Informatics and Mathematical Modelling / Intelligent Signal Processing 1 Morten Mørup Decomposing event related EEG using Parallel Factor Morten Mørup.

Informatics and Mathematical Modelling / Intelligent Signal Processing

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Decomposing event related EEG using Parallel FactorMorten MørupInformatics and Mathematical ModelingIntelligent Signal ProcessingTechnical University of Denmark


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OutlineNon-negativity constrained PARAFAC

Application of PARAFAC to the EEG


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PARAFAC(Harshman & Carrol and Chang 1970)


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Alternating Least Squares (ALS)

ALS corresponds to maximizing the likelihood of a GaussianConsequently, ALS assumes normal distributed noise.


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Gradient descent

Especially good for cost functions without analytical solution.

Let C be the cost function, then update the parameters according to:


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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. c1||A||2+c2||B||2+c3||S||2

Orthogonality constraints, i.e. ATA=I

Non negativity constraint on all modalities

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


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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 updatesUpdate parameters so they remain in the positive domain.

Among various other methods


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Non-negative matrix factorization (NMF)

Generalization to PARAFAC

(Lee & Seung 2001)


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Electroencephalography (EEG)

EEG measures electrical potential at the scalp arisingprimarily from synchronous neuronal activity of pyramidalcells in the brain.

Event related potential (ERP) is EEG measurements time locked to a stimulus event


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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)


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time

time

frequency

Wavelet transform

Complex Morlet wavelet - Real part - Complex part

Absolute value of wavelet coefficient

Captures frequency changes through time


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time

channel

subje

cts

Möcks (1988)Field & Graupe (1991)

time

frequency

channel

Miwakeichi (2004)

PARAFAC Assumption: Same signal havingVarious strength in each subjectmixed in the channels.

PARAFAC Assumption: Same Frequency signature present to variousdegree in time mixed in thechannels.


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The Vector strength

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

Vector strength a measure of coherence


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Visual Paradigm(Herrmann et al. 2004)

Expected result: Coherence around 30-80 Hz, 100 ms,stronger in Objects having LTM representation.


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Inter trial phase coherence (ITPC)

time

freq

uenc

ych

annel

Mørup et al.(article in press, NeuroImage 2005)

subject

Condit

ion

n

e e

en tfcX

tfcXtfcITPC

1

1

,,

,,),,(

Parafac Assumption: Same Frequency signature present to variousdegree 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)


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Proof of normality of ITPC

Bootstrapping:Randomly selectData from the epochsto form new datasets (each epoch might be represented 0, 1 or several times in the datasets). Calculate the ITPC ofeach of these datasets.Evaluate the distributionof these ITPC’s.

Coherent region Incoherent region


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ANOVA Test of difference between conditions over subjects

ANOVA F-test valueANOVA F-test value

Time

Frequency

Channel

F-test value

K

k

S

s

K

k

KSktfcIkstfcITPC

KtfcIktfcIS

tfcZ

1 1

2

1

2

/,,,),,,,(

1/),,(),,,(

),,(

K

k

S

s

S

s

kstfcITPCKS

tfcI

kstfcITPCS

ktfcI

1 1

1

,,,,1

,,

,,,,1

,,,



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5-way analysis



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Time-frequency decomposition of ITPC

Time-frequency

Subject

conditi

on

Channel

Pull paradigm - 6 subjects, 2 condition. Even trials: Right hand was pulled by a weightOdd trials: Left hand was pulled by a weight.


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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,

Informatics and Mathematical Modelling / Intelligent Signal Processing 1 Morten Mørup Decomposing event related EEG using Parallel Factor Morten Mørup.

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