Decomposing event related EEG using Parallel Factor

Decomposing event related EEG using Parallel Factor

Morten Mørup

Informatics and Mathematical Modeling Intelligent Signal Processing

Technical University of Denmark

Outline

 Non-negativity constrained PARAFAC

 Application of PARAFAC to the EEG

(Harshman & Carrol and Chang 1970)

Alternating Least Squares (ALS)

ALS corresponds to maximizing the likelihood of a Gaussian

Consequently, ALS assumes normal distributed noise.

Gradient descent

Especially good for cost functions without analytical solution.

Let C be the cost function, then update the parameters

according to:

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

||A||

+c

||B||

+c

||S||

Orthogonality constraints, i.e. A

A=I

Non negativity constraint on all modalities

(if data is positive and factor components considered purely additive)

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

Non-negative matrix factorization (NMF)

Generalization to PARAFAC

(Lee & Seung 2001)

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

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)

time

time

frequency

Wavelet transform

Complex Morlet wavelet - Real part - Complex part

Absolute value of wavelet coefficient

Captures frequency changes through time

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.

The Vector strength

Vectors coherent, i.e. correlated Vectors incoherent, i.e. uncorrelated

Vector strength a measure of coherence

Visual Paradigm

(Herrmann et al. 2004)

Expected result: Coherence around 30-80 Hz, 100 ms,

stronger in Objects having LTM representation.

Inter trial phase coherence (ITPC)

time frequency

channel

Mørup et al.

(article in press, NeuroImage 2005)

subject

Condition

 

 



 ⁿ

e e

n Xe c f t t f c t X

f c ITPC

1 1

, ,

, ) ,

, , (

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)

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 region Incoherent region

Test of difference between conditions over subjects

Time Frequency Channel

F-test value

   

 

   





 











 _K

k S s

K k

K S k t f c I k s t f c ITPC

K t f c I k t f c I S t

f c Z

1 1

2 1

2

/ , , , ) , , , , (

1 / ) , , ( ) , , , ( )

, , (

   

    



 







K k

S s S s

k s t f c KS ITPC

t f c I

k s t f c S ITPC k t f c I

1 1 1

, , , 1 ,

, ,

, , , 1 ,

, , ,

Mørup et al.

(article in press, NeuroImage 2005)

5-way analysis

Mørup et al.

(article in press, NeuroImage 2005)

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.

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,