1. Analytical Techniques
Hypothesis Driven
Data Driven
Principal Component Analysis (PCA)
Independent Component Analysis (ICA)
Fuzzy Clustering
Others
Structural equation modeling

2. Matrix Notation of fMRI Data
1 voxel
BOLD signal
t=1
t=2
t=3
t=4
. . .
Voxels X
time
Data Matrix
Slice 1

3. Calculating level of Significance
significance: ~ t statistic i/i  X G
+  =
fMRI Data + =
Variability explained by the model
total variability
noise

4. SPM Nomenclature for Design Matrix
G (interesting)
H (non-interesting)
Covariate
Indicator variable G1 Gc Hc H1
Global activity
Linear trends
E.g. dose of drug
subject
Design matrix G

5. Some General Linear Model (GLM) Assumptions:
Design matrix known without error
the design matrix is the same everywhere in the brain
the ’s follow a Gaussian distribution
the residuals are well modeled by Gaussian noise
the voxels are temporally aligned
each time point is independent of the others (time courses of voxels are white)
each voxel is independent of the others

6. Inclusion of Global Signal in Regression
Hypothesis
Regression Coefficients
 < 0!!!
Global
signal
Hypothesis
Test voxel
< 5 degrees difference between Global Signal & Hypothesis !

7. “Reference Function, R”
Inclusion of Global Covariate in Regression: Effect of non orthogonality 1 2 X1
X’1
db1
db2
b = (GTG)-1GTX 2
db2 X1
X’1 1
“Reference Function, R”

8. Consider an fMRI experiment with only 3 time points

9. Consider an fMRI experiment with only 3 time points

10. Analysis of Brain Systems
Correlation viewed as
a projection
reference function R1 R2 R2
Corr(R2, ref)
ref
Corr(R1, ref)
Although R1 and R2 both somewhat correlated with the reference function, they are uncorrelated with each other R1

11. Principal Component Analysis (PCA)
Voxel 1
Voxel 2
Voxel 1
Voxel 2
Voxel 3
PC1 t
Voxel 3
Eigenimage + time course

12. Independent Component Analysis (ICA)
Without knowing position of microphones or what any person is saying, can you isolate each of the voices?

13. Independent Component Analysis (ICA)
Assumption: each sound from speaker unrelated to others (independent)

14. Some ICA assumptions g(C) :
Position of microphones and speakers is constant (mixing matrix constant)
Sources Ergodic
The propagation of the signal from the source to the microphone is instantaneous
Sources sum linearly
Number of microphones equals the number of speakers
In Bell-Sejnowski algorithm, the non-linearity approximates the cdf of the sources
g(C) :

15. Independent Component Analysis (ICA)
Independent Sources
(individuals’ speech)
time
Mixing
matrix =
Data S ?M X
Goal of ICA: given Data (X), can we recover the sources (S), without knowing M?
Independent Components
time =
Data X W
Weight matrix C
g(C) :
‘InfoMax’ algorithm: Iteratively estimate W, so that:
Key point: maximizing H(y) implies that rows of C are maximally independent
Goal of ICA: Find W, so that Kullback-Leibler divergence between f1(C) and f2(S) is minimized ?

16. Independent Component Analysis (ICA)
Non task-related
activations
(e.g. Arousal)
Task
Measured
Signal
Machine Noise
Pulsations
Assumption: spatial pattern from sources of variability unrelated (independent)

17. time M Mixing #1 t = 1 #2 t = 2 n t = n COMPONENT MAPS MEASURED
The fMRI data at each time point is considered a mixture of activations from each component map
Mixing
COMPONENT
MAPS
MEASURED
fMRI SIGNAL #1 S
t = 1
time #2
‘mixing matrix’, M S
t = 2 S n S
t = n

18. Selected Components: Consistently task-related Transiently
Abrupt head
movement
Quasi-periodic
Slowly-varying
Slow head
movement
Activated
Suppressed

19. Increasing spatial independence between components
Comparison of Three Linear Models
PCA (2nd order)
4th order
ICA (all orders)
r = 0.46
r = 0.85
r = 0.92
Increasing spatial independence between components

20. Statistically Independent
Are Two Maps Independent?
0.4, 1.2, 4.3, -6.9, ,
0.1, 1.2, 1.3, -1.9, , ? A B
Statistically Independent = å i q p B A
ICA (all orders)
Identical
2nd-order
statistics
Higher-
order
statistics
Comon’s 4th order = å i B A
PCA (2nd order)
Decorrelated

21. Derived Independent Components
ICA Component
Histogram of voxel values
for component map
z > 1
A component map
specified by voxel values
0.4, 1.2, 4.3, -6.9, ,
Component map
after thresholding
associated time course

22. Unexpected Frontal-cerebellar activation detected with ICA
Self-paced movement
Rest
Movie

23. A Transiently task-related (TTR) component (active during first two trials)
Martin J. McKeown, CNL, Salk Institute,

24. Single trial fMRI Trial 1 ICA component time course Aligned
ICA component spatial distribution
(a)
(b)

25. Single trial fMRI
(c)
(d)
(e)
19-sec
All p < 10-20

26. Assessing Statistical Models
PRESS Statistic: ^
-i
+  G
Eliminate 1 time point =
Data
How well does G-i match data?
Gives some idea of the influence of the ith time point

27. Hybrid Techniques
Hypothesis
Driven
Data Driven
Exp
Con

28. HYBICA: L arm pronation/supination
hypothesis
Hybrid activation

29. Use of HYBICA for Memory Load Hypothesis testing

30. Use of HYBICA for Memory Load Hypothesis testing
Maintenance

31. Use of HYBICA for Memory Load Hypothesis testing