1. Informatics and Mathematical Modelling / Intelligent Signal Processing 1 Morten Mørup Decomposing event related EEG using Parallel Factor Morten Mørup Informatics and Mathematical Modeling Intelligent Signal Processing Technical University of Denmark

2. Informatics and Mathematical Modelling / Intelligent Signal Processing 2 Morten Mørup Outline Non-negativity constrained PARAFAC Application of PARAFAC to the EEG

3. Informatics and Mathematical Modelling / Intelligent Signal Processing 3 Morten Mørup PARAFAC (Harshman & Carrol and Chang 1970)

4. Informatics and Mathematical Modelling / Intelligent Signal Processing 4 Morten Mørup Alternating Least Squares (ALS) ALS corresponds to maximizing the likelihood of a Gaussian Consequently, ALS assumes normal distributed noise.

5. Informatics and Mathematical Modelling / Intelligent Signal Processing 5 Morten Mørup Gradient descent Especially good for cost functions without analytical solution. Let C be the cost function, then update the parameters according to:

6. Informatics and Mathematical Modelling / Intelligent Signal Processing 6 Morten Mørup Why imposing Non-negativity constraints Most PARAFAC algorithms known to have problems of degeneration among the factors Degeneration result of factors counteracting each other. Some solutions: Sparseness/regularization constraints i.e. c 1 ||A|| 2 +c 2 ||B|| 2 +c 3 ||S|| 2 Orthogonality constraints, i.e. A T A=I Non negativity constraint on all modalities (if data is positive and factor components considered purely additive)

7. Informatics and Mathematical Modelling / Intelligent Signal Processing 7 Morten Mørup How to impose non-negativity constraints Active set algorithm (Bro & Jong, 1997) Iteratively optimizes cost function until no variables are negative. Gradient descent with positive updates Update parameters so they remain in the positive domain. Among various other methods

8. Informatics and Mathematical Modelling / Intelligent Signal Processing 8 Morten Mørup Non-negative matrix factorization (NMF) Generalization to PARAFAC (Lee & Seung 2001)

9. Informatics and Mathematical Modelling / Intelligent Signal Processing 9 Morten Mørup Electroencephalography (EEG) EEG measures electrical potential at the scalp arising primarily from synchronous neuronal activity of pyramidal cells in the brain. Event related potential (ERP) is EEG measurements time locked to a stimulus event

10. Informatics and Mathematical Modelling / Intelligent Signal Processing 10 Morten Mørup History of PARAFAC and EEG Harshman (1970) (Suggested its use on EEG) Möcks (1988) (Topographic Component Analysis) ERP of (channel x time x subject) Field and Graupe (1991) ERP of (channel x time x subject) Miwakeichi et al. (2004) EEG of (channel x time x frequency) Mørup et al. (2005) ERP of (channel x time x frequency x subject x condition)

11. Informatics and Mathematical Modelling / Intelligent Signal Processing 11 Morten Mørup time frequency Wavelet transform Complex Morlet wavelet - Real part - Complex part Absolute value of wavelet coefficient Captures frequency changes through time

12. Informatics and Mathematical Modelling / Intelligent Signal Processing 12 Morten Mørup time channel subjects Möcks (1988) Field & Graupe (1991) time frequency channel Miwakeichi (2004) PARAFAC Assumption: Same signal having Various strength in each subject mixed in the channels. PARAFAC Assumption: Same Frequency signature present to various degree in time mixed in the channels.

13. Informatics and Mathematical Modelling / Intelligent Signal Processing 13 Morten Mørup The Vector strength Vectors coherent, i.e. correlated Vectors incoherent, i.e. uncorrelated Vector strength a measure of coherence

14. Informatics and Mathematical Modelling / Intelligent Signal Processing 14 Morten Mørup Visual Paradigm (Herrmann et al. 2004) Expected result: Coherence around 30-80 Hz, 100 ms, stronger in Objects having LTM representation.

15. Informatics and Mathematical Modelling / Intelligent Signal Processing 15 Morten Mørup Inter trial phase coherence (ITPC) time frequency channel Mørup et al. (article in press, NeuroImage 2005) subject Condition Parafac Assumption: Same Frequency signature present to various degree in time, mixed in the channels and present to different degree in each condition and each subject. Factor components only additive (non-negativity constraint) ITPC normal distributed - proven by bootstrapping. The ITPC is the vector strength over trials (epochs)

16. Informatics and Mathematical Modelling / Intelligent Signal Processing 16 Morten Mørup Proof of normality of ITPC Bootstrapping: Randomly select Data from the epochs to form new datasets (each epoch might be represented 0, 1 or several times in the datasets). Calculate the ITPC of each of these datasets. Evaluate the distribution of these ITPC’s. Coherent regionIncoherent region

17. Informatics and Mathematical Modelling / Intelligent Signal Processing 17 Morten Mørup ANOVA Test of difference between conditions over subjects ANOVA F-test value Time Frequency Channel F-test value Mørup et al. (article in press, NeuroImage 2005)

18. Informatics and Mathematical Modelling / Intelligent Signal Processing 18 Morten Mørup 5-way analysis Mørup et al. (article in press, NeuroImage 2005)

19. Informatics and Mathematical Modelling / Intelligent Signal Processing 19 Morten Mørup Time-frequency decomposition of ITPC Time-frequency Subject  condition Channel Pull paradigm - 6 subjects, 2 condition. Even trials: Right hand was pulled by a weight Odd trials: Left hand was pulled by a weight.

20. Informatics and Mathematical Modelling / Intelligent Signal Processing 20 Morten Mørup References Bro, R., Jong, S. D., 1997. A fast non-negativity-constrained least squares algorithm. Journal of Chemometrics 11, 393-401. Carrol, J. D., Chang, J., 1970. Analysis of individual differences in multidimensional scaling via an N.way generalization of 'Eckart- Young' decomposition. Psychometrika 35, 283-319. Field, Aaron S.; Graupe, Daniel “Topographich Component (Parallel Factor) analysis of Multichannel Evoked Potentials: Practical Issues in Trilinear Spatiotemporal Decomposition” Brain Topographa, Vol. 3, Nr. 4, 1991 Harshman, R. A., 1970. Foundation of the PARAFAC procedure: models and conditions for an 'explanatory' multi-modal factor analysis. UCLA Work. Pap. Phon. 16, 1-84. Herrmann, Christoph S; Lenz, Daniel; Junge, Stefanie ; Busch, Niko A; Maess, Burkhard “Memory-matches evoke human gamma- responses” BMC Neuroscience 2004, 5:13 Lee, D. D., Seung, H. S., 2001. Algorithms for non-negative matrix factorization. Advances in Neural information processing 13, Miwakeichi, F., Martinez-Montes, E., Valdes-Sosa, P. A., Nishiyama, N., Mizuhara, H., Yamaguchi, Y., 2004. Decomposing EE data into space-time-frequency components using Parallel Factor Analysis. Neuroimage 22, 1035-1045. Möcks, J., 1988. Decomposing event-related potentials: a new topographic components model. Biol. Psychol. 26, 199-215. Mørup, M., Hansen, L. K., Herrmann, C. S., Parnas, J., Arfred, S. M., 2005. Parallel Factor Analysis as an exploratory tool for wavelet transformed event-related EEG. NeuroImage Article in press,