TEMPORALLY RESOLVED MULTI-WAY COMPONENT ANALYSIS OF DYNAMIC SOURCES IN EVENT-RELATED EEG DATA USING PARAFAC2 Martin Weis, Dunja Jannek, Thomas Guenther, Peter Husar Biosignal Processing Group Ilmenau University of Technology Gustav-Kirchhoff Str. 2, D-98684 Ilmenau, Germany {martin.weis, peter.husar}@tu-ilmenau.de {dunja.jannek, thomas.guenther}@tu-ilmenau.de www.tu-ilmenau.de/bsv Florian Roemer, Martin Haardt Communications Research Laboratory Ilmenau University of Technology Helmholzplatz 2, D-98684 Ilmenau, Germany {florian.roemer, martin.haardt}@tu-ilmenau.de www.tu-ilmenau.de/crl ABSTRACT The identification of signal components in electroencephalographic (EEG) data is a major task in neuroscience. The interest to this area has regained new interest due to the possibilities of multi- dimensional signal processing. In this contribution we analyze event-related multi-channel EEG recordings on the basis of the time-varying spectrum for each channel. To identify the signal com- ponents it is a common approach to use parallel factor (PARAFAC) analysis. However, the PARAFAC model cannot cope with compo- nents appearing time-shifted over the different channels. Further- more, it is not possible to track PARAFAC components over time. We show how to overcome these problems by using the PARAFAC2 decomposition, which renders it an attractive approach for process- ing EEG data with highly dynamic (moving) sources. Additionally, we introduce the concept of PARAFAC2 component amplitudes, which resolve the scaling ambiguity in the PARAFAC2 model and can be used to judge the relevance of the components. 1. INTRODUCTION In this contribution we focus on analyzing measured electroen- cephalographic (EEG) data to identify the components of activity. This analysis can also be used to detect and localize epileptic seizure onset zones on the scalp as well as projections of cognitive pro- cessing like speech or auditory handling. Unfortunately, different sources in the brain can produce the same EEG pattern, which ren- ders them in general non-separable. Source localization algorithms, such as LORETA [15] or dipole fitting methods can resolve this am- biguity by imposing additional assumptions. For further improve- ments of these methods, preprocessing in form of subspace decom- positions, e.g., principle component analysis (PCA), independent component analysis (ICA), singular value decomposition (SVD), or beamforming algorithms [10] have been applied. However, these methods cannot exploit the multi-dimensional nature of the EEG data. Moreover, to obtain matrix decompositions like PCA or ICA, physically unsatisfiable assumptions like orthogonality or indepen- dence have to be imposed. Therefore, tensor decompositions are a more promising approach to handle EEG signals. Especially the well known parallel factor (PARAFAC) analysis is widely used in recent literature, because it is essentially unique under mild con- ditions [2] without any artificial constraints, such as orthogonality. In the last years PARAFAC was applied to EEG signals, e.g., for estimating sources of cognitive processing [13], for the analysis of event-related potentials (ERP) [14], and for epileptic seizure local- ization [18]. In order to resolve the temporal evolution as well as the fre- quency content of the EEG recordings, a time-frequency analysis (TFA) is applied for each channel. Therefore, the data is analyzed over three dimensions, i.e., time, frequency, and space (channels). Different TFA algorithms have been studied for the analysis of EEG signals [7]. The most common method is the continuous wavelet transformation (CWT). However, wavelet analysis may not provide adequate time and frequency resolution for EEG data. In [19] it was shown that the reduced interference distribution (RID) [5] is partic- ularly useful for the TFA of EEG data and its subsequent multi-way component analysis, since it provides an improved time-frequency resolution. The common approaches for the three-way component analy- sis of EEGdata to date are based on the PARAFAC model. How- ever, this model is not able to resolve moving EEG components which appear time-shifted over the different channels. Therefore, the PARAFAC component analysis is only useful in case of static sources. In this contribution we introduce the PARAFAC2 decom- position [9] for the space-time-frequency analysis of EEG data. The PARAFAC2 model supports time-shifted component signals. Fur- thermore, we show how the PARAFAC2 model can be adopted in order to track the different EEG components over time. The PARAFAC2 model is rarely used up to now, i.e., in [3] it is ap- plied to chemometric data including retention time-shifts, and [1] uses it for the time-space-window analysis of EEG and electrocar- diographic (ECG) recordings. The PARAFAC2 model is essentially unique up to scaling and permutation. In order to resolve the scaling ambiguity, we introduce the least squares PARAFAC2 component amplitudes which can be used to judge the influence of the individ- ual components. This paper is organized as follows: In Section 2 we clarify the notation and define the operators and symbols that are used. In Sec- tion 3 we discuss the signal processing steps to analyze EEG sig- nals. Thereby, the Sections 3.1 and 3.2 present the methods for the measurement preprocessing and the time-frequency analysis. Sub- sequently, Section 3.3 describes the three-way component analysis of the different time-frequency distributions using PARAFAC2. In Section 4 we present the results of the event-related EEG analy- sis based on measurements, before drawing the conclusions in Sec- tion 5. 2. NOTATION To facilitate the distinction between scalars, vectors, matrices, and higher-order tensors, we use the following notation: scalars are denoted by lower-case italic letters (a, b, ...), vectors by boldface lower-case italic letters (a, b, ...), matrices by boldface upper-case letters (A, B, ...), and tensors are denoted as upper-case, boldface, calligraphic letters (A, B, ...). This notation is consistently used for lower-order parts of a given structure, unless stated otherwise. For example A ∈ R I 1 ×I 2 ×···×I N represents an N -dimensional tensor of size In along mode n. Its elements are referenced by ai 1 ,i 2 ,...,i N for in =1, 2,...In and n =1, 2,...,N . For matrices we use the superscripts T , H , −1 , + for transposition, Hermitian transposition, matrix inverse, and Moore-Penrose pseudo-inverse, respectively. The tensor operations we use are consistent with [12]. The higher-order norm of a tensor A, symbolized by ‖A‖H, is defined as the square root of the sum of the squared magnitude of all el- ements in A. The n-mode vectors of a tensor A are obtained 18th European Signal Processing Conference (EUSIPCO-2010) Aalborg, Denmark, August 23-27, 2010 © EURASIP, 2010 ISSN 2076-1465 696