Transcript mckeown ppt - Duke-UNC Brain Imaging and Analysis Center
Analytical Techniques
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Hypothesis Driven
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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. Slice 1 Data Matrix X Voxels
Calculating level of Significance
X fMRI Data significance: ~ t statistic i / i G = + total variability = Variability explained by the model + noise
Covariate Indicator variable
SPM Nomenclature for Design Matrix
G 1 G (interesting) G c H 1 H (non-interesting) H c Global activity Linear trends E.g. dose of drug Design matrix G subject
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
Inclusion of Global Signal in Regression
Global Signal Hypothesis Test voxel Regression Coefficients 4 3 -1 -2 2 1 0
Global signal
1
Hypothesis
2 < 0!!!
< 5 degrees difference between Global Signal & Hypothesis !
db 2 2
Inclusion of Global Covariate in Regression: Effect of non orthogonality
X 1 X’ 1 db db 1 2 “ Reference Function, R” 1 2 X 1 X’ 1 b = (G T G) -1 G T X 1
Consider an fMRI experiment with only 3 time points
Consider an fMRI experiment with only 3 time points
Analysis of Brain Systems
reference function R 1 Correlation viewed as a projection
R 2
R 2 Although R1 and R2 both somewhat correlated with the reference function, they are uncorrelated with each other
Corr(R 2, ref) Corr(R 1, ref) ref R 1
Principal Component Analysis (PCA)
Voxel 1 Voxel 2
PC1 t Voxel 3 Eigenimage + time course
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
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Position of microphones and speakers is constant (mixing matrix constant)
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Sources Ergodic
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The propagation of the signal from the source to the microphone is instantaneous
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Sources sum linearly
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Number of microphones equals the number of speakers
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In Bell-Sejnowski algorithm, the non-linearity approximates the cdf of the sources
g(C) :
Independent Component Analysis (ICA)
?M
S = X Mixing matrix Independent Sources (individuals’ speech) = Data time Goal of ICA : given Data (X), can we recover the sources (S), without knowing M?
W X = C Weight matrix Data = Independent Components time ‘InfoMax’ algorithm: Iteratively estimate W, so that: Goal of ICA: Find W, so that Kullback-Leibler divergence between f
W
0 , 1 y (C) and f g(C) 2 (S) is minimized ?
g(C) : Key point :
maximizing H(y) implies that rows of C are maximally independent
Independent Component Analysis (ICA)
Task Non task-related activations (e.g. Arousal) Machine Noise Measured Signal Pulsations 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
COMPONENT MAPS
#1
Mixing
#2 ‘mixing matrix’, M n S S S
MEASURED fMRI SIGNAL
t = 1 S t = 2 t = n
Selected Components:
Consistently task-related Transiently task-related Abrupt head movement Quasi-periodic Slowly-varying Slow head movement Activated Suppressed
Comparison of Three Linear Models PCA (2nd order) 4th order ICA (all orders) r = 0.46
r = 0.85
Increasing spatial independence between components r = 0.92
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 ...
?
Identical 2nd-order statistics A Statistically Independent
i
A i p B i q
B 0 ICA (all orders) Higher order statistics Decorrelated
i A i B i
0
Comon’s 4th order
PCA (2nd order)
Derived Independent Components
ICA Component A component map specified by voxel values 0.4, 1.2, 4.3, -6.9, ... -2.1, 0.2 ...
Histogram of voxel values for component map z > 1 0 Component map after thresholding associated time course
Unexpected Frontal-cerebellar activation detected with ICA Self-paced movement Rest Movie 0 10 20 30 40 50 60
A Transiently task-related (TTR) component (active during first two trials) Martin J. McKeown, CNL, Salk Institute, [email protected]
ICA component time course
(a)
Aligned
Single trial fMRI
Trial 1
ICA component spatial distribution
(b)
(c) (d) (e)
19-sec All p < 10 -20
Single trial fMRI
Assessing Statistical Models
Voxel # fMRI (X) Data Reference function G =
PRESS Statistic:
Eliminate 1 time point G -i = Data + How well does G -i match data?
• Gives some idea of the influence of the i th time point +
Hybrid Techniques
Data Driven Hypothesis Driven
Con Exp Con Exp Con Exp Con Exp
HYBICA:
L arm pronation/supination hypothesis Hybrid activation 0 10 20 30 40 50 60
Use of HYBICA for Memory Load Hypothesis testing
S1
Use of HYBICA for Memory Load Hypothesis testing
Maintenance
Use of HYBICA for Memory Load Hypothesis testing
S2