In this work, I will describe a new statistical tool: the canonical bicoherence, which is a combination of the canonical coherence and the bicoherence. I will provide its definitions, properties, estimation by multitaper methods and statistics, and estimate the variance of the estimates by the weighted jackknife method. I will discuss its applicability and usefulness in nonlinear quadratic phase coupling detection and analysis for multivariate random processes. Furthermore, I will develop the time-varying canonical bicoherence for the nonlinear analysis of non-stationary random processes. In this thesis, the canonical bicoherence is mainly applied in two types of data: a) three-component geomagnetic field data, and b) high-dimensional brain electroencephalogram data. Both results obtained will be linked with physical or physiological interpretations. In particular, this thesis is the first work where the novel method of ``canonical bicoherence'' is introduced and applied to the nonlinear quadratic phase coupling detection and analysis for multivariate random processes. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-10-31 15:03:57.596

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