Cosmo-not: a brief look at methods of analysis in functional MRI and in diffusion tensor imaging...
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Transcript of Cosmo-not: a brief look at methods of analysis in functional MRI and in diffusion tensor imaging...
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Cosmo-not:a brief look at methods of
analysis in functional MRI andin diffusion tensor imaging (DTI)
Paul Taylor
AIMS, UMDNJ
Cosmology seminar, Nov. 2012
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Outline• FMRI and DTI described (briefly)• Granger Causality• PCA• ICA
– Individual, group, covariance networks• Jackknifing/bootstrapping
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The brain in brief (large scales)has many parts-
complexblood vesselsneuronsaqueous tissue(GM, WM, CSF)
activity (examples):hydrodynamicselectrical impulseschemical
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how do different parts/areas work together?A) observe various parts acting together in unison during some
activities (functional relation -> fMRI)B) follow structural connections, esp. due to WM tracts, which affect
random motion in fluid/aqueous tissue (-> DTI, DSI, et al.)
has many parts-complexblood vesselsneuronsaqueous tissue(GM, WM, CSF)
activity (examples):hydrodynamicselectrical impulseschemical
The brain in brief (large scales)
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Biswal et al.(2010, PNAS)
GM ROIs in networks: spatially distinct regions working in concert
Example:Resting statenetworks
Functional (GM)
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Basic fMRI• General topic of functional MRI:
– Segment the brain into ‘functional networks’ for various tasks– Motor, auditory, vision, memory, executive control, etc.– Quantify, track changes, compare populations (HC vs disorder)
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Basic fMRI• General topic of functional MRI:
– Segment the brain into ‘functional networks’ for various tasks– Motor, auditory, vision, memory, executive control, etc.– Quantify, track changes, compare populations (HC vs disorder)
• Try to study which regions have ‘active’ neurons– Modalities for measuring metabolism directly include PET scan
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Basic fMRI• General topic of functional MRI:
– Segment the brain into ‘functional networks’ for various tasks– Motor, auditory, vision, memory, executive control, etc.– Quantify, track changes, compare populations (HC vs disorder)
• Try to study which regions have ‘active’ neurons– Modalities for measuring metabolism directly include PET scan
• With fMRI, use an indirect measure of blood oxygenation
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MRI vs. fMRI
↑ neural activity ↑ blood oxygen ↑ fMRI signal
MRI fMRI
one image
fMRI Blood Oxygenation Level Dependent (BOLD) signal
indirect measure of neural activity
…
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BOLD signal
Source: fMRIB Brief Introduction to fMRI
neural activity ↑ blood flow ↑ oxyhemoglobin ↑ T2* ↑ MR signal
Blood Oxygen Level Dependent signal
time
MxySignal
Mosinθ T2* task
T2* control
TEoptimum
StaskScontrol
ΔS
Source: Jorge Jovicich
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Step 1: Person is told to perform a task,maybe tapping fingers, in a time-varyingpattern:
OFF
ON
30s 30s 30s 30s 30s 30s
Basic fMRI
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Step 2: we measurea signal from eachbrain voxel over time
(example slice oftime series)
signal: basically, local increase in oxygenation: idea thatneurons which are active are hungrier, and demand an increase in food (oxygen)
Basic fMRI
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Step 3: wecompare brainoutput signals tostimulus/inputsignal
looking for: strongsimilarity(correlation)
Basic fMRI
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First Functional Images
Source: Kwong et al., 1992
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Step 4: map out regions of significant correlation (yellow/red)and anti-correlation (blue), which we take to be someinvolved in specific task given (to some degree); these areasare then taken to be ‘functionally’ related networks
Basic fMRI
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Basic fMRI• Have several types of tasks:
– Again: motor, auditory, vision, memory, executive control, etc.– Could investigate network by network…
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Basic fMRI• Have several types of tasks:
– Again: motor, auditory, vision, memory, executive control, etc.– Could investigate network by network…
• Or, has been noticed that correlations among networkROIs exist even during rest– Subset of functional MRI called resting state fMRI (rs-fMRI)– First noticed by Biswal et al. (1995)– Main rs-fMRI signals exist in 0.01-0.1 Hz range– Offer way to study several networks at once
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Basic rs-fMRIe.g., Functional Connectome Project resting state networks (Biswal et al.,
2010):
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Granger Causality• Issue to address: want to find relations
between time series- does one affect anotherdirectly? Using time-lagged relations, can tryto infer ‘causality’ (Granger 1969) (NB: carefulin what one means by causal here…).
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Granger Causality
• Modelling a measured time series x(t) aspotentially autoregressive (first sum) and withtime-lagged contributions of other time seriesy(i)– u(t) are errors/‘noise’ features, and c1 is baseline
• Issue to address: want to find relationsbetween time series- does one affect anotherdirectly? Using time-lagged relations, can tryto infer ‘causality’ (Granger 1969) (NB: carefulin what one means by causal here…).
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Granger Causality• Calculation:
– Residual from:
– Is compared with that of:
– And put into an F-test:
– (T= number of time points, p the lag)– Model order determined with Akaike Info. Criterion or Bayeian
Info. Criterion (AIC and BIC, respectively)
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Granger Causality• Results, for example, in directed graphs:
(Rypma et al. 2006)
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• Principal Component Analysis (PCA): can treatFMRI dataset (3spatial+1time dimensions) as a 2Dmatrix (voxels x time).– Then, want to decompose it into spatial maps (~functional
networks) with associated time series– goal of finding components which explain max/most of
variance of dataset– Essentially, ‘eigen’-problem, use SVD to find
eigenmodes, with associated vectors determining relativevariance explained
PCA
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• To calculate from (centred) dataset M with Ncolumns:– Make correlation matrix:
• C = M MT /(N-1)– Calculate eigenvectors Ei and -values λi from C, and the
principal component is:• PCi = Ei [λI]1/2
PCA
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• To calculate from (centred) dataset M with Ncolumns:– Make correlation matrix:
• C = M MT /(N-1)– Calculate eigenvectors Ei and -values λi from C, and the
principal component is:• PCi = Ei [λI]1/2
• For FMRI, this can yield spatial/temporaldecomposition of dataset, with eigenvectorsshowing principal spatial maps (and associated timeseries), and the relative contribution of eachcomponent to total variance
PCA
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• Graphic example: finding directions of maximumvariance for 2 sources
PCA
(example from web)
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• (go to PCA reconstruction example inaction fromhttp://www.fil.ion.ucl.ac.uk/~wpenny/mbi/)
PCA
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ICA• Independent
component analysis(ICA) (McKeown et al.1998; Calhoun et al.2002) is a method fordecomposing a ‘mixed’MRI signal intoseparate (statistically)independentcomponents.
(McKeown et al. 1998)
(NB: ICA~ known ‘blind source separation’ or ‘cocktail party’ problems)
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ICA• ICA in brief (excellent discussion, see Hyvarinen & Oja 2000):
– ICA basically is undoing Central Limit Theorem• CLT: sum of independent variables with randomness -> Gaussianity• Therefore, to decompose the mixture, find components with
maximal non-Gaussianity– Several methods exist, essentially based on which function is powering
the decomposition (i.e., by what quantity is non-Gaussianity measured):kurtosis, negentropy, pseudo-negentropy, mutual information, max.likelihood/infomax (latter used by McKeown et al. 1998 in fMRI)
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ICA• ICA in brief (excellent discussion, see Hyvarinen & Oja 2000):
– ICA basically is undoing Central Limit Theorem• CLT: sum of independent variables with randomness -> Gaussianity• Therefore, to decompose the mixture, find components with
maximal non-Gaussianity– Several methods exist, essentially based on which function is powering
the decomposition (i.e., by what quantity is non-Gaussianity measured):kurtosis, negentropy, pseudo-negentropy, mutual information, max.likelihood/infomax (latter used by McKeown et al. 1998 in fMRI)
• NB: can’t determine ‘energy’/variances or order of ICs, due toambiguity of matrix decomp (too much freedom to rescalecolumns or permute matrix).– i.e.: relative importance/magnitude of components is not known.
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ICA• Simple/standard representation of matrix
decomposition for ICA of individual dataset:
time ->
voxels ->
=
voxels ->
time ->
# ICs
# ICs
x
Time series of ith component
Spatial map(IC) of ithcomponent
Have to choose number of ICs--often based on ‘knowledge’ of system, or preliminary PCA-variance explained
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ICA• Can do group ICA, with assumptions of some
similarity across a group to yield ‘group level’ spatialmap– Very similar to individual spatial ICA, based on concatenating sets
along time
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ICA• Can do group ICA, with assumptions of some
similarity across a group to yield ‘group level’ spatialmap– Very similar to individual spatial ICA, based on concatenating sets
along time
Subjects and tim
e ->
voxels ->
Subject 1 =
voxels -># ICs
# ICs
x
Time series of ith component, S1
Groupspatial map(IC) of ithcomponent
Subject 2
Subject 3 Time series of ith component, S2
Time series of ith component, S3
Subjects and tim
e ->
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ICA• Group ICA example (visual paradigm)
(Calhoun et al. 2009)
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ICA
(images:Calhoun et al. 2009)
• GLM decomp (~correlation tomodelled/known time course)
vsICA decomp (unknowncomponents-- ‘data driven’,assumptions of indep. sources)
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ICA
(images:Calhoun et al. 2009)
• GLM decomp (~correlation tomodelled/known time course)
vsICA decomp (unknowncomponents-- ‘data driven’,assumptions of indep. sources)
• PCA decomp (ortho.directions of maxvariance; 2nd order)
vsICA decomp (directionsof max independence;higher order)
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Dual Regression
• ICA is useful for finding an individual’s (independent)spatial/temporal maps; also for the ICs which arerepresented across a group.– Dual regression (Beckmann et al. 2009) is a method for taking
that group IC and finding its associated, subject-specific IC.
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Dual Regression
• ICA is useful for finding an individual’s (independent)spatial/temporal maps; also for the ICs which arerepresented across a group.– Dual regression (Beckmann et al. 2009) is a method for taking
that group IC and finding its associated, subject-specific IC.
• 1) ICA decomposition:– >‘group’ time courses
and ‘group’ spatialmaps, independentcomponents (ICs)
(graphics from ~Beckmann et al. 2009)
time
time
time
time
voxels voxels# ICs
# ICs
x
Steps:
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Dual Regression
• 2) Use group ICs asregressors perindividual in GLM– > Time series
associated with thatspatial map
(graphics from ~Beckmann et al. 2009)
time time
voxels
# ICs
# ICs
x
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Dual Regression
• 2) Use group ICs asregressors perindividual in GLM– > Time series
associated with thatspatial map
• 3) GLM regression withtime courses perindividual– > find each subject’s
spatial map of that IC
(graphics from ~Beckmann et al. 2009)
time
time
voxels voxels# ICs
# ICs
x=
time time
voxels
# ICs
# ICs
x
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Covariance networks (in brief)
• Group level analysis tool• Take a single property across whole brain
– That property has different values across brain (persubject) and across subjects (per voxel)
• Find voxels/regions (->network) in which that propertychanges similarly (-> covariance) as one goes fromsubject to subject (-> subject series)
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ICA for BOLD series and FCNsStandard BOLD
analysisSubject series
analysis
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Covariance networks (in brief)
• Group level analysis tool• Take a single property across whole brain
– That property has different values across brain (persubject) and across subjects (per voxel)
• Find voxels/regions (->network) in which that propertychanges similarly (-> covariance) as one goes fromsubject to subject (-> subject series)
• Networks reflect shared information or single influenceat basic/organizational level (discussed further, below).
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Covariance networks (in brief)
• Can use with many different parameters, e.g.:– Mechelli et al. (2005): GMV– He et al. (2007): cortical thickness– Xu et al. (2009): GMV– Zielinski et al. (2010): GMV– Bergfield et al. (2010): GMV– Zhang et al. (2011): ALFF– Taylor et al. (2012): ALFF, fALFF, H, rs-fMRI mean
and std, GMV– Di et al. (2012): FDG-PET
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• A) Start with group of M subjects (for example, fMRI dataset)
Analysis: making subject series
12
3
4 5
+
A
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• A) Start with group of M subjects (for example, fMRI dataset)• B) Calculate a voxelwise parameter, P, producing 3D dataset per subject
Analysis: making subject series
12
3
4 5
+
A
Pi
B
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• A) Start with group of M subjects (for example, fMRI dataset)• B) Calculate a voxelwise parameter, P, producing 3D dataset per subject• C) Concatenate the 3D datasets of whole group (in MNI) to form a 4D ‘subject
series’– Analogous to standard ‘time series’, but now each voxel has M values of P– Instead of i-th ‘time point’, now have i-th subject
Analysis: making subject series
12
3
4 5
+
A
n=12
34
5
C
Pi
B
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Analysis: making subject series
12
3
4 5
+
A
n=12
34
5
C
Pi
B
• A) Start with group of M subjects (for example, fMRI dataset)• B) Calculate a voxelwise parameter, P, producing 3D dataset per subject• C) Concatenate the 3D datasets of whole group (in MNI) to form a 4D ‘subject
series’– Analogous to standard ‘time series’, but now each voxel has M values of P– Instead of i-th ‘time point’, now have i-th subject
• NB: for all analyses, order of subjects is arbitrary and has no effect
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• A) Start with group of M subjects (for example, fMRI dataset)• B) Calculate a voxelwise parameter, P, producing 3D dataset per subject• C) Concatenate the 3D datasets of whole group (in MNI) to form a 4D ‘subject
series’– Analogous to standard ‘time series’, but now each voxel has M values of P– Instead of i-th ‘time point’, now have i-th subject
• NB: for all analyses, order of subjects is arbitrary and has no effect• Can perform usual ‘time series’ analyses (correlation, ICA, etc.) on subject series
Analysis: making subject series
12
3
4 5
+
A
n=12
34
5
C
Pi
B
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Interpreting subject seriescovariance
X1 Y1
Z1
X2 Y2
Z2
X3 Y3
Z3
X4 Y4
Z4
X5 Y5
Z5
Ex.: Consider 3 ROIs (X, Y and Z) in subjects with GMV data Say, values of ROIs X and Y correlate strongly, but neither with Z.
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Interpreting subject seriescovariance
X1 Y1
Z1
X2 Y2
Z2
X3 Y3
Z3
X4 Y4
Z4
X5 Y5
Z5
Ex.: Consider 3 ROIs (X, Y and Z) in subjects with GMV data Say, values of ROIs X and Y correlate strongly, but neither with Z.
--> X and Y form ‘GMV covariance network’
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Interpreting subject seriescovariance
X1 Y1
Z1
Ex.: Consider 3 ROIs (X, Y and Z) in subjects with GMV data Say, values of ROIs X and Y correlate strongly, but neither with Z.
Then, knowing the X-values and one Y-value (since X and Y canhave different bases/scales) can lead us to informed guesses aboutthe remaining Y-values, but nothing can be said about Z-values.
X2 Y2
Z2
X3 Y3
Z3
X4 Y4
Z4
X5 Y5
Z5
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Interpreting subject seriescovariance
X1 Y1
Z1
Then, knowing the X-values and one Y-value (since X and Y canhave different bases/scales) can lead us to informed guesses aboutthe remaining Y-values, but nothing can be said about Z-values.
X2 Y2
Z2
X3 Y3
Z3
X4 Y4
Z4
X5 Y5
Z5
-> ROIs X and Y have information about each other even acrossdifferent subjects, while having little/none about Z.-> X and Y must have some mutual/common influence, which Z maynot.
Ex.: Consider 3 ROIs (X, Y and Z) in subjects with GMV data Say, values of ROIs X and Y correlate strongly, but neither with Z.
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Interpreting covariance networks• Analyzing: similarity of brain structure across subjects.
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Interpreting covariance networks• Analyzing: similarity of brain structure across subjects.• Null hypothesis: local brain structure due to local control, (mainly)
independent of other regions.– -> would observe little/no correlation of ‘subject series’ non-locally
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Interpreting covariance networks• Analyzing: similarity of brain structure across subjects.• Null hypothesis: local brain structure due to local control, (mainly)
independent of other regions.– -> would observe little/no correlation of ‘subject series’ non-locally
• Alt. Hypothesis: can have (1 or many) extended/multi-regioninfluences controlling localities as general feature– -> can observe consistent patterns of properties as correlation of
subject series ‘non-locally’
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Interpreting covariance networks• Analyzing: similarity of brain structure across subjects.• Null hypothesis: local brain structure due to local control, (mainly)
independent of other regions.– -> would observe little/no correlation of ‘subject series’ non-locally
• Alt. Hypothesis: can have (1 or many) extended/multi-regioninfluences controlling localities as general feature– -> can observe consistent patterns of properties as correlation of
subject series ‘non-locally’– -> observed network and property are closely related
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Interpreting covariance networks• Analyzing: similarity of brain structure across subjects.• Null hypothesis: local brain structure due to local control, (mainly)
independent of other regions.– -> would observe little/no correlation of ‘subject series’ non-locally
• Alt. Hypothesis: can have (1 or many) extended/multi-regioninfluences controlling localities as general feature– -> can observe consistent patterns of properties as correlation of
subject series ‘non-locally’– -> observed network and property are closely related– -> one network would have one organizing influence across itself– [-> perhaps independent networks with separate influences might
have low/no correlation; related networks perhaps have somecorrelation].
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Switching gears…
• Statistical resampling: methods for estimatingconfidence intervals for estimates
• Several kinds, two common ones in fMRI arejackknifing and bootstrapping (see, e.g. Efron et al.1982).
• Can use with fMRI, and also with DTI (~for noisyellipsoid estimates-- confidence in fit parameters)
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Jackknifing• Basically, take M acquisitions
e.g., M=12
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Jackknifing
e.g., M=12 MJ=9
[D11 D22 D33 D12 D13 D23] = ....
• Basically, take M acquisitions• Randomly select MJ < M to use
to calculate quantity of interest– standard nonlinear fits
(ellipsoid is defined by 6 parameters of quadratic surface)
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Jackknifing
e.g., M=12 MJ=9
[D11 D22 D33 D12 D13 D23] = ....[D11 D22 D33 D12 D13 D23] = ....[D11 D22 D33 D12 D13 D23] = .... ....
• Basically, take M acquisitions• Randomly select MJ < M to use
to calculate quantity of interest– standard nonlinear fits
• Repeatedly subsample largenumber (~103-104 times)
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Jackknifing• Basically, take M acquisitions• Randomly select MJ < M to use
to calculate quantity of interest– standard nonlinear fits
• Repeatedly subsample largenumber (~103-104 times)
• Analyze distribution of valuesfor estimator (mean) andconfidence interval– sort/%iles
• (not so efficient)– if Gaussian, e.g. µ±2σ
• simple
e.g., M=12 MJ=9
[D11 D22 D33 D12 D13 D23] = ....[D11 D22 D33 D12 D13 D23] = ....[D11 D22 D33 D12 D13 D23] = .... ....
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Jackknifing
M=32 gradients
- quite Gaussian- Gaussianity, σ
increase withdecreasing MJ
- µ changes little
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Jackknifing
M=12 gradients
- not too bad withsmaller M, even
- but could usemin/max fromdistributions for%iles (don’t needto sort)
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Bootstrapping• Similar principal to jackknifing,but need multiple copies of dataset.
e.g., M=12e.g., M=12
e.g., M=12 e.g., M=12
A B
C D
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Bootstrapping• Make an estimate from 12 measures, but randomly selected from
each set:
e.g., M=12e.g., M=12
e.g., M=12 e.g., M=12
A B
C D
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Bootstrapping• Then select another random (complete) set, build a distribution, etc.
e.g., M=12e.g., M=12
e.g., M=12 e.g., M=12
A B
C D
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Summary• There are a wide array of methods applicable to MRI analysis
– Many of them involve statistics and are therefore always believable at facevalue.
– The applicability of the assumptions of the underlying mathematics to thereal situation is always key.
– Often, in MRI, we are concerned with a ‘network’ view of regions workingtogether to do certain tasks.• Therefore, we are interested in grouping regions together per task (as with
PCA/ICA)– New approaches start now to look at temporal variance of networks (using,
e.g., sliding window or wavelet decompositions).– Methods of preprocessing (noise filtering, motion correction, MRI-field
imperfections) should also be considered as part of the methodology.