Transcript DCM - University College London

The General Linear Model (GLM)
Will Penny
Functional Imaging Laboratory (FIL)
Wellcome Trust Centre for Neuroimaging
University College London
May 2009
Overview of SPM
Image time-series
Realignment
Kernel
Design matrix
Smoothing
General linear model
Statistical parametric map (SPM)
Statistical
inference
Normalisation
Gaussian
field theory
p <0.05
Template
Parameter estimates
A very simple fMRI experiment
One session
Passive word
listening
versus rest
7 cycles of
rest and listening
Blocks of 6 scans
with 7 sec TR
Stimulus function
Question: Is there a change in the BOLD response
between listening and rest?
Modelling the measured data
Why?
How?
Make inferences about effects of interest
1. Decompose data into effects and
error
2. Form statistic using estimates of
effects and error
stimulus
function
data
linear
model
effects
estimate
error
estimate
statistic
Voxel-wise time series analysis
model
specification
Time
parameter
estimation
hypothesis
statistic
BOLD signal
single voxel
time series
SPM
Time
=1
BOLD signal
+ 2
x1
+
x2
y  x11  x2 2  e
error
Single voxel regression model
e
Mass-univariate analysis: voxel-wise GLM
p
1
1
1

p
y
N
=
N
X
y  X  e
e ~ N (0, I )
2
+
N
e
Model is specified by
1. Design matrix X
2. Assumptions about e
N: number of scans
p: number of regressors
The design matrix embodies all available knowledge about
experimentally controlled factors and potential confounds.
GLM: mass-univariate parametric analysis
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one sample t-test
two sample t-test
paired t-test
Analysis of Variance (ANOVA)
Factorial designs
correlation
linear regression
multiple regression
F-tests
fMRI time series models
etc…
Parameter estimation
 1 
  +
 2
=
y
Objective:
estimate parameters
to minimize
X
y  X  e
e
N
e
t 1
Ordinary least squares
estimation (OLS)
(assuming i.i.d. error):
ˆ  ( X T X ) 1 X T y
2
t
A geometric perspective on the GLM
Smallest errors (shortest error vector)
when e is orthogonal to X
y
e
yˆ  Xˆ
x2
X e0
T
X ( y  X ˆ )  0
X T y  X T X ˆ
T
x1
Design space
defined by X
ˆ  ( X T X )1 X T y
Ordinary Least Squares (OLS)
Correlated and orthogonal regressors
y
x2*
x2
x1
y  x11  x2  2  e
y  x11  x2* 2*  e
1   2  1
1  1;  2*  1
Correlated regressors =
explained variance is shared
between regressors
When x2 is orthogonalized with
regard to x1, only the parameter
estimate for x1 changes, not that
for x2!
What are the problems of this model?
1.
BOLD responses have a delayed
and dispersed form.
2.
The BOLD signal includes substantial amounts of lowfrequency noise (eg due to scanner drift).
3.
Due to breathing, heartbeat & unmodeled neuronal activity,
the errors are serially correlated. This violates the
assumptions of the noise model in the GLM
HRF
Problem 1: Shape of BOLD response
Solution: Convolution model
Expected BOLD
HRF
Impulses

=
t
f  g (t )   f ( ) g (t   )d
0
The response of a linear time-invariant (LTI) system is the convolution of the input
with the system's response to an impulse (delta function).
expected BOLD response
= input function impulse response function (HRF)
Convolution model of the BOLD response
Convolve stimulus function with
a canonical hemodynamic
response function (HRF):
t
f  g (t )   f ( ) g (t   )d
0
 HRF
Problem 2: Low-frequency noise
Solution: High pass filtering
discrete cosine
transform (DCT) set
High pass filtering: example
blue =
data
black = mean + low-frequency drift
green = predicted response, taking into account
low-frequency drift
red =
predicted response, NOT taking into
account low-frequency drift
Problem 3: Serial correlations
et  aet 1   t
with
 t ~ N (0,  2 )
1st order autoregressive process: AR(1)
N
Cov(e)
autocovariance
function
N
Multiple covariance components
Ci   i V
2
ei ~ N (0, Ci )
enhanced noise model at voxel i
V
= 1
V    jQ j
error covariance components Q
and hyperparameters 
Q1
+ 2
Q2
Estimation of hyperparameters  with ReML (Restricted Maximum Likelihood).
Parameters can then be estimated using Weighted Least Squares (WLS)
T 1
1
T 1
ˆ
  (X V X ) X V y
Let
W W V
T
1
Then
T
T
1
T
T
ˆ
  ( X W WX ) X W Wy
T
1
T
ˆ
  (X X ) X y
s
s
s
where
X s  WX , ys  Wy
s
WLS equivalent to
OLS on whitened
data and design
Contrasts &
statistical parametric maps
c=10000000000
Q: activation during
listening ?
X
Null hypothesis:
1  0
c ˆ
t
T ˆ
Std (c  )
T
Summary
• Mass univariate approach.
• Fit GLMs with design matrix, X, to data at different points in
space to estimate local effect sizes, 
• GLM is a very general approach
• Hemodynamic Response Function
• High pass filtering
• Temporal autocorrelation