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Title: Gaussian process arc lengths, functional regression and applications
Author: Bewsher, Justin Dragon
ISNI:       0000 0004 7966 0465
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2017
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This thesis presents novel functional regression methods for non-linear data and develops core Gaussian Process theory for arc lengths. The need for richer functional regression methods and theory around Gaussian Process arc lengths is established in the introduction. Chapter 1 commences with the requisite background material and Chapter 2 works through the fundamentals of probability, highlighting key messages with a worked example. Gaussian Processes are introduced in Chapter 3, developing the core theory needed to extend the functional methods outlined in Chapter 4 and to prepare for considering the arc length dis- tribution of Gaussian Processes. Three novel Gaussian Process functional regression models are introduced in Chapter 5, building upon existing models and furthering the probabilistic approach to functional regression, namely the Gaussian Process Functional Index Model, the Gaussian Process Functional Additive Model and the Gaussian Process Functional Generalised Additive Model. In each model a kernel function is derived whilst outlining inference and prediction. It is clearly demonstrated that the functional regression models out perform their competitors on synthetic examples. In Chapter 6, the arc length distribution of a Gaussian Process is de- rived. By tackling the arc length integrand, a novel approach to the one dimensional arc length mean is presented. This equips us with the tools to approximate the arc length of a vector Gaussian Process. We derive the arc length distribution for a prior and posterior Gaussian Process in terms of its kernel functions and hyperparamaters. Chapter 7 signals a return to functional regression where two challenging real world functional data sets are tackled. The novel Gaussian Process methods provide competitive results in a number of cases, supporting the potential for the new methods. The thesis concludes with a vision for future work.
Supervisor: Osborne, Michael ; Roberts, Stephen Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available