Unformatted text preview: fMRI Course
Lecture 9: Introduction to Independent Component Analysis Vince D. Calhoun, Ph.D.
Director, Image Analysis & MR Research The Mind Research Network Associate Professor, Electrical and Computer Engineering, Neurosciences & Computer Science The University of New Mexico Overview
• • • • • (Brief) ICA Intro ICA of fMRI Sorting/Calibration Validation Demo singlesubject ICA single 2009 fMRI Course X A S 2 Modeling the Brain?
Results
From “Science with a Smile” by Subramanian Raman Blind Source Separation: The Cocktail Party Problem Modeling Discussion Observations Mixing matrix A Sources • “All models are wrong, but some are useful!” useful!”
• “All models are wrong.” G.E. Box (1976) quoted by Marks Nester in, “An applied statistician’s wrong.” statistician’ creed,” Applied Statistics, 45(4):401410, 1996. creed,” 45(4):401 • “I believe in ignorancebased methods because humans have a lot of ignorance and we ignoranceshould play to our strong suit.” suit.”
• Eric Lander, Whitehead Institute, M.I.T.
3 2009 fMRI Course 4 2009 fMRI Course 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 • Nongaussian sources Uncorrelated: E y1 y2 E y1 E y2 Independent: p y1 , y2 p y1 p y2 ICA vs PCA E h y1 h y2 E h y1 E h y2 PCA finds directions of maximal variance (using second order statistics)
5 2009 fMRI Course ICA finds directions which maximize independence (using higher order statistics) 2009 fMRI Course 6 1 ICA vs. PCA ICA vs. PCA 2009 fMRI Course PCA finds directions of maximal variance (using second order statistics) 7 2009 fMRI Course ICA finds directions which maximize independence (using higher order statistics) 8 ICA Example
Infomax • Mixing simple signals: sinus + chainsaw.
Bell&Sejnowski ‘95 Cardoso ’96 Pearlmutter ’97
(if the nonlinearities in the NN are chosen as the cdf’s) cdf’ Maximum Likelihood
Gaeta ’90, Pearlmutter ‘96 Karhunen ’97 Girolami & Fyfe ‘97 Lee ‘98 Cardoso ’99 Maximum Negentropy
Comon ’94 (represent likelihood by KL distance between observed and factorized density) Nonlinear PCA
Oja ’97, Karhunen & Jautsensalo ’94 Lee ‘98 Blind Deconvolution
Lambert ’96/Bussgangs Lee ’98 Hyvarinen ‘99
(if constrained to be uncorrelated) Min. Mutual Info.
Comon ’94 2009 fMRI Course 2009 From: Chap. 14.6, Friedman, Hastie, Tibshirani: The elements of statistical learning. Tibshirani: 9 2009 fMRI Course 10 ICA Maximizes Nongaussianity FastICA demo (mixtures) mixtures) mean
Y E{Y } variance Y2 E{Y E (Y )}2 skewness 3 Y E Y 3 kurtosis 4 Y E Y 4 3 E Y 2 2 Gaussian supergaussian • Many realworld data sets have realsupergaussian distributions supergaussian
• The random variables take relatively more often values that are very close to zero or are large 2009 fMRI Course 11 2009 fMRI Course 12 2 FastICA demo (whitened) whitened) FastICA demo (step 1) 2009 fMRI Course 13 2009 fMRI Course 14 FastICA demo (step 2) FastICA demo (step 3) 2009 fMRI Course 15 2009 fMRI Course 16 FastICA demo (step 4) FastICA demo (step 5  end) 2009 fMRI Course 17 2009 fMRI Course 18 3 ICA of FMRI
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 Regression Results ˆ ˆ y j 0 i xi j e j i 1
20 M 2009 fMRI Course 19 2009 fMRI Course General Linear Model (GLM)
Voxels Time 2009 fMRI Course 2009 fMRI Course “Activation maps” Corresponding to columns of G The GLM is by far A little more detail
Voxels Time Spatially Independent Components Data(X) = G
Design matrix ˆ β the most common approach to analyzing fMRI data. To use this approach, one needs a model for the fMRI time course Data(X) = R† ˆ W 1 Components (C)
Mixing matrix Time courses {
TC
21
2009 fMRI Course Independent Component Analysis (ICA)
Voxels Time Spatially Independent Components
In spatial ICA, there is no model for the fMRI time course, this is estimated along with the hemodynamic source locations Time courses Data(X) ˆ 1 Components (C) =W
Mixing matrix Time courses
22 ICA Example ICA Halloween (Un)Mixer! (Un)Mixer!
X = A ×
background → Time candle 1 S =
candle 2 candle 3 23 2009 fMRI Course Candle out
24 4 ICA of fMRI
The ICA model assumes the fMRI data, x, is a linear mixture of statistically independent sources, s. GLM ICA +
fMRI data, x Input: Time courses Data Input:
Data x As p s1 , s2 p s1 p s2 Source 1 Time course 1 The goal of ICA is to separate the sources given the mixed data and thus determine the s and A matrices
2009 fMRI Course Output:
A
Activation Maps Output:
Activation Maps (Components) Time Courses s s1 s2 T Source 2 Time course 2 25 2009 fMRI Course 26 ICA of fMRI Data Signal Types
Task related Cardiac Motion Vasomotor oscillation/ High order visual 2009 fMRI Course 27 2009 fMRI Course 28 Motion Artifact Artifact Detection and Reduction
Note: PREPROCESSING MAY DIFFER FOR Art. Hunting Approach Eye movements N/2 Nyquist Ghost Motionrelated signal due to mouth movement from inferior temporal and orbitofrontal regions Hemodynamic Model Source: Christian Beckmann’s “Little Shop of fMRI Horrors”: Beckmann’ Horrors” http://www.fmrib.ox.ac.uk/~beckmann/homepage/academis/littleshop/ http://www.fmrib.ox.ac.uk/~beckmann/homepage/academis/littleshop/ 2009 fMRI Course 29 2009 fMRI Course 30 5 Spatial versus Temporal ICA
• • • Does it matter? Why is spatial ICA more common? Some examples: Temporally and Spatially lowcorrelated Components low SICA SPM TICA 2009 fMRI Course 31 2009 fMRI Course 32 Spatially Dependent Components Temporally Dependent Components SICA SPM TICA SICA SPM TICA 2009 fMRI Course 33 2009 fMRI Course 34 Temporally and Spatially Dependent Components SICA SPM TICA A few points
• ICA is not modelfree! (but does make no modelassumptions about shape of time course) • Fishing vs. Hypothesizing
• ICA is a datadriven approach, but that does not mean datayou should automatically associate it with fishing (flexible analysis vs. hypothesisbased study) analysis hypothesisstudy • The key here is what are the questions being asked, and what is the approach that will be used to answer these questions • There are tools available for asking focused questions about an ICA of fMRI analysis 2009 fMRI Course 35 2009 fMRI Course 36 6 Uses of ICA
• Improving fit to taskrelated components task• Find areas of ‘activation’ which respond in a more activation’ complex way to an external stimulus • Artifact Reduction/Filtering • Examination of temporally coherent, but not necessarily taskrelated components task• Data exploration of unpredicted structure Sorting 2009 fMRI Course 37 2009 fMRI Course 38 Ambiguities of ICA: Sorting/Scaling
• ICA is modeling the data as a linear combination of images and time courses • Why is sorting necessary?
• Permutation ambiguity: X=AS=(AP1)(PS) Types of Sorting
• Temporal Sorting
• Correlation • Multiple regression • Others? (skew, kurtosis, power spectra, etc.) • Why is scaling/calibration necessary?
• Scaling ambiguity: X=AS=(AM1)(MS) • Spatial Sorting
• Correlation (w/ mask or SPM) • Maximum value (w/i mask) (w/i • Multiple regression data tc1 * im1 tc2 * im2 E
1 1 data (a * tc1 ) * * im1 (b * tc2 ) * * im2 E a b 2009 fMRI Course • Multivariate sorting Multi• SVM Approaches (Formisano) (Formisano) 39 2009 fMRI Course 40 Example + Right
+ Left
t (secs)
+ 0 90 180 270 360 2009 fMRI Course 41 2009 fMRI Course 42 7 Number of Components
• • • Too many > oversplitting of the components overToo few > overclumping of the components overHow 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) • PostICA clustering is also used to address this issue Post Number of Components (Order Selection)
1 1 ˆ MDL N M K N L N 1 NK N 1 ln M 2 2 1 AIC N 2 M K N L ˆN 2 1 NK N 1 2 1 N 1...K K N ˆ L N ln 1 N 1 ... K KN 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. 140151, 2001.] Correction for correlated samples [Y. Li, T. Adali, and V. D. Calhoun, "Sample Dependence Correction For Order Selection In FMRI Analysis," in Proc. ISBI, Washington, D.C., 2006.]
2009 fMRI Course 43 2009 fMRI Course 44 Validation
• Algorithm Differences
• Esposito F, Formisano E, Seifritz E, Goebel R, Morrone R, Tedeschi G, Tedeschi Salle FD. 2002. Spatial Independent Component Analysis of Functional MRI TimeSeries: to What Extent Do Results Depend on the Algorithm TimeUsed? Hum Brain Map 16:14957. 16:149 • Algorithm & Preprocessing Differences
• Calhoun, V. D., Adali, T., and Pearlson, G. D. 2004. Independent Components Analysis Applied to fMRI Data: A Generative Model for Validating Results. Journal of VLSI Signal Proc Systems. vol. 37, pp. 281291. 281 • Cluster Validation
• Himberg J, Hyvarinen A, Esposito F. 2004. Validating the Independent Components of Neuroimaging Time Series Via Clustering and Visualization. Neuroimage 22(3):121422. 22(3):1214 • Test/retest Performance
• Nybakken GE, Quigley MA, Moritz CH, Cordes D, Haughton VM, Meyerand ME. 2002. TestRetest Precision of Functional Magnetic TestResonance Imaging Processed With Independent Component Analysis. Neuroradiology 44(5):4036. 44(5):403 2009 fMRI Course 45 2009 fMRI Course 46 “Hybrid” fMRI Experiment Hybrid” Impact of preprocessing/algorithms/etc
Criterion: KullbackLeibler (KL) Kullbackdivergence p ξ D (s u) ps ξ ln s p ξ dξ u Ground Truth • Define sources • Generate sources • For all:
• • • • • Add noise Smooth Reduce (PCA, cluster, etc.) Unmix (Info., fastICA, jade, etc.) Evaluate (KL) Mixed with fMRI Data • min(KL) is winner
V.D.Calhoun, T.Adali, and G.D.Pearlson, "Independent Components Analysis Applied to FMRI Data: A V.D.Calhoun, T.Adali, G.D.Pearlson, Generative Model for Validating Results," Journal of VLSI Signal Proc. Systems, 2004. 2009 fMRI Course 47 2009 fMRI Course 48 8 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, 2005. N. Correa, T. Adali, and V. D. Calhoun "Performance of Blind Source Separation Algorithms for fMRI Analysis," Mag.Res.Imag., 2006 (submitted).
2009 fMRI Course 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, 2005. N. Correa, T. Adali, and V. D. Calhoun "Performance of Blind Source Separation Algorithms for fMRI Analysis," Mag.Res.Imag., 2006 (submitted).
49
2009 fMRI Course 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).
2009 fMRI Course 51 2009 fMRI Course 52 A Few Software Packages
• The ICA:DTU toolbox (http://mole.imm.dtu.dk/toolbox/ica/index.html) http://mole.imm.dtu.dk/toolbox/ica/index.html
• • • matlab three different ICA algorithms fMRI specific with demo data Group ICA of fMRI Toolbox (GIFT)
Left Hemisphere Visual Stimuli Onset • FMRLAB (http://www.sccn.ucsd.edu/fmrlab/) (http://www.sccn.ucsd.edu/fmrlab/
• • • matlab infomax algorithm fMRI specific with additional tools • FMRIB Software Library, which includes the ICA tool MELODIC (http://www.fmrib.ox.ac.uk/analysis/research/mel odic/):
• • • C FastICA+ Complete Package • ICALAB
• • • matlab multiple ICA algorithms not fMRI specific although one fMRI example included • GIFT (http://icatb.sourceforge.net) (http://icatb.sourceforge.net
• • • • • matlab >9 ICA algorithms (more coming) including infomax and fastICA Constrained ICA algorithms Visualization tools and sorting options. Sample data and a stepbystep walk through stepby • AnalyzeFMRI (http://www.stats.ox.ac.uk/~marchini/software.ht ml)
• • R FastICA Voxel 600+ unique downloads
http://icatb.sourceforge.net
Funded by NIH 1 R01 EB 000840 (to V. Calhoun) Major Contributors: Tülay Adalı – University of Maryland Andrzej Cichocki, RIKEN, Japan Jim Pekar – Johns Hopkins Hichem Snoussi – IRCCyN, France
53
2009 fMRI Course • BrainVoyager(http://www.brainvoyager.com/) BrainVoyager(http://www.brainvoyager.com/
• • • Commercial FastICA Complete Package BOLD Signal 2009 fMRI Course 54 9 Demo
• Singlesubject ICA Single• Component Estimation • Defaults file
• Detrend level • Scaling • • • • • • Sorting Component Explorer Orthogonal Viewer Composite Viewer Examine Regression Parameters Artifact Removal Tool 2009 fMRI Course 55 2009 fMRI Course 56 10 ...
View
Full Document
 Spring '10
 Cudeback
 Independent Component Analysis, fMRI Course

Click to edit the document details