Model-free Functional Data Analysis MELODIC Multivariate Exploratory Linear Optimised Decomposition into Independent Components
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(2) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Model-based (GLM) analysis. =. +. • model each measured time-series as a linear combination of signal and noise. • If the design matrix does not capture every signal, we typically get wrong inferences!. Data Analysis. Confirmatory. • “How well does my model fit to the data?”. Problem. Model. Exploratory. • “Is there anything. interesting in the data?”. Data. Problem. Analysis. Analysis. Results. • results depend on the model. Data. Model. Results. • can give unexpected results. EDA techniques. • try to explain / represent the data. • by calculating quantities that summarise the data. • by extracting underlying hidden features that are interesting . • differ in what is considered interesting . • are localised in time and/or space (Clustering). • explain observed data variance (PCA, FDA, FA). • are maximally independent (ICA). • typically are multivariate and linear. CF Beckmann: MELODIC. 2.
(3) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. EDA techniques for FMRI. • are mostly multivariate. • often provide a multivariate linear decomposition: space. space. # maps" # maps". time". time". Scan #k . =". FMRI data". spatial maps. Data is represented as a 2D matrix and decomposed into factor matrices (or modes). What are components?. +. ≈. • express observed data. ×. +. ×. ×. as linear combination of spatio-temporal processes. • techniques differ in the way data is represented by components . PCA for FMRI. 1. assemble all (de-meaned) data as a matrix. 2. calculate the data covariance matrix. 3. calculate SVD. 4. project data onto the Eigenvectors. CF Beckmann: MELODIC. 3.
(4) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. PCA for FMRI. 1. assemble all (de-meaned) data as a matrix. 2. calculate the data covariance matrix. 3. calculate SVD. 4. project data onto the Eigenvectors. PCA for FMRI. # PCs". time". 1. assemble all (de-meaned) data as a matrix. 2. calculate the data covariance matrix. 3. calculate SVD. 4. project data onto the Eigenvectors. PCA for FMRI. 1. assemble all (de-meaned) data as a matrix. 2. calculate the data covariance matrix. 3. calculate SVD. 4. project data onto the Eigenvectors. CF Beckmann: MELODIC. 4.
(5) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Correlation vs. independece. • de-correlated signals. can still be dependent. • higher-order statistics (beyond mean and variance) can reveal these dependencies. Stone et al. 2002. PCA vs. ICA. • PCA finds projections of. maximum amount of variance in Gaussian data (uses 2nd order statistics only). • Independent Component Analysis (ICA) finds projections of maximal independence in nonGaussian data (using higher-order statistics). non-Gaussian data Gaussian data. Spatial ICA for FMRI. space. space. # ICs". =". # ICs". FMRI data". time". time". Scan #k . spatial maps. impose spatial. independence. • data is decomposed into a set of spatially independent maps and a set of time-courses. McKeown et al. . HBM 1998. CF Beckmann: MELODIC. 5.
(6) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. PCA vs. ICA ?. Simulated. Data. (2 components, slightly different timecourses). PCA. • Timecourses. orthogonal. • Spatial maps and timecourses “wrong”. ICA. • Timecourses non-. co-linear. • Spatial maps and timecourses “right”. The overfitting problem. • fitting a noise-free model to noisy observations:. • no control over signal vs. noise (non-interpretable results). • statistical significance testing not possible . GLM analysis . standard ICA (unconstrained). Probabilistic ICA model. • statistical. latent variables model: we observe linear. mixtures of hidden sources in the presence of Gaussian noise. • Issues:. • Model Order Selection: how many components?. • Component estimation: how to find ICs?. • Inference: how to threshold ICs?. CF Beckmann: MELODIC. 6.
(7) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Model Order Selection. • The sample. covariance matrix has a Wishart distribution and we can calculate the empirical distribution function for the eigenvalues. Everson & Roberts, IEEE Trans. Sig. Proc. 48(7), 2000. Empirical distribution function. observed Eigenspectrum. optimal fit. over-fitting. under-fitting. Model Order Selection. 15. under-fitting: the amount of explained data variance is insufficient to obtain good estimates of the signals. 33 optimal fitting: the amount of explained data variance is sufficient to obtain good estimates of the signals while preventing further splits into spurious components. 165. #components observed Eigenspectrum of the data covariance matrix Laplace approximation of the posterior probability of the model order theoretical Eigenspectrum from Gaussian noise. over-fitting: the inclusion of too many components leads to fragmentation of signal across multiple component maps, reducing the ability to identify the signals of interest. Variance Normalisation. • we might choose to. ignore temporal autocorrelation in the EPI time-series but:. • generally need to. normalise by the voxel-wise variance. Estimated voxel-wise std. deviation (log-scale) from FMRI data obtained under rest condition.. CF Beckmann: MELODIC. 7.
(8) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. PCA tissue-type bias. voxel-wise temporal standard deviations. (log) - frequency of voxels appearing in major PCA space. 100. 3. 0. 0. • Without normalisation PCA is biased towards tissue exhibiting strong temporal variation. ICA estimation. • need to find an unmixing matrix such that the dependency between estimated sources is minimised. • need (i) a contrast (objective/cost) function to drive the unmixing which measures statistical independence and (ii) an optimisation technique:. ‣ ‣ ‣ . kurtosis or cumulants & gradient descent (Jade) . maximum entropy & gradient descent (Infomax) . neg-entropy & fixed point iteration (FastICA). Non-Gaussianity. mixing. sources. CF Beckmann: MELODIC. mixtures. 8.
(9) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Non-Gaussianity. mixing. non-Gaussian. Gaussian. ICA estimation. • Random mixing results in more Gaussian-shaped PDFs (Central Limit Theorem). • conversely:. if mixing matrix produces less Gaussian-shaped PDFs this is unlikely to be a random result. ➡ . measure non-Gaussianity. • can use neg-entropy as a measure of nonGaussianity . Hyvärinen & Oja 1997. Thresholding. • classical null-hypothesis testing is invalid. • After estimation, the spatial maps are a projection of data on to the unmixing matrix . • data is assumed to be a linear combination of signals and noise. • the distribution of the. estimated spatial maps is a mixture distribution!. CF Beckmann: MELODIC. 9.
(10) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Alternative Hypothesis Test. • use Gaussian/Gamma. mixture model fitted to the histogram of intensity values (using EM). Full PICA model. Beckmann and Smith. IEEE TMI 2004. Probabilistic ICA. GLM analysis . standard ICA (unconstrained). probabilistic ICA. • designed to address the overfitting problem :. • tries to avoid generation of spurious results. • high spatial sensitivity and specificity. CF Beckmann: MELODIC. 10.
(11) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Applications. EDA techniques can be useful to. • investigate the BOLD response. • estimate artefacts in the data . • find areas of activation which respond in a nonstandard way. • analyse data for which no model of the BOLD response is available. Applications. EDA techniques can be useful to. ‣ investigate the BOLD response. • estimate artefacts in the data . • find areas of activation which respond in a nonstandard way. • . analyse data for which no model of the BOLD response is available. Investigate BOLD response. estimated signal time course. standard hrf model. CF Beckmann: MELODIC. 11.
(12) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Applications. EDA techniques can be useful to. • investigate the BOLD response. ‣ estimate artefacts in the data . • find areas of activation which respond in a nonstandard way. • . analyse data for which no model of the BOLD response is available. Artefact detection. • FMRI data contain a variety of source processes. • Artifactual 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 spatial maps. slice drop-outs. CF Beckmann: MELODIC. 12.
(13) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. gradient instability. EPI ghost. high-frequency noise. CF Beckmann: MELODIC. 13.
(14) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. head motion. field inhomogeneity. eye-related artefacts. Wrap around. CF Beckmann: MELODIC. 14.
(15) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Structured Noise and the GLM. • . structured noise appears:. • in the GLM residuals - inflates variance estimates (more false negatives). • in the parameter estimates (more false positives and/or false negatives). • In either case leads to suboptimal estimates and wrong inference!. Structured noise and GLM Z-stats bias. • Correlations of the. noise time courses with typical FMRI regressors can cause a shift in the histogram of the Zstatistics . • Thresholded maps. will have wrong false-positive rate . CF Beckmann: MELODIC. 15.
(16) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Denoising FMRI. before denoising. • Example:. left vs right hand finger tapping. LEFT - RIGHT RIGHT. LEFT contrast. after denoising. Apparent variability. McGonigle et al.: 33 Sessions under motor paradigm. ‘de-noising’ data by regressing out noise:. reduced ‘apparent’ session variability . Applications. EDA techniques can be useful to. • investigate the BOLD response. • estimate artefacts in the data . ‣ find areas of activation which respond in a nonstandard way. • . analyse data for which no model of the BOLD response is available. CF Beckmann: MELODIC. 16.
(17) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Example of nonstandard temporal response. • use EPI readout gradient noise to evoke auditory responses in patients before cochlear implant surgery . Bartsch et al. HBM 2004. Applications. EDA techniques can be useful to. • investigate the BOLD response. • estimate artefacts in the data . • find areas of activation which respond in a nonstandard way. ‣ analyse data for which no model of the BOLD response is available. PICA on resting data. • perform ICA on null data. and compare spatial maps between subjects/scans. • ICA maps depict spatially localised and temporally coherent signal changes . Example: ICA maps 1 subject at 3 different sessions. CF Beckmann: MELODIC. 17.
(18) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Multi-Subject ICA. Multi-level GLMs. group. subject 1. subject 2. .... subject N. Higher-level:. between subjects. .... First-level: . within session. Different ICA models. Single-Session ICA. each ICA component comprises:. spatial map & timecourse. Multi-Session or Multi-Subject ICA:. Concatenation approach. each ICA component comprises:. spatial map & timecourse. (that can be split up into subject-specific chunks). Multi-Session or Multi-Subject ICA:. Tensor-ICA approach. each ICA component comprises:. spatial map, session-long-timecourse. & subject-strength plot. CF Beckmann: MELODIC. 18.
(19) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Different ICA models. Single-Session ICA. each ICA component comprises:. spatial map & timecourse. Multi-Session or Multi-Subject ICA:. Concatenation approach. good when:. each subject has DIFFERENT timeseries. e.g. resting-state FMRI . Multi-Session or Multi-Subject ICA:. Tensor-ICA approach. good when:. each subject has SAME timeseries. e.g. activation FMRI. ICA Group analysis. #. su. bj. ec. ts ". Extend single ICA to higher dimensions. ts ". ⊗. # ICs". Scan #k . =". FMRI data". space. # ICs". time". time". #. su. bj. ec. space space. spatial maps. Tri-linear model. Bi-linear model. Tensor-ICA multi-session example. • 10 sessions under. motor paradigm (right index finger tapping) . • Group-level GLM results (mixedeffects). McGonigle et al.. NI 2000. CF Beckmann: MELODIC. 19.
(20) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Tensor-ICA multi-session example. • tensor-ICA map. Tensor-ICA multi-session example. • estimated. de-activation . Tensor-ICA multi-session example. • residual stimulus. correlated motion. tensor-PICA map. Z-stats map (session 9). CF Beckmann: MELODIC. Group-GLM (ME) Z-stats map. 20.
(21) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Tensor-ICA multi-subject example. • reaction-time task (index vs. sequential vs. random finger movement. • r > 0.84 regression fit of GLM model against time course. • GLM on associated timecourse shows significant (p<0.01) differences: I < S < R. Tensor-ICA. • simultaneously estimates processes in the temporal & spatial & session/subject domain. • full mixed-effects model. • can use parametric stats on estimated time courses and session/subject modes. • robust statistics: can deal with outlier processes. • MELODIC 3. ICA-based methodology for multi-subject RSN analysis. • Why not just run ICA on each subject separately?. • Correspondence problem (of RSNs across subjects). • Different splittings sometimes caused by small changes in the data (naughty ICA!). • Instead - start with a group-average ICA. • But then need to relate group maps back to the individual subjects. CF Beckmann: MELODIC. 21.
(22) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Methodology for multi-subject RSN analysis 1: Concat-ICA. • Concatenate all subjects data temporally • Then run ICA. • More appropriate than tensor ICA (for RSNs). (Actually: group-based PCA reduction on each subject, then concat, then PCA reduction). #components. voxels. . .. .. . .. .. =. voxels. ×. #components. group mixing matrix. time. time. Subject 2. Subject 1. group ICA maps. time. Methodology for multi-subject RSN analysis2: Dual Regression. • For each subject separately:. • Take all Concat-ICA spatial maps and use with the. subject s data in GLM: Gives subject-specific timecourses. Take these timecoures and use with the subject s data in GLM: Gives subject-specific spatial maps (that look similar to the group maps but are subjectspecific). • . • Can now do voxelwise testing across subjects, separately for each original group ICA map. • Can choose to look at strength-and-shape differences. Methodology for multi-subject RSN analysis2: Dual Regression. CF Beckmann: MELODIC. ×. time courses 1. voxels. #components. time. =. time courses 1. time. FMRI data 1. time. #components. #components. voxels. =. group ICA maps. 4D data to find subjectspecific spatial ICs associated with the group ICs. voxels. • regress these back into the. FMRI data 1. subject s 4D data to find subject-specific timecourses. #components. voxels. time. • Regress spatial ICs into each. ×. spatial maps 1. 22.
(23) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. Methodology for multi-subject RSN analysis2: Dual Regression. • Collect maps and perform voxel-wise test (e.g. randomisation test on GLM). #EVs. =. voxels. #EVs. #subjects. collected components. #subjects. voxels. ×. group difference . . . .. • Can now do voxelwise testing across subjects, separately for each original group ICA map. • Can choose to look at strength-and-shape differences. 1. Separate-subject ICAs. E.g. SOG-ICA - BrainVoyager (Esposito, NeuroImage, 2005). • Separate ICA for each subject. • can add robustness via multiple runs ( ICASSO ). • Cluster components across subjects, to achieve robust matching of any given RSN across subjects. • Good:. Keeps benefits of single-subject ICA - better modelling/ ignoring of structured noise in data. • Bad:. Correspondence problem, in particular different splittings in different subjects caused by even small changes in the data; PCA bias!. 2. Group-ICA and Back-Projection. E.g. GICA - GIFT (Calhoun, HBM, 2001). • Separate PCA (dimensionality reduction) for each subject. • Concat PCA-output across subjects and do group-ICA. • Back-project (invert) ICA results to get individual subject maps. • Good: No correspondence problem. • Bad: . • Lose benefits of single-subject ICA. • PCA-bias. CF Beckmann: MELODIC. 23.
(24) SPM 2011 — 14. Kurs zur funktionellen Bildgebung. 22 Sept. 2011. 3. Group-ICA and Dual-Regression. E.g. MELODIC+dual-reg - FSL (Beckmann, OHBM, 2009). • Group-average PCA (dimensionality reduction). • Project each subject onto reduced group-average-PCA-space. • Concat PCA-output across subjects and do group-ICA. • Regress group-ICA maps onto individual subject datasets to get individual subject maps. • Good: No correspondence problem, no group/PCA bias. • Bad: Lose benefits of single-subject ICA. That s all folks. CF Beckmann: MELODIC. 24.
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