1. Cosmo-not:
a brief look at methods of
analysis in functional MRI and
in diffusion tensor imaging (DTI)
Paul Taylor
AIMS, UMDNJ
Cosmology seminar, Nov. 2012

2. Outline
• FMRI and DTI described (briefly)
• Granger Causality
• PCA
• ICA
– Individual, group, covariance networks
• Jackknifing/bootstrapping

3. The brain in brief (large scales)
has many parts-
complex
blood vessels
neurons
aqueous tissue
(GM, WM, CSF)
activity (examples):
hydrodynamics
electrical impulses
chemical

4. The brain in brief (large scales)
has many parts-
complex
blood vessels
neurons
aqueous tissue
(GM, WM, CSF)
activity (examples):
hydrodynamics
electrical impulses
chemical
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.)

5. Functional (GM)
Example:
Resting state
networks
Biswal et al.
(2010, PNAS)
GM ROIs in networks: spatially distinct regions working in concert

6. 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)

7. 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

8. 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

9. MRI vs. fMRI
MRI fMRI
one image
…
fMRI
Blood Oxygenation Level Dependent (BOLD) signal
indirect measure of neural activity
↑ neural activity  ↑ blood oxygen  ↑ fMRI signal

10. BOLD signal
Blood Oxygen Level Dependent signal
neural activity  ↑ blood flow  ↑ oxyhemoglobin  ↑ T2*  ↑ MR signal
Mxy
Signal
Mo
sinθ T2* task
T2* control
Stask
Scontrol ΔS
TEoptimum time
Source: fMRIB Brief Introduction to fMRI Source: Jorge Jovicich

11. Basic fMRI
Step 1: Person is told to perform a task,
maybe tapping fingers, in a time-varying
pattern:
ON
OFF
30s 30s 30s 30s 30s 30s

12. Basic fMRI
Step 2: we measure
a signal from each
brain voxel over time
signal: basically,
local increase in
oxygenation: idea that
neurons which are
active are hungrier, and
demand an increase in
food (oxygen)
(example slice of
time series)

13. Basic fMRI
Step 3: we
compare brain
output signals to
stimulus/input
signal
looking for: strong
similarity
(correlation)

14. First Functional Images
Source: Kwong et al., 1992

15. Basic fMRI
Step 4: map out regions of significant correlation (yellow/red)
and anti-correlation (blue), which we take to be some
involved in specific task given (to some degree); these areas
are then taken to be ‘functionally’ related networks

16. Basic fMRI
• Have several types of tasks:
– Again: motor, auditory, vision, memory, executive control, etc.
– Could investigate network by network…

17. 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 network
ROIs 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

18. Basic rs-fMRI
e.g., Functional Connectome Project resting state networks (Biswal et al.,
2010):

19. Granger Causality
• Issue to address: want to find relations
between time series- does one affect another
directly? Using time-lagged relations, can try
to infer ‘causality’ (Granger 1969) (NB: careful
in what one means by causal here…).

20. Granger Causality
• Issue to address: want to find relations
between time series- does one affect another
directly? Using time-lagged relations, can try
to infer ‘causality’ (Granger 1969) (NB: careful
in what one means by causal here…).
• Modelling a measured time series x(t) as
potentially autoregressive (first sum) and with
time-lagged contributions of other time series
y(i)
– u(t) are errors/‘noise’ features, and c1 is baseline

21. 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)

22. Granger Causality
• Results, for example, in directed graphs:
(Rypma et al. 2006)

23. PCA
• Principal Component Analysis (PCA): can treat
FMRI dataset (3spatial+1time dimensions) as a 2D
matrix (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 relative
variance explained

24. PCA
• To calculate from (centred) dataset M with N
columns:
– 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

25. PCA
• To calculate from (centred) dataset M with N
columns:
– 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/temporal
decomposition of dataset, with eigenvectors
showing principal spatial maps (and associated time
series), and the relative contribution of each
component to total variance

26. PCA
• Graphic example: finding directions of maximum
variance for 2 sources
(example from web)

27. PCA
• (go to PCA reconstruction example in
action from
http://www.fil.ion.ucl.ac.uk/~wpenny/mbi/)

28. ICA
• Independent
component analysis
(ICA) (McKeown et al.
1998; Calhoun et al.
2002) is a method for
decomposing a ‘mixed’
MRI signal into
separate (statistically)
independent
components.
(NB: ICA~ known ‘blind
source separation’ or
‘cocktail party’ problems) (McKeown et al. 1998)

29. 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)

30. 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 to
ambiguity of matrix decomp (too much freedom to rescale
columns or permute matrix).
– i.e.: relative importance/magnitude of components is not known.

31. ICA
• Simple/standard representation of matrix
decomposition for ICA of individual dataset:
voxels -> # ICs voxels ->
# ICs
time ->
time ->
= x
Spatial map
(IC) of ith
component
Time series of ith component
Have to choose number of ICs--often based on ‘knowledge’
of system, or preliminary PCA-variance explained

32. ICA
• Can do group ICA, with assumptions of some
similarity across a group to yield ‘group level’ spatial
map
– Very similar to individual spatial ICA, based on concatenating sets
along time

33. ICA
• Can do group ICA, with assumptions of some
similarity across a group to yield ‘group level’ spatial
map
– Very similar to individual spatial ICA, based on concatenating sets
along time
voxels -> # ICs voxels ->
# ICs
Subjects and time ->
Subjects and time ->
Subject 1 = x
Group
spatial map
Subject 2 Time series of ith component, S1 (IC) of ith
component
Subject 3 Time series of ith component, S2
Time series of ith component, S3

34. ICA
• Group ICA example (visual paradigm)
(Calhoun et al. 2009)

35. ICA
• GLM decomp (~correlation to
modelled/known time course)
vs
ICA decomp (unknown
components-- ‘data driven’,
assumptions of indep. sources)
(images:Calhoun et al. 2009)

36. ICA
• GLM decomp (~correlation to • PCA decomp (ortho.
modelled/known time course) directions of max
vs variance; 2nd order)
ICA decomp (unknown vs
components-- ‘data driven’, ICA decomp (directions
assumptions of indep. sources) of max independence;
higher order)
(images:Calhoun et al. 2009)

37. Dual Regression
• ICA is useful for finding an individual’s (independent)
spatial/temporal maps; also for the ICs which are
represented across a group.
– Dual regression (Beckmann et al. 2009) is a method for taking
that group IC and finding its associated, subject-specific IC.

38. Dual Regression
• ICA is useful for finding an individual’s (independent)
spatial/temporal maps; also for the ICs which are
represented across a group.
– Dual regression (Beckmann et al. 2009) is a method for taking
that group IC and finding its associated, subject-specific IC.
Steps:
• 1) ICA decomposition: voxels # ICs voxels
time
# ICs
time
– >‘group’ time courses
and ‘group’ spatial x
time
maps, independent
time
components (ICs)
(graphics from ~Beckmann et al. 2009)

39. Dual Regression
• 2) Use group ICs as time # ICs time
regressors per
# ICs
voxels
individual in GLM x
– > Time series
associated with that
spatial map
(graphics from ~Beckmann et al. 2009)

40. Dual Regression
• 2) Use group ICs as time # ICs time
regressors per
# ICs
voxels
individual in GLM x
– > Time series
associated with that
spatial map
• 3) GLM regression with voxels # ICs voxels
time courses per
# ICs
time
time
individual = x
– > find each subject’s
spatial map of that IC
(graphics from ~Beckmann et al. 2009)

41. Covariance networks (in brief)
• Group level analysis tool
• Take a single property across whole brain
– That property has different values across brain (per
subject) and across subjects (per voxel)
• Find voxels/regions (->network) in which that property
changes similarly (-> covariance) as one goes from
subject to subject (-> subject series)

42. ICA for BOLD series and FCNs
Standard BOLD Subject series
analysis analysis

43. Covariance networks (in brief)
• Group level analysis tool
• Take a single property across whole brain
– That property has different values across brain (per
subject) and across subjects (per voxel)
• Find voxels/regions (->network) in which that property
changes similarly (-> covariance) as one goes from
subject to subject (-> subject series)
• Networks reflect shared information or single influence
at basic/organizational level (discussed further, below).

44. 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

45. Analysis: making subject series
• A) Start with group of M subjects (for example, fMRI dataset)
A
2
1 3
+
4 5

46. Analysis: making subject series
• A) Start with group of M subjects (for example, fMRI dataset)
• B) Calculate a voxelwise parameter, P, producing 3D dataset per subject
A B
2
1 3
Pi
+
4 5

47. Analysis: making subject series
• 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
A B C
2
1 3 n=1
Pi
2
+ 3
4
4 5 5

48. Analysis: making subject series
• 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
A B C
2
1 3 n=1
Pi
2
+ 3
4
4 5 5

49. Analysis: making subject series
• 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
A B C
2
1 3 n=1
Pi
2
+ 3
4
4 5 5

50. Interpreting subject series
covariance
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.
X1 Y1 X2 Y2 X3 Y3 X4 Y4 X5 Y5
Z1 Z2 Z3 Z4 Z5

51. Interpreting subject series
covariance
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.
X1 Y1 X2 Y2 X3 Y3 X4 Y4 X5 Y5
Z1 Z2 Z3 Z4 Z5
--> X and Y form ‘GMV covariance network’

52. Interpreting subject series
covariance
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.
X1 Y1 X2 Y2 X3 Y3 X4 Y4 X5 Y5
Z1 Z2 Z3 Z4 Z5
Then, knowing the X-values and one Y-value (since X and Y can
have different bases/scales) can lead us to informed guesses about
the remaining Y-values, but nothing can be said about Z-values.

53. Interpreting subject series
covariance
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.
X1 Y1 X2 Y2 X3 Y3 X4 Y4 X5 Y5
Z1 Z2 Z3 Z4 Z5
Then, knowing the X-values and one Y-value (since X and Y can
have different bases/scales) can lead us to informed guesses about
the remaining Y-values, but nothing can be said about Z-values.
-> ROIs X and Y have information about each other even across
different subjects, while having little/none about Z.
-> X and Y must have some mutual/common influence, which Z may
not.

54. Interpreting covariance networks
• Analyzing: similarity of brain structure across subjects.

55. 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

56. 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-region
influences controlling localities as general feature
– -> can observe consistent patterns of properties as correlation of
subject series ‘non-locally’

57. 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-region
influences 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

58. 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-region
influences 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 some
correlation].

59. Switching gears…
• Statistical resampling: methods for estimating
confidence intervals for estimates
• Several kinds, two common ones in fMRI are
jackknifing and bootstrapping (see, e.g. Efron et al.
1982).
• Can use with fMRI, and also with DTI (~for noisy
ellipsoid estimates-- confidence in fit parameters)

60. Jackknifing
• Basically, take M acquisitions
e.g., M=12

61. Jackknifing
• Basically, take M acquisitions
e.g., M=12
• Randomly select MJ < M to use
MJ=9
to calculate quantity of interest
– standard nonlinear fits
(ellipsoid is defined
by 6 parameters of [D11 D22 D33 D12 D13 D23] = ....
quadratic surface)

62. Jackknifing
• Basically, take M acquisitions
e.g., M=12
• Randomly select MJ < M to use
MJ=9
to calculate quantity of interest
– standard nonlinear fits
• Repeatedly subsample large
number (~103-104 times)
[D11 D22 D33 D12 D13 D23] = ....
[D11 D22 D33 D12 D13 D23] = ....
[D11 D22 D33 D12 D13 D23] = ....
....

63. Jackknifing
• Basically, take M acquisitions
e.g., M=12
• Randomly select MJ < M to use
MJ=9
to calculate quantity of interest
– standard nonlinear fits
• Repeatedly subsample large
number (~103-104 times)
• Analyze distribution of values
for estimator (mean) and [D11 D22 D33 D12 D13 D23] = ....
confidence interval [D11 D22 D33 D12 D13 D23] = ....
– sort/%iles
• (not so efficient) [D11 D22 D33 D12 D13 D23] = ....
– if Gaussian, e.g. µ±2σ ....
• simple

64. Jackknifing
- quite Gaussian
- Gaussianity, σ
increase with
decreasing MJ
- µ changes little
M=32 gradients

65. Jackknifing
- not too bad with
smaller M, even
- but could use
min/max from
distributions for
%iles (don’t need
to sort)
M=12 gradients

66. Bootstrapping
• Similar principal to jackknifing,but need multiple copies of dataset.
A B
e.g., M=12 e.g., M=12
C D
e.g., M=12 e.g., M=12

67. Bootstrapping
• Make an estimate from 12 measures, but randomly selected from
each set:
A B
e.g., M=12 e.g., M=12
C D
e.g., M=12 e.g., M=12

68. Bootstrapping
• Then select another random (complete) set, build a distribution, etc.
A B
e.g., M=12 e.g., M=12
C D
e.g., M=12 e.g., M=12

69. Summary
• There are a wide array of methods applicable to MRI analysis
– Many of them involve statistics and are therefore always believable at face
value.
– The applicability of the assumptions of the underlying mathematics to the
real situation is always key.
– Often, in MRI, we are concerned with a ‘network’ view of regions working
together 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.