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In this work, the authors study the classification of electroencephalogram (EEG) signals for the determination of the state of sleep of a patient. They employ the power spectral density (PSD) matrices as the feature for the distinction between different classes of EEG signals. This not only allows us to examine the power spectrum contents of each signal as well as the correlation between the multi-channel signals, but also complies with what clinical experts use in their visual judgement of EEG signals. To establish a metric facilitating the classification, the authors exploit the specific geometric properties, and develop, with the aid of fibre bundle theory, an appropriate metric in the Riemannian manifold described by the PSD matrices. To use this new metric effectively for the EEG signal classification, the authors further need to find a weighting for the PSD matrices so that the distances of similar features are minimised whereas those for dissimilar features are maximised. A closed form of this weighting matrix is obtained by solving an equivalent convex optimisation problem. The effectiveness of using these new metrics is examined by applying them to a collection of recorded EEG signals for sleep pattern classification based on the k-nearest neighbour decision algorithm with excellent outcome.