Despite empirical mode decomposition (EMD) becoming a de facto standard for time-frequency analysis of nonlinear and non-stationary signals, its multivariate extensions are only emerging; yet, they are a prerequisite for direct multichannel data analysis. An important step in this direction is the computation of the local mean, as the concept of local extrema is not well defined for multivariate signals. To this end, we propose to use real-valued projections along multiple directions on hyperspheres (n-spheres) in order to calculate the envelopes and the local mean of multivariate signals, leading to multivariate extension of EMD. To generate a suitable set of direction vectors, unit hyperspheres (n-spheres) are sampled based on both uniform angular sampling methods and quasi-Monte Carlo-based low-discrepancy sequences. The potential of the proposed algorithm to find common oscillatory modes within multivariate data is demonstrated by simulations performed on both hexavariate synthetic and real-world human motion signals.

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Abstract-Brain electrical activity recorded via electroencephalogram (EEG) is the most convenient means for brain-computer interface (BCI), and is notoriously noisy. The information of interest is located in well defined frequency bands, and a number of standard frequency estimation algorithms have been used for feature extraction. To deal with data nonstationarity, low signal-to-noise ratio, and closely spaced frequency bands of interest, we investigate the effectiveness of recently introduced multivariate extensions of empirical mode decomposition (MEMD) in motor imagery BCI. We show that direct multichannel processing via MEMD allows for enhanced localization of the frequency information in EEG, and, in particular, its noise-assisted mode of operation (NA-MEMD) provides a highly localized time-frequency representation. Comparative analysis with other state of the art methods on both synthetic benchmark examples and a well established BCI motor imagery dataset support the analysis.Index Terms-Brain-computer interface (BCI), electroencephalogram (EEG), empirical mode decomposition, motor imagery paradigm, noise assisted multivariate extensions of empirical mode decomposition (NA-MEMD).

A noise-assisted approach in conjunction with multivariate empirical mode decomposition (MEMD) algorithm is proposed for the computation of empirical mode decomposition (EMD), in order to produce localized frequency estimates at the accuracy level of instantaneous frequency. Despite many advantages of EMD, such as its data driven nature, a compact decomposition, and its inherent ability to process nonstationary data, it only caters for signals with a sufficient number of local extrema. In addition, EMD is prone to mode-mixing and is designed for univariate data. We show that the noiseassisted MEMD (NA-MEMD) approach, which utilizes the dyadic filter bank property of MEMD, provides a solution to the above problems when used to calculate standard EMD. The method is also shown to alleviate the effects of noise interference in univariate noise-assisted EMD algorithms which directly add noise to the data. The efficacy of * Corresponding author. 1350007-1 N. ur Rehman et al.the proposed method, in terms of improved frequency localization and reduced modemixing, is demonstrated via simulations on electroencephalogram (EEG) data sets, over two paradigms in brain-computer interface (BCI).

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Abstract. Established complexity measures typically operate at a single scale and thus fail to quantify inherent long-range correlations in real-world data, a key feature of complex systems. The recently introduced multiscale entropy (MSE) method has the ability to detect fractal correlations and has been used successfully to assess the complexity of univariate data. However, multivariate observations are common in many real-world scenarios and a simultaneous analysis of their structural complexity is a prerequisite for the understanding of the underlying signal-generating mechanism. For this purpose, based on the notion of multivariate sample entropy, the standard MSE method is extended to the multivariate case, whereby for rigor, the intrinsic multivariate scales of the input data are generated adaptively via the multivariate empirical mode decomposition (MEMD) algorithm. This allows us to gain better understanding of the complexity of the underlying multivariate real-world process, together with more degrees of freedom and physical interpretation in the analysis. Simulations on both synthetic and real-world biological multivariate data sets support the analysis.