The General Linear Model
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Transcript of The General Linear Model
The General Linear Model
Guillaume FlandinWellcome Trust Centre for Neuroimaging
University College London
SPM fMRI CourseLondon, October 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
A geometric perspective on the GLM
ye
Design space defined by X
x1
x2 ˆ Xy
Smallest errors (shortest error vector)when e is orthogonal to X
Ordinary Least Squares (OLS)
0eX T
XXyX TT
0)ˆ( XyX T
yXXX TT 1)(ˆ
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.
Contrasts &statistical parametric
maps
Q: activation during listening ?
c = 1 0 0 0 0 0 0 0 0 0 0
Null hypothesis: 01
)ˆ(
ˆ
T
T
cStdct
X
stimulus function
1. Decompose data into effects and error
2. Form statistic using estimates of effects and error
Make inferences about effects of interest
Why?
How?
datastatisti
c
Modelling the measured data
linearmode
l
effects estimate
error estimate
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
withttt aee 1 ),0(~ 2 Nt
1st order autoregressive process: AR(1)
)(eCovautocovariance
function
N
N
Problem 3: Serial correlations
discrete cosine transform (DCT) set
High pass filtering
∫ t
dtgftgf0
)()()(
Problem 1: BOLD responseSolution: Convolution model
expected BOLD response = input function impulse response function (HRF)
=
Impulses HRF Expected BOLD