Modelling with Independent Components

Transcription

1 Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB) Department of Clinical Neurology University of Oxford Christian F Beckmann beckmann@fmriboxacuk IPAM Mathematics in Brain Imaging - 22/07/04

2 Outline Outline 1 Exploratory Data Analysis Principles of EDA Principal Component Analysis Independent Component Analysis 2 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model 3 Application of (P)ICA to FMRI data Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks 4 Tensor-PICA c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

3 Review of the GLM Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis model each measured time-series as a linear combination of signal and noise: x i = Yβ i + η i = β i + If the design matrix does not model all signal, we get wrong inferences! c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

4 Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis Classical vs Exploratory Data Analysis Classical Data Analysis (eg GLM) How well does the model fit the data Problem Data Model Analysis Conclusions results depend on the model test specific hypothesis Exploratory Data Analysis (eg ICA) Is there anything interesting in the data Problem Data Analysis Model Conclusions results depend on the data can give surprising results c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

5 Exploratory Data Analysis techniques Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis try to explain / represent the data by calculating quantities which summerise data by extracting underlying (hidden) variables that are interesting differ in what is considered to be interesting signals which explain (co-)variances (PCA, FDA, FA) signals which have large (co-)variances with eg a design matrix (PLS, CVA) signals which are clustered in space/time (clustering) signals which are statistically independent / maximally non-gaussian (ICA) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

6 Exploratory Data Analysis techniques Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis often are multivariate often provide a multivariate linear decomposition space # maps space time Scan #k FMRI data = time # maps spatial maps X = R r a r b r + η Data are represented as a 2D matrix and decomposed into factor matrices A and B, representing the characteristics of R processes in time and across space c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

7 Principal Component Analysis (PCA) Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis Example finds new variables which are linear combinations of the observed data along axis of maximum variation scan 1 scan 2 w 1 = arg max w =1 E{(w t x) 2 }, w k = arg max w =1 E{(w t (x i<k wt i xw i)) 2 } c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

8 Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis Principal Component Analysis for FMRI calculate the data covariance matrix calculate the full set of Eigenvectors calculate the Eigenimages by projecting the data onto the Eigenvectors Example c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

9 Exploratory Data Analysis Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis Statistical independence and correlation de-correlated signals can still be dependent higher-order statistics (beyond mean and variance) can reveal those dependencies Stone, Trends Cog Sci, 6(2):59 64 (2002) Example y 2 = sin(z) 2 y = sin(z) Plot x vs y r= x = cos(z) high order correlations r= x 2 = cos(z) xy x 2 y c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

10 PCA versus ICA Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis Principal Component Analysis (PCA) finds directions of maximal variance in Gaussian data (uses second-order statistics) Independent Component Analysis (ICA) finds directions of maximal independence in non-gaussian data (higher-order statistics) Gaussian data !1!2!3!4 PCA!5!5!4!3!2! c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

11 PCA versus ICA Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis Principal Component Analysis (PCA) finds directions of maximal variance in Gaussian data (uses second-order statistics) Independent Component Analysis (ICA) finds directions of maximal independence in non-gaussian data (higher-order statistics) non-gaussian data !1!2!3!4 PCA!5!5!4!3!2! c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

12 PCA versus ICA Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis Principal Component Analysis (PCA) finds directions of maximal variance in Gaussian data (uses second-order statistics) Independent Component Analysis (ICA) finds directions of maximal independence in non-gaussian data (higher-order statistics) non-gaussian data !1!2!3!4 ICA!5!5!4!3!2! c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

13 Spatial ICA for FMRI Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis the data is represented as a 2D matrix and decomposed into a set of spatially independent component maps and a set of associated time-courses space # ICs space time Scan #k FMRI data = time # ICs Components Noise-free generative model impose spatial independence McKeown etal, Human Brain Mapping, 6(5): (1998) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

14 The Overfitting Problem Principles of EDA Principal Component Analysis From PCA to ICA Independent Component Analysis Example: visual stimulation, b/w reversing checkerboard (8Hz) GLM results (using FEAT) caused by fitting a noise free model to noisy data in the absence of a noise model, everything is significant! std ICA results (all maps with r > 03 temp corr between time-course and design) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

15 Probabilistic ICA Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model statistical latent variables model: we observe linear mixtures of hidden sources x i = As i + η i If η i N [0, σ 2 Σ i ] we can use voxel-wise pre-whitening (eg Woolrich etal, NeuroImage, 14(6): (2001)) If η i N [0, σ 2 I] then R X AA t + σ 2 I as N, ie for isotropic Gaussian noise the eigenspectrum is raised by σ 2 we can estimate the model order from the Eigenspectrum of the data covariance matrix R X but R X = R XQ for any Q with QQ t = I, ie R X is rotational invariant c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

16 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Rotational invariance: the geometry of PCA and ICA 08 s 2 08 s 2 PC PC 2 s 1 0 s independent, uniformly distributed sources s 1 and s linear mixtures of sources with principal directions c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

17 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Rotational invariance: the geometry of PCA and ICA 08 s 1 IC 1 08 IC 2 s s 1 IC 2 IC 1 04 s PCA solution (projection on to PC 1 and PC 2 ) ICA solution (rotation of PCA solution) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

18 Variance-normalisation Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Example: FMRI resting state data Voxel wise standard deviation need to normalise by the voxel-wise variance this amounts to modelling the spatial covariance matrix as as diagonal: V 1/2 = diag(σ 1,, σ N ) Estimated voxel-wise noise standard deviation (log-scale) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

19 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Probabilistic 15 IllustrationICA (I) 34 Σ i reestimate temporally whitened data original data variance normalised data prior infor mation noise estimate IC maps + unmixing spatially whitened data PPCA estimate model order Rx standard c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004 Mixture Model Modelling with Independent Components

20 Incorporating prior information Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model use regularised PCA (or FDA) to regularise time courses ( Ramsay & Silverman, Functional Data Analysis (1997)) signal+noise sub-space is determined from data cov matrix: Rx = i w i (x i x )(x i x ) t (typically w i = 1 N ) ij w i w j m ij (x i x j )(x i x j ) t + ij w i w j (1 m ij )(x i x j )(x i x j ) t, the matrix M = (m ij ); m ij [0, 1] defines a weighted graph of N nodes: can be used to spatially regularise PPCA c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

21 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Model order selection (Probabilistic PCA) the sample covariance matrix has a Wishart distribution and we can calculate the empirical distribution function for the eigenvalues Everson & Roberts, IEEE TSP, 48(7): (2000) Example: 2 signals in noise AR 0 noise + signal AR 16 noise + signal AR 4 noise + signal null data + signal c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

22 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Model order selection (Probabilistic PCA) Example: 2 signals in noise use a probabilistic PCA model and calculate (approximate) the Bayesian evidence for the model order Minka, TR 514 MIT Media Lab (2000) AR 0 noise + signal AR 16 noise + signal AR 4 noise + signal null data + signal c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

23 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model different Bayesian estimators at different points in the processing chain different estimators give similar results Laplace approximation of the Bayesian evidence is most robust 36Example: Discussion 10 signals in Gaussian noise 98 Key: ( ) Eigenspectrum ( ) Lap ( ) BIC ( ) MDL ( ) AIC original data original data after variancenormalisation after variancenormalisation and adjustment of the eigenspectrum variance-normalised data c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

24 Component estimation Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model estimate an unmixing matrix W = A such that the statistical dependency between the estimated sources ŝ i = Wx i is minimised use (i) a contrast function and (ii) an optimisation technique: kurtosis or cumulants & gradient descent (Jade) maximum entropy & gradient descent (Infomax) neg-entropy & fixed-point iteration (FastICA) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

25 non-gaussianity is interesting Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model original sounds mixtures mixing mixtures are more Gaussian c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

26 Component estimation Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model random mixing results in more Gaussian shaped pdfs (Central Limit Theorem) if an unmixing matrix produces non Gaussian signals, this is unlikely to be a random result use neg-entropy as a measure of non-gaussianity: J (s) = H(s gauss ) H(s) allows for the identification of exactly those source processes which violate standard GLM assumptions can use fast approximations: J (s) p i κ i (E{g i (s)} E{g i (s gauss )}) Hyvärinen & Oja, Neural Computation, 9(7): (1997) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

27 Σ i temporally whitened data Exploratory Data Analysis Probabilistic ICA (II) Estimating the model order Estimating Independent variance Components Statistical Inference on IC maps Full PICA model normalised data infor mation noise estimate + spatially whitened data unmixing IC maps Z stat map standard deviation of η form voxel-wise Z-statistics using the estimated standard deviation of the noise PICA prob map c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

28 Thresholding IC maps Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model estimated maps have been optimised to violate the noise model null-hypothesis test is invalid thresholding based on Z-transforming across the spatial domain gives wrong false-positives rate example histogram and fit to single Gaussian right tail left tail c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

29 Thresholding IC maps Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model under the model: Ŝ ML = Â X = Â AS + Â E, ie the estimated spatial maps contain a linear projection of the noise the distribution of the estimated spatial maps is a mixture distribution use Gaussian / Gamma mixture model for each spatial map s r : p(s r θ) = π r,1 N [s r ; µ r,1, σ 2 r,1] + π r,2 G + [s r µ r,1 ; µ r,2, σ r,2 ] + π r,3 G [ s r + µ r,1 ; µ r,3, σ r,3 ] c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

30 Thresholding IC maps Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model fit using Expectation Maximisation (EM) different ways of thresholding: posterior probabilities, NHT, FDR no multiplecomparison problem Example c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

31 Thresholding IC maps Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model fit using Expectation Maximisation (EM) different ways of thresholding: posterior probabilities, NHT, FDR no multiplecomparison problem Example c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

32 Why Gaussian/Gamma mixtures? Example: Gaussian MM fit (2,3 mixtures) Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Gaussian densities for non-background classes are suboptimal: non-background densities are typically not symmetric c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

33 Why Gaussian/Gamma mixtures? Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Exampe: Gaussian/Gamma MM fit (2,3 mixtures) Gaussian/Gamma MM is more robust wrt specification of the right number of mixtures c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

34 The effect of variance-normalisation Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model IC histogram (without variance-norm) without variance normalisation TP:423 TN:15880 FP:845 FN: IC histogram (with variance-norm) with variance normalisation TP:423 TN:16716 FP:9 FN: c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

35 15 Probabilistic Illustration Independent Component Analysis for FMRI 34 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Σ i reestimate noise estimate IC maps standard deviation of η temporally whitened data + unmixing spatially whitened data original data variance normalised data PPCA estimate model order Rx Z stat map Mixture Model prior infor mation full PICA model implemented as Melodic, part of FMRIB s Software Library (FSL) Beckmann & Smith, IEEE TMI, 23(2): (2004) PICA map prob maps Figure 17: Schematic illustration of the analysis steps involved in estimating the PICA model c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

36 Receiver-Operator Characteristics Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model simulated FMRI data PICA vs GLM at different activation levels and different thresholds plot of true-positives rate vs false-positives rate % Table 41: Temporal accuracy at different activation levels: correlation between the extracted time courses07 from PICA and the true signal time courses07 over 150 runs % Results % 1% 3% 5% vis 033 ± ± ± ± 000 aud 029 ± ± ± ± 000 3% 09 5% Key: ( ) PICA ( ) GLM visual act cluster aud act cluster c CF Beckmann - IPAM: Mathematics in Brain 05 Imaging

37 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Example: visual stimulation, b/w reversing checkerboard (8Hz) GLM results dim-est PICA results (constrained) standard ICA (unconstrained) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

38 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Example: visual stimulation, b/w reversing checkerboard (8Hz) PICA maps show primary visual cortex and V3 (MT) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

39 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Can we still estimate spatially correlated signals? spatial correlation between 2 sources s 1 and s 2 : ρ(s 1, s 2 ) = in the presence of noise: s t 1 s 2 N Var(s 1 ) Var(s 2 ) ρ(s 1 + η 1, s 2 + η 2 ) = s1 ts 2, N Var(s 1 ) + σ1 2 Var(s 2 ) + σ2 2 ie for sparse signals in noise, imposing orthogonality (de-correlating estimated signals) is not necessarily restrictive c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

40 Estimating the model order Estimating Independent Components Statistical Inference on IC maps Full PICA model Example: 2 correlated sources original sources sources with noise de-correlated sources thresholded sources ρ = 05 ρ < 01 ρ = 0 ρ 05 c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

41 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks Investigating the temporal characteristics of the BOLD response pain study: 14 short bursts of painful heat estimated (top) and expected (bottom) temporal response to stimulation GLM result using canonical model GLM result using estimated model Wise & Tracey, 2000 c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

42 Application of (P)ICA to FMRI data Tensor-PICA Summary Detecting artefacts in FMRI data Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks FMRI data contain a variety of source processes Artefactual sources typically have unknown spatial and temporal extent and cannot easily be modelled accurately exploratory techniques do not require a priori knowledge of time-courses and/or spatial maps c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

43 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks slice-dropout (scanner instability) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

44 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks EPI ghost (phantom) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

45 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks EPI ghost (in human subject with head motion) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

46 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks high-frequency noise (mainly in ventricles) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

47 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks head motion c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

48 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks B 0 field inhomogeneity c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

49 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks eye-related artefacts (eyeblink, eyball motion?) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

50 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks wrap-around in FOV due to interaction with the EPI ghost Wrap around c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

51 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks Data from McGonigle etal, NeuroImage, 11: (2000) 33 sessions under visual stimulation - some data was discarded stimulus-correlated motion GLM 15motion estimates FE PICA 1 15 x standard ICA (7 sources with significant temp correlation) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

52 FIGURE 4: Application of (P)ICA to FMRI data nerates considerable acoustic noise by steep gradient pulses at unrestricted Auditory signal as expected vs extracted by regularized MELODIC Tensor-PICA oimaging, unshielded EPI-sounds inevitably activate the auditory system and (averaged over n=30 healthy subjects / 10 repetitions each) Summary etection of concomitant or successive stimulations Instead of vanishing the sion [eg, 1], our goal was to utilize it for the functional neuroimaging of audition hieved by either introducing aberrant gradient switches [2-4], amplitudes or [5] or by sampling physiologically delayed hemodynamic BOLD-responses to the termittently delayed but otherwise unmodified gradient noise [6] Here, we othesis- as well as data-driven detection of modulations in the BOLD signal of the holding read-outs (R/Os) from the gradient-train of an EPI pulse sequence Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks Example: Scanning for the scanner utilise the effective fluctuation of the EPI sequence noise to scan for residual auditory responses in patients Bartsch etal, HBM (2004) METHODS & MATERIAL MODIFIED EPI GRADIENT-TRAIN WITH READ-OUT OMISSIONS & EXPECTED AUDITORY BOLD SIGNAL MODULATIONS: mbers of 1-8 read-outs may were improve of the EPI gradient-train with R/O omissions evoked appropriate significant The modified efitting no scans read by including a temporal derivative Due to the nature grees of autocorrelation between successive R/Os and unknown characteristics signal in theofauditory cortex which were detected in 27 of the 30 healthy subjects (90% I] which was!"#$%&'"()% +,=700 * hearing data-driven(41-96db) analysis by regularized MELODIC seems preferable Even -( when ordinary MELODIC Their extracted time-courses were appropriately directed and highl fluctuations vestibular schwannomas or consecutive postoperative deafness, large-vestibularexemplify the spatial and temporal correspondence of their results for one representat oam as well as muffle erpes-simplex-encephalitis; Figures 4 6), regularized MELODIC was sensitive the heuristic prior that the signal of interest is presumed to vary slo By encoding oid startle motions nd atreliably predict audition (up to 90 db, approximately, compared extracted to standard smoother time-courses (Figure 3) Their correlation with the synthetic regr on) bw=1860hz/px, Thus, normal omittinghearing R/Os from EPI is suited to assess audition in clinical settings within a double-gamma HRF were highly significant and above 074 (Figure 4) Thus, MELO ealthy, time [scan-bins] ut specific task compliance No sophisticated equipment other than the by MR-scanner, FDA principles is suited to optimize the model of the HRF while protecting agains ique; gap=25%) were ed) TRUFI-localizers ence, and fully automated, user-independent data-driven analysis is required model via appropriate model-order selection [9] in the scanner ases of three different bular schwannoma, a FIGURE 4: FIGURE 6: esiduum of a herpesimposing PRIOR TEMPORAL CONTRAINTS TO DATA-DRIVEN PRight ROBABILISTIC INDEPENDENT COMPONENT ANALYSISdeafness vestibular schwannoma / postoperative Herpes-simplex-encephalitis EGULARIZED (T1-w MELODIC SE): post Gadolinium; regularized MELODIC*: p>050) (PD-w SE & T2-w FLAIR; regularized MELODIC*:( R p>050) Within the PICA framework, estimation of maximally non-gaussian sources is based on an Eigenvalue decomposition of the data-covariance-matrix (RX) For FMRI data in general and for this study in particular, the nd consisted of slicefiltering (15 times the signal of interest is assumed to vary rather slowly over time Thus, high-frequencies yield rather non-pausible cified as a stimulus of spatial maps and time-courses, and their estimation is dispensable To favour smooth time-courses, MELODIC (HRF; [7]) The scans was regularized by incorporating principles of functional data analysis (FDA, [8]): Identification of initial Eigenvectors was temporally constraint by a suitably chosen set of regularly spaced (TR/5) cubic B-spline basis gnal in a steady-state functions Thereby, resulting ICA time-courses - estimated as linear combinations of the Eigenvectors - were type of FEAT-design restricted to contain primarily lower frequencies Final PICA maps generated by ordinary as well as regularized persion derivative For MELODIC were thresholded using an alternative hypothesis test based on Gaussian/Gamma mixture models [9] art of FSL) was used FIGURE 2: FIGURE 3: Temporal correlation between time-courses Time-courses [TRs] extracted by ordinary (A) vs regularized MELODIC (B) extracted by ordinary MELODIC and FEAT temporally constrained by smooth principal components (r=093, p<0001) *component of interest overlaid on smoothed data!"#$%&'()* *+&,-&-"$*/0(1*)2,(1"(0+*345 Herpes-simplex-encephalitis c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004 postoperative deafness

53 Application of (P)ICA to FMRI data Tensor-PICA Summary PICA on resting data Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks Example: 1 subject, 3 sessions perform ICA on null data and compare spatial maps between subjects/scans ICA maps depict spatially localised and temporally coherent signal changes that are confounding effects for the GLM c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

54 Application of (P)ICA to FMRI data Tensor-PICA Summary Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks RSN classification (7 normals): 4 consistent maps primary and secondary sensory, anterior insula, pain visual cortex medial occipital visual, lateral occipital, medial parietal posterior parietal, prefrontal: attention, working memory DeLuca etal, ISMRM (2004) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

55 Application of (P)ICA to FMRI data Tensor-PICA Summary Simultaneous EEG/FMRI Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks record single bipolar EEG channel recording during FMRI estimate subject specific alpha power und use for GLM PICA vs GLM Goldman & Cohen, HBM (2003) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

56 Application of (P)ICA to FMRI data Tensor-PICA Summary Simultaneous EEG/FMRI Investigating the BOLD response Artefact detection Estimating difficult activation pattern Investigation into resting-state networks task ERP data temporal ICA FMRI data spatial ICA and project maps to scalp (need forward model only) match ERP and FMRI sources using the scalp spatial maps Loftus etal, HBM (2003) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

57 Application of (P)ICA to FMRI data Tensor-PICA Summary Generative model PARAFAC Tensor-PICA estimation Example Tensor-PICA: multi-way generalisation of PICA subject time space Scan #k FMRI data = time # maps subject # maps space spatial maps x ijk = R r a ir b jr c kr + η ijk Data are represented as a 3D array and decomposed into factor matrices A, B and C c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

58 PARAFAC Application of (P)ICA to FMRI data Tensor-PICA Summary Generative model PARAFAC Tensor-PICA estimation Example as a symmetric least-square problem this is known as PARAFAC (Parallel Factor Analysis) and can be solved using Alternating Least Squares (ALS), ie by iterating least-squares solutions for X i = Bdiag(a i )C t + E i i X j = Cdiag(b j )A t + E j j X k = Adiag(c k )B t + E k k requires system variation (no co-linearity in A, B or C) treats all modes the same c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

59 Application of (P)ICA to FMRI data Tensor-PICA Summary Tensor-PICA: estimation Generative model PARAFAC Tensor-PICA estimation Example rewrite: X IK J = (C A)B t + E can be treated as a 2-stage estimation problem: 1 PICA estimation of B from X IK J by estimating M as the mixing matrix 2 rank-1 Eigen-decomposition of each column M (r), reshaped into a I K matrix, in order to find the underlying factor matrices such that M = (C A) Jointly estimates R modes which describe signal characteristics in the temporal, spatial and subject/session domain c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

60 Application of (P)ICA to FMRI data Tensor-PICA Summary Generative model PARAFAC Tensor-PICA estimation Example Example 10 sessions under motor paradigm (right index finger tapping) 2000 McGonigle etal, NeuroImage 11: , Group-level mixed-effects results associated time course c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

61 Application of (P)ICA to FMRI data Tensor-PICA Summary Generative model PARAFAC Tensor-PICA estimation Example Tensor PICA: primary activation associated time course normalised response over sessions contra-lateral primary motor/sensory; SMA; bi-lateral secondary somatosensory; anterior lobe of cerebellum c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

62 Application of (P)ICA to FMRI data Tensor-PICA Summary Generative model PARAFAC Tensor-PICA estimation Example Tensor PICA: primary de-activation associated time course normalised response over sessions ipsi-lateral primary motor/somatosensory c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

63 Application of (P)ICA to FMRI data Tensor-PICA Summary Generative model PARAFAC Tensor-PICA estimation Example Tensor PICA: artefacts Tensor-PICA associated time course 0 group level GLM normalised response over sessions lower-level GLM (session 9) stimulus-correlated motion (strong in 2 sessions) c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

64 Conclusions Application of (P)ICA to FMRI data Tensor-PICA Summary Conclusions Acknowledgements exploring your data is important in order to get a better understanding don t just look at post-thresholded stats images! model-free analysis is complementary to GLM - make use of it PCA/ICA techniques are easy to use - results are often less easy to interpret, though probabilistic ICA can produce plausible activation maps and associated time-courses c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004

65 Application of (P)ICA to FMRI data Tensor-PICA Summary Acknowledgements Conclusions Acknowledgements MELODIC software available as part of FSL Caitlinn Loftus (HFI Melbourne) Andreas Bartsch (BJMU Würzburg) Robin Goldmann (Columbia Univ) & Mark Cohen (UCLA) Dave McGonigle (CNRS Paris) Pedro Valdés-Sosa (CNC Havana) Stephen Smith, Mark Jenkinson, Mark Woolrich, Marilena DeLuca (FMRIB) GlaxoSmithKline c CF Beckmann - IPAM: Mathematics in Brain Imaging 2004