Transcript mbi2004 3852

FMRI Time Series Analysis
Mark Woolrich & Steve Smith
Oxford University Centre for
Functional Magnetic Resonance Imaging of the Brain
(FMRIB)
FMRI Time Series Analysis
Overview
• Noise modelling (autocorrelation)
• Signal modelling:
• Complex parameterised HRF model
• Optimised basis functions for HRF modelling
Reminder - Multiple Regression Model (GLM)
Temporal Autocorrelation
FMRI Noise
Power Spectral
Density
• Time series from each voxel contains low frequency
drifts and high frequency noise
• Drifts are scanner-related and physiological (cardiac
cycle, breathing etc)
• Both high and low frequency noise hide activation
High-pass Filtering
• Removes the worst of the low frequency trends
High-pass
High-Frequency Noise
• Unless high-frequency noise is modelled or corrected for:
– Incorrect stats (probably false positives)
– Inefficient stats (false negatives)
Temporal Filtering and the GLM
SY  SXB  Se
e ~ N(0,  2V )
S is a matrix for temporal filtering
Prewhitening
Precolouring
(Bullmore et al ‘96)
S = K-1
Best Linear Unbiased Estimator
(Worsley et al ‘95)
S=L
(L is a low pass filter)
Smothers intrinsic autocorrelation
Different Regressors
Boxcar
Fixed ISI singleevent with jitter
Randomized ISI
single-event
Regressor
Low-pass
FFT
Low-pass
Low-pass
Sensitivity
FMRIB's Improved Linear
Modelling (FILM)
Performs prewhitening LOCALLY:
• Fit the GLM and estimate the raw
autocorrelation on the residuals
• Spectrally and spatially smooth
autocorrelation estimate
• Construct prewhitening filter to "undo"
autocorrelation
• Use filter on data and design matrix and refit
FMRIB's Improved Linear
Modelling (FILM)
Performs prewhitening LOCALLY:
• Fit the GLM and estimate the raw
autocorrelation on the residuals
• Spectrally and spatially smooth
autocorrelation estimate
• Construct prewhitening filter to "undo"
autocorrelation
• Use filter on data and design matrix and refit
Spectral Smoothing
Raw autocorrelation
Autocorrelation estimate
IFFT
FFT
Power Spectral Density
Tukey taper
smoothed
Spatial Smoothing
Autocorr
EPI
Non-linear Spatial Smoothing
Gaussian spatial
smoother with weights:
wij  exp{( I i  I j ) 2 / 2t 2 }
Ii= EPI signal intensity
t = brightness threshold
Unbiased Statistics
• P-P plots for FILM on 6 null datasets
Boxcar
Single Event
Session Effects Investigation
• 3 paradigms x 33 “identical” sessions (McGonigle 2000)
• Variety of 1st-level analyses - use group-level mixed-effects-Z to judge
efficiency of first-level analysis
Session Effects
ME-Z Plots
Autocorrelation Conclusions
• Precolouring is nearly as sensitive as prewhitening for
boxcar designs
• Single-event designs require prewhitening for increased
sensitivity
• Local autocorrelation estimation using a Tukey taper with
nonlinear spatial smoothing produces close to zero bias
when prewhitening
• More advanced: need spatiotemporal noise model:
– Model-based (Woolrich)
– Model-“free” (Beckmann)
Signal Modelling
• Start with stimulation timings
• Several conditions (original Evs)?
• Convolve with HRF to blur and delay
• What choice of HRF?
• Does it vary across subjects?
• Does it vary across the brain?
• “Advanced” issues:
• Allow signal height to change over time (dynamic)?
• Use nonlinear convolution (events interact)?
• Spatiotemporal modelling
HRF Modelling
Linear (time invariant) System
Experimental Stimulus, e.g.
boxcar
Parameterised HRF, e.g.
Gamma function

hit

st
Assumed response
git  st  hit
HRF Parameterisation
Half-cosine parameterisation
?
Prior samples
HRF Parameterisation
Half-cosine parameterisation
Model Selection
Model 1 (no undershoot): c2 = 0
Model 2 (undershoot): c2 ≠ 0
?
Automatic Relevance
Determination (ARD) Prior
(Mackay 1995)
c2 ~ N (0,  1 )
• Relevance of a parameter c2 is automatically
determined by the parameter 
 
then
c2  0 with high precision : Model 1
 0
then
c2
is non-zero : Model 2
MCMC
ARD of Undershoot
Simulated data with no undershoot
No ARD prior
True value
ARD prior
True value
Prior
samples
HRF Results
Boxcar
Jittered single-event
Boxcar
Jittered Single-event
Randomised single-event
Randomised Single-event
Response fits
Marginal
posterior
samples Posterior
samples
Posterior
samples
Posterior
samples
HRF Basis Sets
Basis Sets for HRF Modelling
• Basis functions in the GLM:
instead of one fixed HRF we can
have several
• Here an F across all 3 betas finds
the best linear combination of the
3 HRFs
• 3 Original EVs now become 6
(2 submodels each with 3 HRFs)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
4 HRF basis
functions
partial
model fits
full
model fits
Basis Sets for HRF Modelling
• Instead of a parameterised HRF
MCMC
• We can use a linear basis set to
span the space of expected HRF
shapes
• WHY? : We can then use the
yit   ahi xht  eit
h
basis set in an easier to infer GLM
Variational
Bayes
Generating HRF Basis Sets
(1) Take samples of
the HRF
Generating HRF Basis Sets
(1) Take samples of
the HRF
(2) Perform SVD
Generating HRF Basis Sets
(1) Take samples of
the HRF
(2) Perform SVD
(3) Select the top
eigenvectors as the optimal
basis set
Unconstrained Basis Set
BUT: The basis set spans a wider range of HRF shapes
than we want to allow:
HRF samples from prior
Unconstrained basis set
Constrained Basis Set
We can regress the HRF samples back on to the basis set
HRF samples from prior
Basis set
Constrained Basis Set
and fit a multivariate normal to the basis set parameter space
p(ahit ) ~ MVN (m, C )
HRF samples from prior
Basis set
Constrained Basis Set
This contrains the basis set to give only sensible looking HRF
shapes
HRF samples from prior
Samples from basis set
Unconstrained
Constrained
Using the Constrained Basis Set
• The multivariate normal on the basis set parameters can
then be used as a prior on those parameters in the GLM
yit   ahit xht  eit
h
p(ahit ) ~ MVN (m, C )
Variational Bayes
Constrained basis set
Constrained Basis Set - Results
Constraining Basis Sets Results
in Increased Sensitivity
Acknowledgements
FMRIB Analysis Group
UK EPSRC, MRC, MIAS-IRC