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Independent Component Analysis

Independent Component Analysis

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Independent Component Analysis

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  1. Independent Component Analysis NiC fMRI-methodology Journal Club 05 Dec. 2008

  2. There will be math … Overview • Theory • PCA • ICA • Application • Data reduction • Spatial vs. Temporal ICA • Group-ICA

  3. Non-fMRI example • The cocktail-party problem • Many people are speaking in a room • Given a number of microphonerecordings from different locations,can the various sourcesbe reconstructed?

  4. Non-fMRI example Sources • Signals are collected in row-vectors • r1 = 0.8∙s1+0.6∙s2 • r2 = 0.2∙s1+0.5∙s2 Recordings

  5. Non-fMRI example s2 r2 s1 r1

  6. Non-fMRI example • Assumptions • Recordings consist of linearly transformed sources (‘weighted averages’) • Original sources are completely unrelated • Uncorrelated • Independent • Then • Sources can be estimated by linearly (back-)transforming the recordings • The optimal transformation makes the source estimates ‘as unrelated as possible’ • Minimize correlations • Maximize independence • In fMRI • Sources of interest are the various ‘brain systems’ • Recordings are the acquisition volumes and/or voxel time courses

  7. Non-fMRI example S1 R1 S2 R2

  8. Theory PCA ICA

  9. PCA • Recordings typically are not uncorrelated • Goal: find a transformation matrix V such that transformed signals Y=V∙R are uncorrelated • If the transformed signals are properly normalized (|yi|=1)

  10. PCA • Q: how do we find V such that(note that R∙RT equals the covariance matrix) • A: use the singular value decomposition (SVD)such thatresulting in a solution

  11. PCA • In pictures, PCA performes ‘sphering’ • Fit the data cloud with an ellipsoidal function • Rotate the ellipsoid such that its axes coincide with the coordinate axes • Squeeze/stretch the ellipsoid to make it spherical Recordings Rotation Squeezing/stretching r2 x2 y2 r1 x1 y1

  12. By means of a linear transformation of the recordings r1 and r2, we found signals y1 and y2 that are normalized and completely uncorrelated Yet, these signals do not at all correspond with the original sources s1 and s2, and are obviously still mixtures However, any rotation will preserve uncorrelatedness So, how the determine the ‘best’ rotation? PCA y2 y1

  13. ICA • Statistical independence • Knowledge of the value of y1 for a sample does not affect the statistical distribution of the values of y2 • Equivalently, the joint probability distribution is the product of the marginal probability ditributions • As a result, loosely speaking, all non-gaussian ‘features’ of the probability distribution are either caused by P(y1) or P(y2), but not by both, and therefore lie parallel with the coordinate axes

  14. ICA • Independence implies uncorrelatedness(But the reverse is not true!) • Therefore, maximally independent signals are also likely approximately uncorrelated • This suggests performing PCA first, to decorrelate the signals, and determining a suitable rotation next that optimizes independence but (automatically) preserves uncorrelatedness

  15. A simple idea:Penalize all data points with a penalty P=(y12∙y22) No penalty if y1=0 or y2=0 (i.e., for points on any axis) Positive penalty for points in any of the four quadrants Increasing penalty for points that are further away from the axes ICA y2 y1

  16. Minimize the penalty over a search space that consists of all rotations The solution is determined up to Component order Component magnitude Component sign ICA z2 z1

  17. ICA • Average penalty P • Rotations preserve the euclidean distance to the origin • It follows that minimizing P is equivalent to maximizing K • K is closely related to kurtosis (‘peakiness’)

  18. ICA • Central limit theorem • A mixture of a sufficiently large number of independent random variables (each with finite mean and variance) will be approximately normally distributed • This suggests that the original sources are more non-gaussian than the mixtures • Maximum non-gaussianity is the criterion to use to determine the optimal rotation! • This cannot be successful if sources are normally distributed • QuartiMax employs kurtosis as a measure of non-gaussianity

  19. ICA • If some sources are platykurtotic, the method may fail • Therefore, maximize

  20. ICA • Due to the fourth power, the method is sensitive to outliers • Therefore, choose any other function G • G(y) = y4 • G(y) = log(cosh(y)) • G(y) = 1-exp(-y2) • FastICA [Hyvärinen]

  21. ICA • Some non-gaussian distributions will happen to have K=0 • Therefore, use mutual information expressed in terms of negentropy • InfoMax [Bell & Sejnovski]

  22. ICA • Entropy-calculations require approximation of the probability density function P itself • Therefore, expand negentropy in terms of lower-order cumulants (generalized variance/skewness/kurtosis/…) • JADE [Cardoso]

  23. ICA MatLab code % Generate signals S = wavread('D:\sources.wav')'; A = [0.8,0.6;0.2,0.5]; R = A*S; clear S,A; % PCA [UL,LAMBDA,UR] = svd(R,'econ'); V = inv(LAMBDA)*UL'; Y = V*R; % ICA Z = rotatefactors(Y','Method','quartimax')'; Z = icatb_fastICA(Y,'approach','symm','g','tanh'); [d1,d2,Z] = icatb_runica(Y); Z = icatb_jade_opac(Y);

  24. In practice, all mentioned algorithms have been reported to perform satisfactorily QuartiMax FastICA InfoMax JADE ICA QuartiMax InfoMax FastICA JADE [?]

  25. In practice, all mentioned algorithms have been reported to perform satisfactorily QuartiMax FastICA InfoMax JADE ICA QuartiMax InfoMax FastICA JADE

  26. Application Data reduction Spatial vs. Temporal ICA Group-ICA

  27. Data reduction • fMRI-data are gathered in a matrix Y Y

  28. Data reduction • fMRI-data are gathered in a matrix Y • Data are decomposed into principal components by means of SVD time comp Y Ux comp time Λ Ut =∙∙ voxels voxels comp comp

  29. Data reduction • fMRI-data are gathered in a matrix Y • Data are decomposed into principal components by means of SVD • Only the first few strongest components are retained time comp Y Ux comp time Λ Ut =∙∙ voxels voxels comp comp

  30. λ1× × λ2× × λ3× × Data reduction • fMRI-data are gathered in a matrix Y • Data are decomposed into principal components by means of SVD • Only the first few strongest components are retained • Each component is the product of • a coefficient • a spatial map • a time course … residuals Y +

  31. Temporal vs. Spatial ICA Temporal ICA • PCA decomposition results in • Uncorrelated spatial maps and • Uncorrelated time courses • ICA rotation results in • Maximally independent time courses: tICA or • Maximally independent spatial maps: sICA • Some methods employ criteria in both domains Acquisitions Voxel j Component map Voxel i Spatial ICA Voxels Acquisition j Component time course Acquisition i

  32. Temporal vs. Spatial ICA • PCA • tICA • sICA Y Ux = ∙ ∙ St Ut Λ Y Ux Vx = ∙ ∙ ∙ =∙ Λ A St St Y Sx Sx = ∙ ∙ ∙ =∙ A Λ Ut St St Vt

  33. Temporal vs. Spatial ICA • Temporal ICA • Components have independent temporal dynamics:“Strength of one component at a particular moment in time does not provide information on the strength of other components at that moment” • Components may be correlated/dependent in space • Popular for cocktail party problem • Spatial ICA • Components have independent spatial distributions:“Strength of one component in particular voxel does not provide information on the strength of other components in that voxel” • Components may be correlated/dependent in time • Popular for fMRI

  34. Temporal vs. Spatial ICA Calhoun et al., 2001

  35. λi× × Group-ICA • Some problems at the subject level become even more pressing at the group level • Independent components have no natural interpretation • Independent components have no meaningful order • The magnitude of independent component maps and time courses is undetermined • The sign of independent component maps and time courses is arbitrary • For group level analyses, some form of ‘matching’ is required • Assumptions • Equal coefficients across subjects? • Equal distributions across subjects? • Equal dynamics across subjects?

  36. Group-ICA • Method I: Averaging • E.g.: Schmithorst et al., 2004 • Principle • Average the data sets of all subjects before ICA • Perform ICA on the mean data • Key points • All subjects are assumed to have identical components • Equal coefficients > homogeneous population • Equal distributions > comparable brain organization • Equal dynamics > fixed paradigm; resting state impossible • Statistical assessment at group level • Enter ICA time courses into linear regression model (back-projection)

  37. Group-ICA • Method II: Tensor-ICA • E.g.: Beckmann et al., 2005 • Principle • Stack subjects’ data matrices along a third dimension • Decompose data tensor into product of (acquisition-dependent) time course, (voxel-dependent) maps, and (subject-dependent) loadings • Key points • Components may differ only in strength • Unequal coefficients > inhomogeneous population • Equal distributions > comparable brain organization • Equal dynamics > fixed paradigm; resting state impossible • Statistical assessment at group level • Components as a whole

  38. Group-ICA • Method III: Spatial concatenation • E.g.: Svensén et al., 2002 • Principle • Concatenate subjects’ data matrices along the spatial dimension • Perform ICA on aggregate data matrix • Partition resulting components into individual maps • Key points • Components may differ in strength and distribution • Unequal coefficients > inhomogeneous population • Unequal distributions > brain plasticity • Equal dynamics > fixed paradigm; resting state impossible • Statistical assessment at group level • Voxel-by-voxel SPMs

  39. Group-ICA • Method IV: Temporal concatenation • E.g.: Calhoun et al., 2001 • Principle • Concatenate subjects’ data matrices along the time dimension • Perform ICA on aggregate data matrix • Partition resulting components into individual time courses • Key points • Components may differ in strength and dynamics • Unequal coefficients > inhomogeneous population • Equal distributions > comparable brain organization • Unequal dynamics > flexible paradigm • Statistical assessment at group level • Components as a whole, from time course spectrum/power • Voxel-by-voxel SPMs, from back-projection (careful with statistics!)

  40. Group-ICA • Method V: Retrospective matching • Subjective or (semi)automatic matching on basis of similarity between distribution maps and/or time courses • Also used to test the reproducibility of some stochastic ICA algorithms • Various principles, various authors • Principle • Perform ICA on individual subjects • Match similar individual components one-on-one across subjects • Key points • Components may differ in strength and dynamics • Unequal coefficients > inhomogeneous population • Unequal distributions > brain plasticity • Unequal dynamics > flexible paradigm • Statistical assessment at group level • Voxel-by-voxel SPMs (careful with scaling and bias!)

  41. Conclusion • ICA has the major advantage that it requires minimal assumptions or prior knowledge • However, interpretation of the meaning of components occurs retrospectively and may be ambiguous • Unfortunately, methods and statistics are not fully characterized yet and still quite heavily under development • Therefore - IMHO – independent component analysis is an excellent tool for exploratory experiments, but should not be your first choice for confirmatory studies