Quaternion Singular Spectrum Analysis of Electroencephalogram With Application in Sleep Analysis

Abstract

Abstract-A novel quaternion-valued singular spectrum analysis (SSA) is introduced for multichannel analysis of electroencephalogram (EEG). The analysis of EEG typically requires the decomposition of data channels into meaningful components despite the notoriously noisy nature of EEG -which is the aim of SSA. However, the singular value decomposition involved in SSA implies the strict orthogonality of the decomposed components, which may not reflect accurately the sources which exhibit similar neural activities. To allow for the modelling of such co-channel coupling, the quaternion domain is considered for the first time to formulate the SSA using the augmented statistics. As an application, we demonstrate how the augmented quaternionvalued SSA (AQSSA) can be used to extract the sources, even at a signal-to-noise ratio as low as -10 dB. To illustrate the usefulness of our quaternion-valued SSA in a rehabilitation setting, we employ the proposed SSA for sleep analysis to extract statistical descriptors for five-stage classification (Awake, N1, N2, N3 and REM). The level of agreement using these descriptors was 74% as quantified by the Cohen's kappa.

Ensemble SVM Method for Automatic Sleep Stage Classification

“…They range from simple linear discriminant analysis (LDA) to non-linear and highly complex Gaussian mixture model-based classifiers [20,21]. For classification of different sleep states, several traditional methods, namely LDA [18,22], neural networks [23,24], support vector machines (SVM) [25,9,26,27,6], knearest neighbor (k-NN) [16], hidden Markov model [28], fuzzy systems [29,30], etc., are proposed for distinguishing between different sleep stages. The common to all traditional classifiers is that they have only one classifier.…”

Quaternion Common Spatial Patterns

“…Remark 11: Although CSP, SUTCCSP, and A-QCSP methods provided substantial agreements (τ > 0.61), our proposed algorithm G-QCSP outperformed the others by offering almost perfect classification (τ = 0.82) and the highest sensitivity of 9 To confirm the robustness of the proposed method, further experiments were performed to consider negative values in the mixing matrices, see Appendix B. 10 Note that Equation (7) was proposed for quaternion-valued CSP algorithms.…”

“…Furthermore, to cater for quaternion-valued signals with non-circular probability distribution, we exploit the latest advances in quaternion statistics, so-called 'augmented statistics' [7]. Modelling based on augmented quaternion has been previously deployed in biomedical applications such as ocular artefacts [8], sleep EEG analysis [9], and other applications [10]. In the same spirit, we introduce an augmented quaternion CSP (A-QCSP) and a generalised quaternion CSP (G-QCSP) to exploit the inherent coupling between four channels and cater for non-circular datasets.…”

Quaternion adaptive line enhancer

“…Although in some cases such a 3-D shift, ∆ q , for at least one signal cycle is practically easy, such as those for a prescribed hand movement trajectory in an action research arm test (ARAT) [20], in theory, it may pose a challenging problem. An effective option, used in this study, is to apply quaternion singular spectrum analysis (QSSA) introduced in [18] and use the signal obtained from the first eigentriple (i.e. the eigenvalue and eigenvectors of the corresponding augmented quaternion singular value decomposition within the QSSA operation) as the 3-D baseline.…”

“…Quaternions have found applications in computer graphics, for the modelling of three-dimensional (3-D) rotations [6], in robotics [7], molecular modelling [8], processing colour images [9], hyper-complex digital filters [10], texture segmentation [11], source separation [12], watermarking [13], spectrum estimation [14] quaternion singular value decomposition and in the MUSIC algorithm to process polarized waves [15], [16], quaternion least squares [8], [17], and quaternion singular spectrum analysis [18]. In [4] the formulation for a quaternion LMS adaptive filtering has also been provided and used for the processing of quaternion valued signals.…”