the details.

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