Abstract
One of the earlier works on eigen-based techniques for the hyper-complex domain of quaternions was on “quaternion principal component analysis of colour images”. The results of this work are still instructive in many aspects. First, it showed how naturally the quaternion domain accounts for the coupling between the dimensions of red, blue and green of an image, hence its suitability for multichannel processing. Second, it was clear that there was a lack of eigen-based techniques for such a domain, which explains the non-trivial gap in the literature. Third, the lack of such eigen-based quaternion tools meant that the scope and the applications of quaternion signal processing were quite limited, especially in the field of biomedicine. And fourth, quaternion principal component analysis made use of complex matrix algebra, which reminds us that the complex domain lays the building blocks of the quaternion domain, and therefore any research endeavour in quaternion signal processing should start with the complex domain. As such, the first contribution of this thesis lies in the proposition of complex singular spectrum analysis. That research provided a deep understanding and an appreciation of the intricacies of the complex domain and its impact on the quaternion domain. As the complex domain offers one degree of freedom over the real domain, the statistics of a complex variable x has to be augmented with its complex conjugate x*, which led to the term augmented statistics. This recent advancement in complex statistics was exploited in the proposed complex singular spectrum analysis. The same statistical notion was used in proposing novel quaternion eigen-based techniques such as the quaternion singular spectrum analysis, the quaternion uncorrelating transform, and the quaternion common spatial patterns. The latter two methods highlighted an important gap in the literature – there were no algebraic methods that solved the simultaneous diagonalisation of quaternion matrices. To address this issue, this thesis also presents new fundamental results on quaternion matrix factorisations and explores the depth of quaternion algebra. To demonstrate the efficacy of these methods, real-world problems mainly in biomedical engineering were considered. First, the proposed complex singular spectrum analysis successfully addressed an examination of schizophrenic data through the estimation of the event-related potential of P300. Second, the automated detection of the different stages of sleep was made possible using the proposed quaternion singular spectrum analysis. Third, the proposed quaternion common spatial patterns facilitated the discrimination of Parkinsonian patients from healthy subjects. To illustrate the breadth of the proposed eigen-based techniques, other areas of applications were also presented, such as in wind and financial forecasting, and Alamouti-based communication problems. Finally, a preliminary work is made available to suggest that the next step from this thesis is to move from static models (eigen-based models) to dynamic models (such as tracking models).