analytical techniques
DESCRIPTION
Analytical Techniques. Hypothesis Driven. Data Driven Principal Component Analysis (PCA) Independent Component Analysis (ICA) Fuzzy Clustering. Others Structural equation modeling. Matrix Notation of fMRI Data. 1 voxel. BOLD signal. t=1. t=2. t=3. t=4. Voxels. X. Data Matrix. - PowerPoint PPT PresentationTRANSCRIPT
Analytical Techniques
• Data Driven• Principal Component Analysis (PCA)• Independent Component Analysis (ICA)• Fuzzy Clustering
• Others• Structural equation modeling
• Hypothesis Driven
Matrix Notation of fMRI Data1 voxel
t=1
t=2
t=3
t=4. . .
Data Matrix
Voxels
time XSlice 1
BOLD signal
Calculating level of Significance
GX
fMRI Data =
significance: ~ t statistic i/i
+
total variability = Variability explained by the model
+ noise
SPM Nomenclature for Design Matrix
G (interesting)
Design matrix G
G1 Gc
H (non-interesting)
Indicator variable
Covariate
E.g. dose of drug
H1
subject
HcGlobal activity
Linear trends
Some General Linear Model (GLM) Assumptions:
• Design matrix known without error
• the ’s follow a Gaussian distribution
• the design matrix is the same everywhere in the brain
• the residuals are well modeled by Gaussian noise
• each voxel is independent of the others
• the voxels are temporally aligned
• each time point is independent of the others (time courses of voxels are white)
Hypothesis
Test voxel
Global Signal
Inclusion of Global Signal in Regression
< 5 degrees difference between Global Signal & Hypothesis !
1 2-2
-1
0
1
2
3
4
Regression Coefficients
< 0!!!Globalsignal
Hypothesis
Inclusion of Global Covariate in Regression:Effect of non orthogonality
1
2X1
X’1
db1
db2
1
2
“Reference Function, R”
db2
X1
X’1
b = (GTG)-1GTX
Consider an fMRI experiment with only 3 time points
Consider an fMRI experiment with only 3 time points
Analysis of Brain Systems
reference function R1
R2
Although R1 and R2 both somewhat correlated with the reference
function, they are uncorrelated with each other
ref
R1
R2
Corr(R1, ref)
Corr(R2, ref)
Correlation viewed as
a projection
Principal Component Analysis (PCA)
Voxel 1Voxel 2
Vox
el 3
PC1
Voxel 1 Voxel 2
Voxel 3
Eigenimage + time course
t
Independent Component Analysis (ICA)
Without knowing position of microphones or what any person is saying, can you isolate each of the voices?
Independent Component Analysis (ICA)
Assumption: each sound from speaker unrelated to others (independent)
Some ICA assumptions
• 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) :
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
Goal of ICA: Find W, so that Kullback-Leibler divergence between f1(C) and f2(S) is minimized ?g(C) y ,0
)(
W
yHg(C) :
‘InfoMax’ algorithm: Iteratively estimate W, so that:
Key point: maximizing H(y) implies that rows of C are maximally independent
Measured Signal
Task Non task-relatedactivations
(e.g. Arousal)
PulsationsMachine Noise
Independent Component Analysis (ICA)
Assumption: spatial pattern from sources of variability unrelated (independent)
The fMRI data at each time point is considered a mixture of activations from each component map
n
COMPONENTMAPS
MEASURED fMRI SIGNAL
‘mixing matrix’,
M
#1
#2
t = 1
t = 2
t = n
Mixing
time
Selected Components:Consistentlytask-related
Transientlytask-related
Quasi-periodic Slowly-varying Slow headmovement
Abrupt headmovement
ActivatedSuppressed
PCA (2nd order) 4th order ICA (all orders)
Comparison of Three Linear Models
r = 0.46 r = 0.85 r = 0.92
Increasing spatial independence between components
Are Two Maps Independent?0.4, 1.2, 4.3, -6.9, ... -2.1, 0.2 ...0.1, 1.2, 1.3, -1.9, ... -0.1, 4.2 ...
?
A BStatistically
Independent
Decorrelated
Higher-order
statistics
Identical2nd-orderstatistics
0i
qi
pi BA
ICA (all orders)
Comon’s 4th order
0i
ii BAPCA (2nd
order)
0.4, 1.2, 4.3, -6.9, ... -2.1, 0.2 ...
A component map specified by voxel values
Histogram of voxel values for component map
0
z > 1
associated time course
Derived Independent Components
Component map after thresholding
ICA Component
0 10 20 30 40 50 60
RestSelf-paced movement Movie
Unexpected Frontal-cerebellar activation detected with ICA
A Transiently task-related (TTR) component (active during first two trials)
Martin J. McKeown, CNL, Salk Institute, [email protected]
Single trial fMRI
ICA component time course Aligned ICA component spatial distribution
(a)
(b)
Trial 1
Single trial fMRI
All p < 10-20
(c)
(d)
(e)
19-sec
PRESS Statistic:
Assessing Statistical Models
Data
Eliminate 1 time point
=
+ -iG ^
Reference function
fMRI (X) Data
Voxel #
time = +
G
How well does G-i match data?
• Gives some idea of the influence of the ith time point
Hybrid Techniques
Data Driven
Hypothesis
Driven
Exp Exp Exp Exp
Con Con ConCon
0 10 20 30 40 50 60
HYBICA: L arm pronation/supination
hypothesis
Hybrid activation
S1
Use of HYBICA for Memory Load Hypothesis testing
Maintenance
Use of HYBICA for Memory Load Hypothesis testing
S2
Use of HYBICA for Memory Load Hypothesis testing