Complex independent component analysis of frequency-domain electroencephalographic data

Abstract

Independent component analysis (ICA) has proven useful for modeling brain and electroencephalographic (EEG) data. Here, we present a new, generalized method to better capture the dynamics of brain signals than previous ICA algorithms. We regard EEG sources as eliciting spatio-temporal activity patterns, corresponding to, e.g. trajectories of activation propagating across cortex. This leads to a model of convolutive signal superposition, in contrast with the commonly used instantaneous mixing model. In the frequency-domain, convolutive mixing is equivalent to multiplicative mixing of complex signal sources within distinct spectral bands. We decompose the recorded spectraldomain signals into independent components by a complex infomax ICA algorithm. First results from a visual attention EEG experiment exhibit: (1) sources of spatio-temporal dynamics in the data, (2) links to subject behavior, (3) sources with a limited spectral extent, and (4) a higher degree of independence compared to sources derived by standard ICA. q

Complex‐Valued Adaptive Signal Processing

“…Since boundedness is deemed as important for the performancein particular stability-of nonlinear signal processing algorithms, a common practice has been to define functions that do not satisfy the analyticity requirement but are bounded (see e.g., [9,36,45,67,103]). This has been the main motivation in the definition of split-and fully-complex functions given in Definition 1.…”

“…We can use the first-order Taylor series expansion to derive the relative gradient update rule [21] for complex matrix variables, which is usually directly extended to the complex case without a derivation [9,18,34]. To write the relative gradient rule, we consider an update of the parameter matrix W in the invariant form G(W)W [21].…”

“…A number of algorithms have used complex split nonlinear functions such that the real and imaginary parts (or the magnitude and phase) of the argument are processed separately through real-valued nonlinear functions [96,103]. Another approach processes the magnitude of the argument by a real-valued function [9,13], thus limiting the algorithm to circular sources. These approaches, while yielding satisfactory performance for a class of problems, are not effective in generating the higher order statistics required to establish independence for all possible source distributions.…”

“…This is the form of the score function proposed in [9] where all sources are assumed to be circular. If the real and imaginary parts of a given source are mutually independent, the score function takes the form…”

“…We include the performances of complex nonlinear decorrelations (C-ND) [3] using the 2asinh(u) þ u nonlinearity for the sub-Gaussian sources, and the performances of complex FastICA [13] using the log nonlinearity, the kurtosis maximization (KM) algorithm [69], JADE with the version that uses simultaneous diagonalization of N cumulant matrices [22], and for the circular generalized Gaussian sources, complex Infomax using the nonlinear function that assumes circularity given in [9]. Since we have not considered density adaptation, all sources in a given run are generated from the same distribution, and as a result comparisons with SUT are not included since for SUT, all sources have to have distinct spectral coefficients.…”