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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. - PowerPoint PPT Presentation
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Informatics and Mathematical Modelling / Intelligent Signal Processing
1Morten Mørup
Decomposing event related EEG using Parallel FactorMorten MørupInformatics and Mathematical ModelingIntelligent Signal ProcessingTechnical University of Denmark
Informatics and Mathematical Modelling / Intelligent Signal Processing
2Morten Mørup
OutlineNon-negativity constrained PARAFAC
Application of PARAFAC to the EEG
Informatics and Mathematical Modelling / Intelligent Signal Processing
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PARAFAC(Harshman & Carrol and Chang 1970)
Informatics and Mathematical Modelling / Intelligent Signal Processing
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Alternating Least Squares (ALS)
ALS corresponds to maximizing the likelihood of a GaussianConsequently, ALS assumes normal distributed noise.
Informatics and Mathematical Modelling / Intelligent Signal Processing
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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
,,,
Mørup et al.(article in press, NeuroImage 2005)
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5-way analysis
Mørup et al.(article in press, NeuroImage 2005)
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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,