ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August 8, st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Analyzing the Wavelet Transformed EEG using Non-negative Matrix and Tensor Factorization An introduction to ERPWAVELAB

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Parts of the work done in collaboration with Lars Kai Hansen, Professor Department of Signal Processing Informatics and Mathematical Modeling, Technical University of Denmark Sidse M. Arnfred, Dr. Med. PhD Cognitive Research Unit Hvidovre Hospital University Hospital of Copenhagen

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August 8, st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark The continuous wavelet transform and measures of the event related ERP in the time-frequency domain Introduction to NMF and extensions to tensor decompositions (PARAFAC & TUCKER) Accessing significance A demonstration of ERPWAVELAB Discussion OUTLINE

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark time frequency Continuous Wavelet transform Complex Morlet wavelet - Real part - Complex part Absolute value of wavelet coefficient Captures frequency changes through time  

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Continuous Wavelet transform (continued) epoch channel time-frequency epoch channel time

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark The Vector strength Vectors coherent, i.e. correlated Vectors incoherent, i.e. uncorrelated Vector strength a measure of coherence

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Measures of the event related ERP in the time-frequency domain ERSP WTav ITPC avWT

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Measures of the event related ERP in the time-frequency domain (cont.) Since scalp works as low pass filter it is customary to normalize X before calculating each measure. Frequently used normalizations are: (where t b are points in the baseline region and T b the total number of baseline samples)

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark ERPWAVELAB demonstration, tutorial dataset 1 ERSP WTav avWT INDUCED ITPC

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark f’=f, t’=t

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Multi-channel decomposition time-frequency activities appear to be similar across channels but varying in strength. Motivates to decompose the activity into similar time-frequency signatures varying in strength in the recording channels. Thus, this form of decomposition is primarily useful for data exploratory purposes giving very easy summaries of what types of activities are present in the data. Only when the measures of interest can be assumed linear and no cancellation between sources are present the decomposition can also reveal the underlying true sources. ≈

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Factor Analysis Spearman ~1900 VWH  d  V tests x subjects  W tests x intelligences H intelligencesxsubject tests Subjects tests Int. Non-negative Matrix Factorization (NMF): VWH s.t. W i,d,H d,j 0 (~1970 Lawson, ~1995 Paatero, ~2000 Lee & Seung)

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Non-negative matrix factorization (NMF) NMF: V  WH s.t. W i,d,H d,j  0 Multiplicative updates: Let C be a given cost function Positive term Negative term

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Non-negative matrix factorization (NMF) (Lee & Seung ) NMF gives Part based representation (Lee & Seung – Nature 1999)

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark The NMF decomposition is not unique Simplical Cone NMF only unique when data adequately spans the positive orthant (Donoho & Stodden ) z y x Convex Hull z y x Positive Orthant z y x Simplical Cones

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Sparse Coding NMF (SNMF) (Eggert & Körner, 2004) (Mørup & Schmidt, 2006)

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Why sparseness? Ensures uniqueness Eases interpretability (sparse representation  factor effects pertain to fewer dimensions) Can work as model selection (Sparseness can turn off excess factors by letting them become zero) Resolves over complete representations (when model has many more free variables than data points)

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark time-frequency Subjects/Condition/Trials channel Often extra modalities such as subjects, conditions and trials are present, consequently the data forms a tensor.  Need for tensor decomposition

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Higher Order Non-negative Matrix Factorization Factor AnalysisPARAFACTUCKER =

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Uniqueness Although PARAFAC in general is unique under mild conditions, the proof of uniqueness by Kruskal is based on k-rank*. However, the k-rank does not apply for non-negativity**. TUCKER model is not unique, thus no guaranty of uniqueness. Imposing sparseness useful in order to achieve unique decompositions Tensor decompositions known to have problems with degeneracy, however when imposing non-negativity degenerate solutions can’t occur*** * ) k-rank: The maximum number of columns chosen by random of a matrix certain to be linearly independent. **) L.-H. Lim and G.H. Golub, ***) See L.-H. Lim -

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Example why Non-negative PARAFAC isn’t unique

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark PARAFAC model estimation Thus, the PARAFAC model is by the matricizing operation estimated straight forward from regular NMF estimation

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark TUCKER TUCKER model estimation

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Algorithm outline (TUCKER) (PARAFAC follows by setting C = I )

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Accessing Significance Comparison to known distribution Bootstrapping Cross validation

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Comparison to known distribution (Mardia, Directional Statistics) Rayleigh distributed Red: Theoretical mean value of N -½ Black: Mean value estimated by bootstrapping Normal distributed Random ITPC and ERPCOH corresponds to a random walk in the complex plane thus is Rayleigh distributed.

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Bootstrapping 1) Randomly select Data from the epochs to form new datasets (each epoch might be represented 0, 1 or several times in the datasets). 2) Calculate the measure of interest for each of these datasets. 3) Evaluate the values found to the distribution of values found by the bootstrap datasets.

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Cross validation Split dataset into exploratory and confirmatory datasets. Find significant activity in exploratory dataset See if this activity is also significant in confirmatory dataset Correcting for multiple comparison by bootstrapping very expensive and often too conservative. Thus Cross validation useful.

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark – Dataset generation – Single subject analysis Artifact rejection in the time frequency domain NMF decomposition Cross coherence tracking – Multi subject analysis Clustering Analysis of Variance (ANOVA) Tensor decomposition ERPWAVELAB Tutorial: The toolbox is free to download from

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark Epilog: Some 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 ITPC (channel x time x frequency x subject x condition)

ERPWAVELAB 1st International Summer School in Biomedical Engineering1st International Summer School in Biomedical Engineering August st International Summer School in Biomedical Engineering Morten Mørup Technical University of Denmark References Bro, R., Multi-way Analysis in the Food Industry: Models, algorithms and Applications. Amsterdam, Copenhagen. Bro, R.,Jong, S. D., A fast non-negativity-constrained least squares algorithm. J. Chemom. 11, 393–401 Carroll, J. D. and Chang, J. J. Analysis of individual differences in multidimensional scaling via an N-way generalization of "Eckart-Young" decomposition, Psychometrika —319 Delorme, A.,Makeig, S., EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis. J Neurosci Methods 134, 9-21 Donoho, D. and Stodden, V. When does non-negative matrix factorization give a correct decomposition into parts? NIPS2003 Eggert, J. and Korner, E. Sparse coding and NMF. In Neural Networks volume 4, pages , 2004 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 Fiitzgerald, D. et al. Non-negative tensor factorization for sound source separation. In proceedings of Irish Signals and Systems Conference, 2005 Kruskal, J.B. Three-way analysis: rank and uniqueness of trilinear decompostions, with application to arithmetic complexity and statistics. Linear Algebra Appl., 18: , 1977 Harshman, R. A. Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis},UCLA Working Papers in Phonetics —84 Herrmann, C. S., Grigutsch, M.,Busch, N. A., EEG oscillations and wavelet analysis. Lathauwer, Lieven De and Moor, Bart De and Vandewalle, Joos MULTILINEAR SINGULAR VALUE DECOMPOSITION.SIAM J. MATRIX ANAL. APPL.2000 (21)1253–1278 Lee, D.D. and Seung, H.S. Algorithms for non-negative matrix factorization. In NIPS, pages , 2000 Lee, D.D and Seung, H.S. Learning the parts of objects by non-negative matrix factorization, NATURE 1999 Lim, Lek-Heng - Lim, L.-H. and Golub, G.H., "Nonnegative decomposition and approximation of nonnegative matrices and tensors," SCCM Technical Report, 06-01, forthcoming, Mardia, K. V.,Jupp, P. E., Directional Statistics. WILEY & SONS Miwakeichi, F., Martinez-Montes, E., Valdes-Sosa, P. A., Nishiyama, N., Mizuhara, H., Yamaguchi, Y., Decomposing EEG data into space-time-frequency components using Parallel Factor Analysis. Neuroimage 22, Möcks, J., Decomposing event-related potentials: a new topographic components model. Biol. Psychol. 26, Mørup, M and Hansen, L.K and Herman, C.S. and Parnas, Josef and Arnfrede, Sidse M. “Parallel Factor Analysis as an exploratory tool for wavelet transformed event –related EEG” NeuroImage 20, (2006) Mørup, M. and Hansen, L.K.and Arnfred, S.M.Decomposing the time-frequency representation of EEG using nonnegative matrix and multi-way factorization Technical report, Institute for Mathematical Modeling, Technical University of Denmark, 2006a Mørup, M. and Schmidt, M.N. Sparse non-negative matrix factor 2-D deconvolution. Technical report, Institute for Mathematical Modeling, Technical University of Denmark, 2006b Mørup, M. and Hansen, L.K.and Arnfred, S.M. Algorithms for Sparse Higher Order Non-negative Matrix Factorization (HONMF), Technical report, Institute for Mathematical Modeling, Technical University of Denmark, 2006e Tamara G. Kolda Multilinear operators for higher-order decompositions technical report Sandia national laboratory 2006 SAND Tucker, L. R. Some mathematical notes on three-mode factor analysis Psychometrika —311 Welling, M. and Weber, M. Positive tensor factorization. Pattern Recogn. Lett. 2001