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WIREs Data Mining Knowl Discov
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Tensor decompositions and data fusion in epileptic electroencephalography and functional magnetic resonance imaging data

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Electroencephalography (EEG) and functional magnetic resonance imaging (fMRI) record a mixture of ongoing neural processes, physiological and nonphysiological noise. The pattern of interest, such as epileptic activity, is often hidden within this noisy mixture. Therefore, blind source separation (BSS) techniques, which can retrieve the activity pattern of each underlying source, are very useful. Tensor decomposition techniques are very well suited to solve the BSS problem, as they provide a unique solution under mild constraints. Uniqueness is crucial for an unambiguous interpretation of the components, matching them to true neural processes and characterizing them using the component signatures. Moreover, tensors provide a natural representation of the inherently multidimensional EEG and fMRI, and preserve the structural information defined by the interdependencies among the various modes such as channels, time, patients, etc. Despite the well‐developed theoretical framework, tensor‐based analysis of real, large‐scale clinical datasets is still scarce. Indeed, the application of tensor methods is not straightforward. Finding an appropriate tensor representation, suitable tensor model, and interpretation are application dependent choices, which require expertise both in neuroscience and in multilinear algebra. The aim of this paper is to provide a general guideline for these choices and illustrate them through successful applications in epilepsy. WIREs Data Mining Knowl Discov 2017, 7:e1197. doi: 10.1002/widm.1197

Illustration of the most important tensor decompositions: (a) CPD, (b) BTD‐(Lr ,Lr ,1), (c) Tucker, (d) CMTF. CPD, canonical polyadic decomposition; BTD, block term decomposition; CMTF, coupled matrix‐tensor factorization.
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(L,L,1)‐Block term decomposition of the 10‐second long EEG segment visualized in Figure . Low‐rank components can model the temporal evolution of the seizure. In the examples shown, two components are extracted, a rank‐1 component and a low‐rank component with L = 2. (a) The frequency and the temporal modes are chosen to be low rank to explore whether the seizure changes in its spectral content. (b) The spatial and temporal modes are chosen to be low rank to explore whether the seizure spreads through the brain. EEG, electroencephalography.
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CPD model selection for the first 2‐second long segment of EEG data visualized in Figure , tensorized using wavelet transformation. The core consistency (top) and explained variance (bottom) are shown for five randomly initialized CPD models with different ranks ranging from 1 to 20. CPD, canonical polyadic decomposition; EEG, electroencephalography.
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The appropriate tensor model depends on the chosen tensor representation, as the exact same source pattern can have different rank using different representations. For example, a sinusoidal source pattern (left panel) is approximately rank‐1 using wavelet transformation (top middle), while its corresponding Hankel matrix (bottom middle) is rank‐2, as shown by the singular value spectrum of the matrices (right top and bottom, respectively).
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Tensor‐based EEG and fMRI data analysis include the following steps: data tensorization, tensor model selection and computation, and interpretation. (a) A 10‐second long EEG segment with clear oscillatory ictal pattern. (b) Third‐order EEG tensor with dimensions channels × time × frequency, obtained by wavelet transform. (c) Schematic of a CPD model with R = 2. (d) Signatures of the components extracted from the first 2 second of the ictal pattern. The channel signatures are conveniently visualized as topographical images interpolated over a two dimensional (2D) image of the head from a top viewpoint in radiological convention, i.e., the left of the patient is in the right of the image. Red, green, and blue colors indicate positive, zero, and negative values, respectively. EEG, electroencephalography; fMRI, functional magnetic resonance imaging; CPD, canonical polyadic decomposition.
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