This Week. Last Week. Independent Component Analysis. ICA vs. PCA. ICA vs PCA. Blind Source Separation: The Cocktail Party Problem

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1 1 Last Week Functional Connectivity This Week (Brief) ICA Intro ICA of fmri Sorting/Calibration Validation Demo single-subject subject ICA 2 Blind Source Separation: The Cocktail Party Problem Observations Mixing matrix A Sources Independent Component Analysis Goal: Separate sources from a linear mixture Model: X=AS X: Mixture A: Mixing coefficients S: Sources S= W X W= A 1 Assumptions Linear mixing Independence of sources Non-gaussian sources 3 4 ICA vs PCA { 1 2} = { 1} { 2} ( ) = ( ) ( ) E{ h( y1) h( y2) } = E{ h( y1) } E{ h( y2) } Uncorrelated: E yy E y E y Independent: p y, y p y p y ICA vs. PCA PCA finds directions of maximal variance (using second order statistics) ICA finds directions which maximize independence (using higher order statistics) 5 PCA finds directions of maximal variance (using second order statistics) 6

2 7 ICA vs. PCA ICA Example Mixing simple signals: sinus + chainsaw. ICA finds directions which maximize independence (using higher order statistics) From: Chap. 14.6, Friedman, Hastie, Tibshirani: The elements of statistical learning. 8 Infomax Bell&Sejnowski 95 Lee 98 Blind Deconvolution Lambert 96/Bussgangs Maximum Negentropy Comon 94 Lee 98 Cardoso 96 Pearlmutter 97 (if the nonlinearities in the NN are chosen as the cdf s) Lee 98 Hyvarinen 99 (if constrained to be uncorrelated) Cardoso 99 (represent likelihood by K-L L distance between observed and factorized density) Maximum Likelihood Gaeta 90, Pearlmutter 96 Karhunen 97 Girolami & Fyfe 97 Nonlinear PCA Oja 97, Karhunen & Jautsensalo 94 Min. Mutual Info. Comon 94 mean Y = E{ Y} ICA Maximizes Nongaussianity variance 2 σ Y = E{ Y 2 E( Y)} κ 3 = 4 κ 4 ( Y) = E{ Y } 2 E{ Y } 3 3 Y E Y Many real-world data sets have supergaussian distributions The random variables take relatively more often values that are very close to zero or are large skewness ( ) { } Gaussian kurtosis supergaussian ( ) FastICA demo (mixtures) FastICA demo (whitened) 11 12

3 13 FastICA demo (step 1) FastICA demo (step 2) 14 FastICA demo (step 3) FastICA demo (step 4) FastICA demo (step 5 - end) Information Maximization (Infomax) x W, wo g ( ) y = g ( Wx) H[ y] I[ y] N p ( ) i= 1E log i u = i W W W g i u i We see: if the density of the signal estimate u i matches the corresponding derivatives of the nonlinearity g i, the marginal entropy terms will vanish. Thus, maximizing H[y] will minimize I[y] i.e., maximize independence [Bell et al 1995] 17

4 19 Overview (Brief) ICA Intro ICA of fmri Sorting/Calibration Validation Demo single-subject subject ICA 1. Model (1 or more Regressors) 2. Data General Linear Model or xi ( j) y( j) 3. Fitting the Model to the Data at each voxel M ( ) = ˆ β0 + ˆ βi i ( ) + ( ) y j x j e j i= 1 Regression Results 20 Time Time Voxels Voxel Data(X) = G Independent Component Analysis (ICA) Data(X) = General Linear Model (GLM) Design matrix Wˆ 1 Mixing matrix Activation maps Corresponding to columns of G The GLM is by far the most common approach to analyzing fmri ˆβ data. To use this approach, one needs a model for the fmri time course Time courses Spatially Independent Components Components (C) Time courses In spatial ICA, there is no model for the fmri time course, this is estimated along with the hemodynamic source locations Time Voxels A little more detail Data(X) = R TC Wˆ 1 Mixing matrix { Spatially Independent Components Components (C) Time courses ICA of fmri ICA Example The ICA model assumes the fmri data, x,, is a linear mixture of statistically independent sources, s. x= As p( s1, s2) = p( s1) p( s2) The goal of ICA is to separate the sources given the mixed data and thus determine the s and A matrices fmri data, x Source 1 s = T [ s1 s2] Source 2 Time course 1 Time course 2 + A 23 24

5 25 ICA Halloween (Un)Mixer( Un)Mixer! X = A S background ICA of fmri Data Time = candle 1 candle 2 candle 3 Candle out 26 Signal Types Motion Artifact Task related Cardiac Motion Vasomotor oscillation/ High order visual Motion-related signal due to mouth movement from inferior temporal and orbitofrontal regions Hemodynamic Model Artifact Detection and Reduction Note: PREPROCESSING MAY DIFFER FOR Art. Hunting Approach Eye movements N/2 Nyquist Ghost Spatial versus Temporal ICA Does it matter? Why is spatial ICA more common? Some examples: Source: Christian Beckmann s Little Shop of fmri Horrors :

6 31 Temporally and Spatially low-correlated Components SICA SPM TICA Spatially Dependent Components SICA SPM TICA Temporally Dependent Components SICA SPM TICA Temporally and Spatially Dependent Components SICA SPM TICA Uses of ICA of fmri Improving fit to task-related components Find areas of activation which respond in a more complex way to an external stimulus Artifact Reduction/Filtering Examination of temporally coherent, but not necessarily task-related components Data exploration of unpredicted structure Overview (Brief) ICA Intro ICA of fmri Sorting/Calibration Validation Demo single-subject subject ICA 35 36

7 37 Ambiguities of ICA: Scaling 1 1 data = ( a * tc1)* * im1 + ( b* tc2)* * im2 + E a b Z-Score Each image and time course are divided by its standard deviation Percent Signal Change Time courses are regressed onto original data (at voxels which are maximal for a given component) They are scaled to reflect percent signal change from the mean Images are also scaled such that the maximal voxel value contains the maximal percent signal change value Ambiguities of ICA: Permutation 1 1 data = ( a * tc1)* * im1 + ( b* tc2)* * im2 + E a b Selecting the component of interest: Spatial map (e.g. default mode analysis, smoothness) Time courses (e.g. regression with model, spectral power) Parametric analysis of regression parameters (e.g. interactions with variable of interest) Multivariate (SVM approach, Formisano) Frequency analysis e.g. high frequency artifact Spatial edge artifact 38 Types of Sorting Temporal Sorting Correlation Multiple regression Others? (skew, kurtosis, power spectra, etc.) Spatial Sorting Correlation (w/ mask or SPM) Maximum value (w/i( mask) Multiple regression Multi-variate sorting SVM Approaches (Formisano( Formisano) Right Left t (secs) Example Number of Components Too many -> > over-splitting of the components Too few -> > over-clumping of the components How to choose? Between 20 and 40 appears to be a reasonable choice for typical fmri experiment Tools for estimating this number are available in GIFT and other ICA software programs (AIC/MDL/BIC) Post-ICA clustering is also used to address this issue 41 42

8 43 Number of Components (Order Selection) ( ) ( ) ( ˆ ) 1 AIC N = 2M K N L θ N NK + ( N 1) 2 ( ) ( ) ( ˆ 1 1 MDL N = M K N L θ N ) + 1+ NK + ( N 1) lnm 2 2 L ( ˆ θ N ) 1 ( λ 1... ) K N N λ + K = ln 1 ( λn λk) K N M=number of voxels K=number of time points N=number of sources λ=eigenvalues from PCA [V. D. Calhoun, T. Adali, G. D. Pearlson, and J. J. Pekar, "A Method for Making Group Inferences From Functional MRI Data Using Independent Component Analysis," Hum. Brain Map., vol. 14, pp , 2001.] Correction for correlated samples [Y. Li, T. Adali, and V. D. Calhoun, "Sample Dependence Correction For Order Selection In FMRI Analysis," in press Hum. Brain Map.] 44 Overview (Brief) ICA Intro ICA of fmri Sorting/Calibration Validation Demo single-subject subject ICA Validation Algorithm Differences Esposito F, Formisano E, Seifritz E, Goebel R, Morrone R, Tedeschi G, Salle FD Spatial Independent Component Analysis of Functional MRI Time-Series: to What Extent Do Results Depend on the Algorithm Used? Hum Brain Map 16: Algorithm & Preprocessing Differences Calhoun, V. D., Adali, T., and Pearlson, G. D Independent Components Analysis Applied to fmri Data: A Generative Model for Validating Results. Journal of VLSI Signal Proc Systems. vol. 37, pp Cluster Validation Himberg J, Hyvarinen A, Esposito F Validating the Independent Components of Neuroimaging Time Series Via Clustering and Visualization. Neuroimage 22(3): Test/retest Performance Nybakken GE, Quigley MA, Moritz CH, Cordes D, Haughton VM, Meyerand ME Test-Retest Precision of Functional Magnetic Resonance Imaging Processed With Independent Component Analysis. Neuroradiology 44(5): Hybrid fmri Experiment Impact of preprocessing/algorithms/etc Ground Truth Mixed with fmri Data Criterion: Kullback-Leibler (KL) divergence ( ) ( ) ( ) ln p s ξ D su = ps ξ d p ( ) ξ u ξ Define sources Generate sources For all: Add noise Smooth Reduce (PCA, cluster, etc.) Unmix (Info., fastica, jade, etc.) Evaluate (KL) min(kl) ) is winner 47 V.D.Calhoun, T.Adali,, and G.D.Pearlson,, "Independent Components Analysis Applied to FMRI Data: A Generative Model for Validating Results," Journal of VLSI Signal Proc. Systems,

9 49 Comparison of Different Algorithms Consistency of Infomax N. Correa, T. Adali, Y. Li, and V. D. Calhoun, "Comparison of Blind Source Separation Algorithms for FMRI Using a New Matlab Toolbox: GIFT," in Proc. ICASSP, Philadelphia, PA, N. Correa, T. Adali, and V. D. Calhoun "Performance of Blind Source Separation Algorithms for fmri Analysis," Mag.Res.Imag., 2006 (submitted). N. Correa, T. Adali, Y. Li, and V. D. Calhoun, "Comparison of Blind Source Separation Algorithms for FMRI Using a New Matlab Toolbox: GIFT," in Proc. ICASSP, Philadelphia, PA, N. Correa, T. Adali, and V. D. Calhoun "Performance of Blind Source Separation Algorithms for fmri Analysis," Mag.Res.Imag., 2006 (submitted). 50 Clustering of five algorithms using ICASSO Three Review Articles Transient task Default mode Temporal Right task Left task Infomax, FICA1, FICA2, FICA3, JADE N. Correa, T. Adali, and V. D. Calhoun "Performance of Blind Source Separation Algorithms for fmri Analysis," Mag.Res.Imag., 2006 (submitted) A Few Software Packages The ICA:DTU toolbox ( matlab three different ICA algorithms fmri specific with demo data FMRIB Software Library, which includes the ICA tool MELODIC ( odic/): C FastICA+ Complete Package AnalyzeFMRI ( ml) R FastICA BrainVoyager( Commercial FastICA Complete Package FMRLAB ( matlab infomax algorithm fmri specific with additional tools ICALAB matlab many ICA algorithms not fmri specific although one fmri example included GIFT ( matlab 9 ICA algorithms (more coming) including infomax and fastica Visualization tools and sorting options. Sample data and a step-by by-step walk through Group ICA of fmri Toolbox (GIFT) 400+ unique downloads Funded by NIH 1 R01 EB (to V. Calhoun) Major Contributors: Tülay Adalı University of Maryland Andrzej Cichocki, RIKEN, Japan Jim Pekar Johns Hopkins Hichem Snoussi IRCCyN, France Voxel BOLD Signal Left Hemisphere Visual Stimuli Onset 53 54

10 55 Overview (Brief) ICA Intro ICA of fmri Sorting/Calibration Validation Demo single-subject subject ICA