The General Linear Model Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM fMRI Course London, May 2012
Jan 18, 2018
The General Linear Model
Guillaume FlandinWellcome Trust Centre for Neuroimaging
University College London
SPM fMRI CourseLondon, May 2012
Normalisation
Statistical Parametric MapImage time-series
Parameter estimates
General Linear ModelRealignment Smoothing
Design matrix
Anatomicalreference
Spatial filter
StatisticalInference
RFT
p <0.05
Passive word listeningversus rest
7 cycles of rest and listening
Blocks of 6 scanswith 7 sec TR
Question: Is there a change in the BOLD response between listening and rest?
Stimulus function
One session
A very simple fMRI experiment
Time
BOLD signal
Time
single voxeltime series
Voxel-wise time series analysis
ModelspecificationParameterestimationHypothesis
Statistic
SPM
BOLD signal
Time =1 2+ +
error
x1 x2 e
Single voxel regression model
exxy 2211
Mass-univariate analysis: voxel-wise GLM
=
e+y
X
N
1
N N
1 1p
pModel is specified by1. Design matrix X2. Assumptions
about eN: number of scansp: number of regressors
eXy
The design matrix embodies all available knowledge about experimentally controlled factors and potential
confounds.
),0(~ 2INe
• one sample t-test• two sample t-test• paired t-test• Analysis of Variance
(ANOVA)• Analysis of Covariance
(ANCoVA)• correlation• linear regression• multiple regression
GLM: a flexible framework for parametric analyses
Parameter estimation
eXy
= +
e
2
1
Ordinary least squares estimation
(OLS) (assuming i.i.d. error):
yXXX TT 1)(ˆ
Objective:estimate parameters to minimize
N
tte
1
2
y X
Problems of this model with fMRI time series
1. The BOLD response has a delayed and dispersed shape.
2. The BOLD signal includes substantial amounts of low-frequency 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.
Boynton et al, NeuroImage, 2012.
Scaling
Additivity
Shiftinvariance
Problem 1: BOLD responseHemodynamic response function (HRF):
Linear time-invariant (LTI) system:
u(t) x(t)hrf(t)
𝑥 (𝑡 )=𝑢 (𝑡 )∗h𝑟𝑓 (𝑡 )
¿∫𝑜
𝑡
𝑢 (𝜏 ) h𝑟𝑓 (𝑡−𝜏 )𝑑𝜏
Convolution operator:
Problem 1: BOLD responseSolution: Convolution model
Convolution model of the BOLD responseConvolve stimulus function with a canonical hemodynamic response function (HRF):
HRF
∫ t
dtgftgf0
)()()(
blue = datablack = mean + low-frequency driftgreen = predicted response, taking into account low-frequency driftred = predicted response, NOT taking into account low-frequency drift
Problem 2: Low-frequency noise Solution: High pass filtering
discrete cosine transform (DCT) set
)(eCovautocovariance
function
N
N
Problem 3: Serial correlations
𝑒 𝑁 (0 ,𝜎2 𝐼 )i.i.d:
Multiple covariance components
= 1 + 2
Q1 Q2
Estimation of hyperparameters with ReML (Restricted Maximum Likelihood).
V
enhanced noise model at voxel i
error covariance components Q and hyperparameters
jj
ii
QV
VC
2
),0(~ ii CNe
1 1 1ˆ ( )T TX V X X V y
Weighted Least Squares (WLS)
Let 1TW W V
1
1
ˆ ( )ˆ ( )
T T T T
T Ts s s s
X W WX X W Wy
X X X y
Then
where
,s sX WX y Wy
WLS equivalent toOLS on whiteneddata and design
A mass-univariate approach
Time
Summary
Estimation of the parameters
= +𝜀𝛽
𝜀 𝑁 (0 ,𝜎 2𝑉 )
�̂�=(𝑋𝑇𝑉 −1 𝑋)−1 𝑋𝑇𝑉 −1 𝑦
noise assumptions:
WLS:
�̂�1=3.9831
�̂�2−7={0.6871 ,1.9598 ,1.3902 , 166.1007 , 76.4770 ,− 64.8189 }
�̂�8=131.0040
=
�̂� 2=�̂�𝑇 �̂�𝑁−𝑝�̂� 𝑁 (𝛽 ,𝜎2(𝑋𝑇 𝑋 )−1 )
Summary
References
Statistical parametric maps in functional imaging: a general linear approach, K.J. Friston et al, Human Brain Mapping, 1995.
Analysis of fMRI time-series revisited – again, K.J. Worsley and K.J. Friston, NeuroImage, 1995.
The general linear model and fMRI: Does love last forever?, J.-B. Poline and M. Brett, NeuroImage, 2012.
Linear systems analysis of the fMRI signal, G.M. Boynton et al, NeuroImage, 2012.