General Linear
Model
Beatriz Calvo
Davina Bristow
Overview
 Summary of regression
 Matrix formulation of multiple regression
 Introduce GLM
 Parameter Estimation
Residual sum of squares
 GLM and fMRI
 fMRI model
Linear Time Series
Design Matrix
Parameter estimation
 Summary
Summary of Regression
 Linear regression models the linear relationship between a
single dependent variable, Y, and a single independent
variable, X, using the equation:
Y = βX + c + ε
 The regression coefficient, β, reflects how much of an
effect X has on Y
 ε is the error term and is assumed to be independently,
identically, and normally distributed (mean 0 and variance
σ2)
Summary of Regression
 Multiple regression is used to determine the effect of a
number of independent variables, X1, X2, X3 etc, on a
single dependent variable, Y
 The different X variables are combined in a linear way and
each has its own regression coefficient:
Y = β1X1 + β2X2 +…..+ βLXL + ε
 The β parameters reflect the independent contribution of
each independent variable, X, to the value of the
dependent variable, Y.
 i.e. the amount of variance in Y that is accounted for by
each X variable after all the other X variables have been
accounted for
Matrix formulation
Multiplying matrices reminder:
abc x G
H
def
I
Gxa+Hxb+Ixc
=
Gxd+Hxe+Ixf
Matrix Formulation
 Write out equation for each observation of variable Y from 1 to J:
Y1 = X11β1 +…+X1lβl +…+ X1LβL + ε1
Yj = Xj1β1 +…+Xjlβl +…+ XjLβL + εj
YJ = XJ1β1 +…+XJlβl +…+ XJLβL + εJ
Can
turn these simultaneous equations into matrix form to get a single
equation:
Y1
Yj
YJ
Y
Observed data
=
X11 … X1l … X1L
Xj1 … X1l … X1L
X11 … X1l … X1L
β1
βj
βJ
+
ε1
εj
εJ
=
X
β
+
ε
Design Matrix
x
Parameters
Residuals/Error
General Linear Model
 This is simply an extension of multiple regression
 Or alternatively Multiple Regression is just a simple form
of the General Linear Model
 Multiple Regression only looks at ONE dependent (Y)
variable
 Whereas, GLM allows you to analyse several
dependent, Y, variables in a linear combination
 i.e. multiple regression is a GLM with only one Y variable
 ANOVA, t-test, F-test, etc. are also forms of the GLM
GLM - continued..
 In the GLM the vector Y, of J observations of a single Y
variable, becomes a MATRIX, of J observations of N
different Y variables
 An fMRI experiment could be modelled with matrix Y of
the BOLD signal at N voxels for J scans
 However SPM takes a univariate approach, i.e. each
voxel is represented by a column vector of J fMRI signal
measurements, and it processed through a GLM
separately
 (this is why you then need to correct for multiple
comparisons)
GLM and fMRI
How does the GLM apply to fMRI experiments?
Y
=
X
.
Observed data:
Design matrix:
SPM uses a mass
univariate
approach – that is
each voxel is
treated as a
separate column
vector of data.
Y is the BOLD
signal at various
time points at a
single voxel
Several components which
explain the observed data, i.e.
the BOLD time series for the
voxel
Timing info: onset vectors,
Omj, and duration vectors, Dmj
HRF, hm, describes shape of
the expected BOLD response
over time
Other regressors, e.g.
realignment parameters
β
+
ε
Parameters:
Error:
Define the
contribution of each
component of the
design matrix to the
value of Y
Estimated so as to
minimise the error, ε,
i.e. least sums of
squares
Difference
between the
observed
data, Y, and
that predicted
by the model,
Xβ .
Not assumed
to be spherical
in fMRI
Parameter estimation
 In linear regression the parameter β is estimated so that
the best prediction of Y can be obtained from X
 i.e. sums of squares of difference between predicted
values and observed data, (i.e. the residuals, ε) is
minimised
 Remember last week’s talk & graph!
 The method of estimating parameters in GLM is
essentially the same, i.e. minimising sums of squares
(ordinary least squares), it just looks more complicated
Last week’s graph
ε
= ỹ , predicted value
= y i , true value
y = βx + c
ε = residual error
Residual Sums of Squares
 Take a set of parameter estimates, β
 Put these into the GLM equation to obtain estimates of Y from X, i.e.
fitted values, Y :
Y
=X xβ
 The residual errors, e, are the difference between the fitted and
actual values:
e=Y-Y
 Residual sums of squares is:
= Y - Xβ
S = ΣjJej2
 When written out in full this gives:
S = ΣjJ(Yj - Xj1β
1 -…- XjLβ
2
)
L
Minimising S

S is minimised
when the
gradient of this
curve is zero
e = Y - Xβ
S = ΣjJej2
sums of squares (S)
 If you plot the sum
of squares value
for different
parameter, β ,
estimates you get
a curve
S = Σ(Y Xβ )2
Gradient = 0
min S
parameter estimates(B)
Minimising S cont.
 so to calculate the values of β which gives you the
least sums of squares you must find the partial derivative
of
S = ΣjJ(Yj - Xj1β
 Which is
∂S/∂β
1 -…- XjLβ
= 2Σ(-Xjl)(Yj – Xj1β
1-…-
2
)
L
XjLβ
L)
and solve this for ∂S/∂β
=0
 In matrix form of the residual sum of squares is
S = eTe
this is equivalent to ΣjJej2
(remember how we multiply matrices)
 e=Y-Xβ
therefore S = (Y - X β
)T(Y - X β
)
Minimising S cont.
 Need to find the derivative and solve for ∂S/∂β
=0
 The derivative of this equation can be rearranged to give
XTY = (XTX)β
when the gradient of the curve = 0, i.e. S is minimised
 This can be rearranged to give:
β
= XTY(XTX)-1
 But a solution can only be found, if (XTX) is invertible
because you need to divide by it, which in matrix terms is
the same as multiplying by the inverse!
GLM and fMRI
How does the GLM apply to fMRI experiments?
Y
=
X
.
Observed data:
Design matrix:
SPM uses a mass
univariate
approach – that is
each voxel is
treated as a
separate column
vector of data.
Y is the BOLD
signal at various
time points at a
single voxel
Several components which
explain the observed data, i.e.
the BOLD time series for the
voxel
Timing info: onset vectors,
Omj, and duration vectors, Dmj
HRF, hm, describes shape of
the expected BOLD response
over time
Other regressors, e.g.
realignment parameters
β
+
ε
Parameters:
Error:
Define the
contribution of each
component of the
design matrix to the
value of Y
Estimated so as to
minimise the error, ε,
i.e. least sums of
squares
Difference
between the
observed
data, Y, and
that predicted
by the model,
Xβ .
Not assumed
to be spherical
in fMRI
fMRI models
 Completed the experiment, after preprocessing, the data
are ready for STATS.
 STATS: (estimate parameters, β, inference)
indicating evidence against the Ho of no effect at
each voxel are computed->an image of this statistic
is produce
This statistical image is assessed (other talk will
explain that)
Example: 1 subject. 1 session
Moving finger vs rest
7 cycles of rest and moving
Each epoch 6 scans
Whole brain acquisition data
Responses at voxel (x, y, z)
Time series of BOLD response in one voxel
Question:
Is there any change in the
BOLD response between
moving and rest?
Time seconds
Linear Time Series Model
 TIME SERIES:
consist on the sequential
measures of fMRI data signal
intensities over the period of the
experiment
 The same temporal model is
used at each voxel
 Mass-univariated model and
perform the same analysis at
each voxel
 Therefore, we can describe the
complete temporal model for
fMRI data by looking at how it
works for the data from a voxel.
Time
Single Voxel Time Series
Y: My data/ observations
My Data
 Time series of N observations
Y1,…,Ys,…,Yn.
N= scan number
 Acquired at one voxel
at times ts, where S=1:N
Time
Single Voxel Time Series
Y1
Ys
YN
Model specification
 The overall aim of regressor generation is to come
up with a design matrix that models the expected
fMRI response at any voxel as a linear
combinations of columns.
 Design matrix – formed of several components
which explain the observed data.
 Two things SPM need to know to construct the
design matrix:
 Specify regressors
 Basis functions that explain my data
Model specification …
Specify regressors X
Timing information consists of onset vectors
Omj and duration vectors Dm
Other regressors e.g. movement parameters
Include as many regressors as you consider
necessary to best explain what’s going on.
Basis functions that explain my data (HRF)
Expected shape of the BOLD response due to
stimulus presentation
GLM and fMRI data
 Model the observed time series at each voxel as a linear
combination of explanatory functions, plus an error term
Ys= β1 X1(tS)+ …+ βl Xl(tS)+ …+ βL XL(tS) + εs
 Here, each column of the design matrix X contains the
values of one of the continuous regressors evaluated at
each time point ts of fMRI time series
 That is, the columns of the design matrix are the discrete
regressors
GLM and fMRI data …
 Consider the equation for all time points, to give a set of equations
Y1= β1 X1(t1)+ …+ βl Xl(t1)+ …+ βL XL(t1) + ε1
Ys= β1 X1(tS)+ …+ βl Xl(tS)+ …+ βL XL(tS) + εs
YN= β1 X1(tN)+ …+ βl Xl(tN)+ …+ βL XL(tN) + εN
In matrix form:
Y1
Ys
YN
In matrix notation:
=
X1(t1)
X1(tS)
X1(tN)
Xl(t1)
Xl(tS)
Xl(tN)
Y=Xβ+ε
XL(t1)
XL(tS)
XL(tN)
β1
βl
βL
+
εN
εs
εN
Getting the design matrix
Observations
Regressors
β1
+
β2
+
Time
=
Intensity
y=
x1
x2
ε
Errors are
normally and
independently
and identical
distributed
Getting the design matrix …
Observations
Regressors
+
=β
1
Time
Error
β2
+
Intensity
y=
β1x1
+
β2x2 +
ε
Design matrix
Observations
Regressors
x
=
Y
=
Error
X
β1
+
β2
β
+
ε
Design matrix
Observations
Regressors
Error
L
l
l
l
Y
=
N
X
X1(t1)
X1(tS)
X1(tN)
Y1
Ys
YN
β2
β
β1
N
N: nuber of scans
P: number of regressors
Xl
L
(t1) X (t1)
Xl(tS) XL(tS)
Xl(tN) XL(tN)
L
Model is specified by:
+
ε
εN
εs
εN
β1
βl
βL
N
Y=Xβ+ε
•design matrix
•Assumptions about ε
Parametric estimation
ε=Y-Y
=
+
= Y - Xβ
β2
β1
S = ΣtJεt2
+
The error is minimal when
β
Y
ε
X
Estimate parameters
β
= XTY(XTX)-1
Least squares
Parameter estimates
(Get this by putting into matrix form
and finding derivative)
Summary
 The General Linear Model allows you to find the
parameters, β, which provide the best fit with your data, Y
 The optimal parameters estimates, β, are found by
minimising the Sums of Squares differences between your
predicted model and the observed data
 The design matrix in SPM contains the information about
the factors, X, which may explain the observed data
 Once we have obtained the βs at each voxel we can use
these to do various statistical tests
but that is another talk….
THE END
Thank you to
Lucy, Daniel and Will
and to
Stephan for his chapter and slides about GLM
and to
Adam for last year’s presentation
Links:
http://www.fil.ion.ucl.ac.uk/~wpenny/notes03/slides/glm/slide1.htm
http://www.fil.ion.ucl.ac.uk/spm/HBF2/pdfs/Ch7.pdf
http://www.fil.ion.ucl.ac.uk/~wpenny/mfd/GLM.ppt