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Brain Electrophysiological Signal Processing: Preprocessing

ME (Signal Processing), IISc: Neural Signal Processing, Spring 2014. Brain Electrophysiological Signal Processing: Preprocessing. Kaushik Majumdar Indian Statistical Institute Bangalore Center kmajumdar@isibang.ac.in.

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Brain Electrophysiological Signal Processing: Preprocessing

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  1. ME (Signal Processing), IISc: Neural Signal Processing, Spring 2014 Brain Electrophysiological Signal Processing: Preprocessing Kaushik Majumdar Indian Statistical Institute Bangalore Center kmajumdar@isibang.ac.in

  2. Benbadis and Rielo, 2008: http://emedicine.medscape.com/article/1140247-overview Heart-Rate and Muscle Artifacts in EEG ME (Signal Processing), IISc: Neural Signal Processing

  3. Preprocessing • Visual Inspection • Filtering • Principal Component Analysis (PCA) • Independent Component Analysis (ICA) ME (Signal Processing), IISc: Neural Signal Processing

  4. Matrix Representation of Multi-Channel EEG • M is an m x n matrix, whose m rows represent m EEG channels and n columns represent n time points. • Often during EEG processing we are to find a matrix W such that WM is the processed signal. ME (Signal Processing), IISc: Neural Signal Processing

  5. Majumdar, under preparation, 2013 EOG Identification by Principal Component Analysis (PCA) ME (Signal Processing), IISc: Neural Signal Processing

  6. PCA Algorithm (cont.) ME (Signal Processing), IISc: Neural Signal Processing

  7. PCA Algorithm (cont.) PCA Rotation and (Stretching or Contracting) ME (Signal Processing), IISc: Neural Signal Processing

  8. Wallstrom et al., Int. J. Psychophysiol., 53: 105-119, 2004 Performance of PCA in EOG Removal EOG ME (Signal Processing), IISc: Neural Signal Processing

  9. Independent Component Analysis (ICA) • In PCA data components are assumed to be mutually orthogonal, which is too restrictive. PCA components Original data sets ME (Signal Processing), IISc: Neural Signal Processing

  10. ICA (cont.) • PCA will give poor results if the covariance matrix has eigenvalues close to each other. ME (Signal Processing), IISc: Neural Signal Processing

  11. ICA as Blind Source Separation (BSS) S1 S4 Four musicians are playing in a room. From the outside only music can be heard through four microphones. No one can be seen. How the music heard from outside can be decomposed into four sources? S2 S3 1 2 4 3 ME (Signal Processing), IISc: Neural Signal Processing

  12. Mathematical Formulation A is mixing matrix, x is sensor vector, s is source vector and n is noise, which is to be eliminated by filtering. ME (Signal Processing), IISc: Neural Signal Processing

  13. Mathematical Formulation (cont.) Given find such that Any estimation technique of is called an ICA technique or BSS technique in general. ME (Signal Processing), IISc: Neural Signal Processing

  14. Hyvarinen and Oja, Neural Networks, 13: 411-430, 2000 ICA Algorithm: FastICA Whitening: • Normalization (make mean zero). • Make variance one i.e., E expectation, x is the vector of signals and I is identity matrix. ME (Signal Processing), IISc: Neural Signal Processing

  15. FastICA (cont.) B is orthogonal matrix and D is diagonal matrix of E will satisfy Whitening complete ME (Signal Processing), IISc: Neural Signal Processing

  16. Non-Gaussianity • ICA is appropriate only when probability distribution of the data set is non-Gaussian. • Gaussian distribution is of the form ME (Signal Processing), IISc: Neural Signal Processing

  17. Entropy of Gaussian Variable • A Gaussian variable has the largest entropy among a class of random variables with equal variance (for a proof see Cover & Thomas, Elements of Information Theory). Here we will give an intuitive argument. ME (Signal Processing), IISc: Neural Signal Processing

  18. Entropy of a Random Variable X More information Less (zero) information ME (Signal Processing), IISc: Neural Signal Processing

  19. Gaussian Random Variable Has Highest Entropy: Intuitive Proof • By Central Limit Theorem (CLT) the mean of a class of random variables (class is signified by uniform variance) follows normal distribution as the number of members in the class tends to infinity (i.e., becomes very large). • Infinite observations hold infinite or maximum amount of information. ME (Signal Processing), IISc: Neural Signal Processing

  20. Intuitive Proof (cont.) • Therefore a random variable with normal distribution has the highest information content. • So it has the highest entropy. If each variable in a class of random variables admits only finite number of nonzero values, the one with uniform distribution will have the highest entropy. ME (Signal Processing), IISc: Neural Signal Processing

  21. Non-Gaussianity as Negentropy H is entropy and J negentropy. J is to be maximized. When J is maximum y is reduced to a component. This can be shown by calculating the kurtosis for component and sum of components including the said component (See Hyvarinen & Oja, 2000, P. 7). ME (Signal Processing), IISc: Neural Signal Processing

  22. Steps of FastICA after Whitening g is in the form of either of the two ME (Signal Processing), IISc: Neural Signal Processing

  23. Exercise • FastICA has been implemented in EEGLAB (in runica function). Remove artifacts from sample EEG data using the ICA implementation in EEGLAB. ME (Signal Processing), IISc: Neural Signal Processing

  24. Concept of Independence in PCA and ICA • In PCA independence means orthogonality i.e., pairwise dot product is zero. • In ICA independence is statistical independence. Let x, y be random variables, p(x) is probability distribution function of x and p(x,y) is joint probability distribution function of (x,y). If p(x,y) = p(x).p(y) holds we call x and y are statistically independent. ME (Signal Processing), IISc: Neural Signal Processing

  25. Independence (cont.) • If vectors v1 and v2 are orthogonal they are independent. Say not, then a1v1 + a2v2 = 0 implies, a1v1.v1 + a2v2.v1 = 0 or a1 = 0. Similarly a2 = 0. • If v1 = cv2 then both of them must have same probability distribution or p(v1,v2) = p(v1) = p(v2). If v1 and v2 are linearly independent p(v1,v2) = p(v1).p(v2) may or may not hold. • If p(v1,v2) = p(v1).p(v2) holds then v1 and v2 are linearly independent. ME (Signal Processing), IISc: Neural Signal Processing

  26. Conditions for ICA Applicability • Sources are statistically independent. • Propagation delays in the mixing medium are negligible. Sources are time varying. Mixing medium delays may affect sources in different locations differently and thereby corrupting their temporal structures. • Number of sources = number of sensors. ME (Signal Processing), IISc: Neural Signal Processing

  27. References • Benbadis and Rielo, EEG artifacts, eMedicine, available online at http://emedicine.medscape.com/article/1140247-overview, 2008. • Hyvarinen and Oja, Independent component analysis: algorithms and applications, Neural Networks, vol. 13, p. 411-431, 2000. • Majumdar, A Brief Survey of Quantitative EEG Analysis (under preparation), Chapter 2. ME (Signal Processing), IISc: Neural Signal Processing

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