The General Linear Model. The Simple Linear Model Linear Regression.
The General Linear Model
description
Transcript of The General Linear Model
![Page 1: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/1.jpg)
The General Linear Model
Guillaume FlandinWellcome Trust Centre for Neuroimaging
University College London
SPM Short CourseLondon, 21-23 Oct 2010
![Page 2: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/2.jpg)
Normalisation
Statistical Parametric MapImage time-series
Parameter estimates
General Linear ModelRealignment Smoothing
Design matrix
Anatomicalreference
Spatial filter
StatisticalInference
RFT
p <0.05
![Page 3: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/3.jpg)
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
![Page 4: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/4.jpg)
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
![Page 5: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/5.jpg)
Time
BOLD signal
Time
single voxeltime series
Voxel-wise time series analysis
ModelspecificationParameterestimationHypothesis
Statistic
SPM
![Page 6: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/6.jpg)
BOLD signal
Time =1 2+ +
error
x1 x2 e
Single voxel regression model
exxy 2211
![Page 7: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/7.jpg)
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
![Page 8: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/8.jpg)
• 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..
GLM: mass-univariate parametric analysis
![Page 9: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/9.jpg)
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
![Page 10: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/10.jpg)
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
![Page 11: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/11.jpg)
What are the problems of this model?
1. BOLD responses have a delayed and dispersed form.
HRF
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
![Page 12: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/12.jpg)
t
dtgftgf0
)()()(
Problem 1: Shape of BOLD responseSolution: Convolution model
expected BOLD response = input function impulse response function (HRF)
=
Impulses HRF Expected BOLD
![Page 13: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/13.jpg)
Convolution model of the BOLD responseConvolve stimulus function with a canonical hemodynamic response function (HRF):
HRF
t
dtgftgf0
)()()(
![Page 14: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/14.jpg)
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
![Page 15: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/15.jpg)
discrete cosine transform (DCT) set
High pass filtering
![Page 16: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/16.jpg)
withttt aee 1 ),0(~ 2 Nt
1st order autoregressive process: AR(1)
)(eCovautocovariance
function
N
N
Problem 3: Serial correlations
![Page 17: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/17.jpg)
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 Qand hyperparameters
jj
ii
QV
VC
2
),0(~ ii CNe
![Page 18: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/18.jpg)
1 1 1ˆ ( )T TX V X X V y
Parameters can then be estimated using 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
![Page 19: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/19.jpg)
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
![Page 20: The General Linear Model](https://reader036.fdocuments.in/reader036/viewer/2022062323/56816755550346895ddc0b3a/html5/thumbnails/20.jpg)
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