1. Overview of SPM p <0.05 Statistical parametric map (SPM)
Image time-series
Kernel
Design matrix
Realignment
Smoothing
General linear model
Statistical
inference
Gaussian
field theory
Normalisation
p <0.05
Template
Parameter estimates

2. The General Linear Model (GLM)
Frederike Petzschner
Translational Neuromodeling Unit (TNU) Institute for Biomedical Engineering, University of Zurich & ETH Zurich
With many thanks for slides & images to:
FIL Methods group, Virginia Flanagin and Klaas Enno Stephan
This Talk will be concerned with the General Linear Model, which is the core of SPM.
So far you have been concerned with Preprocessing the data such that all the functional images are aligned and the registered to the high resolution anatomical images, the are nicely smoothed and the timing is adjusted.
Now we can use this data to actually ask some questions.
That is we go back to our original experiment

3. Image a very simple experiment…
One session
7 cycles of rest and listening
Blocks of 6 scans with 7 sec TR
The toy example that we are going to use is one of the first fMRI studies which explains why the TR is so long
And it tested listening to words versus rest.
So the experimental manipulation was a block design of blocks of listening versus blocks of rest.
time

4. Image a very simple experiment…
What we measure.
single voxel
time series
Time
What we know.
Know we have acquires a series of images of the course of this manipulation, And in fact we have not acquired images but small volumes so three dimensional data plus we have a forth dimension which is time.
In the GLM we do a voxel-wise analysis
That is we look a the time course of one particular voxel.
For instance if we take a voxel in the rare part of the brain.. We could measure a time series like that over the course of our experimental manipulation
What we are interested in is to find out wheher there is a significant change in the BOLD response due to our experimental manipulation that is between listening and rest.
For the particular voxel we are looking it this is very roughly speaking not the case
time
Question: Is there a change in the BOLD response between listening and rest?

5. Image a very simple experiment…
What we measure.
single voxel
time series
Time
What we know.
But if we look at another voxel. E.g. in the auditory cortex we might actually get something like this.
This looks much more like a change based on listening and rest.
time
Question: Is there a change in the BOLD response between listening and rest?

6. You need a model of your data…
linear model
effects estimate 
statistic
error estimate
But in order to determine whether there is a significant difference we need to model the data based on our knowledge about them and see whether there is a change that is caused by our experimental manipulation.
This model then will give us an effects estimate (beta values) und an estimate of the error with which we can before all kinds of analysis
Question: Is there a change in the BOLD response between listening and rest?

7. = + + Explain your data… error 1 2 x1 x2 e Time BOLD signal
as a combination of experimental manipulation,confounds and errors
error = +
1 2 +
Time
Error is the thing that is not explained
And informative part
Easier in matrix form x1 x2 e
BOLD signal
regressor
Single voxel regression model:

8. = + + Explain your data… error 1 2 x1 x2 e Time BOLD signal
as a combination of experimental manipulation,confounds and errors
error = +
1 2 +
Time
This is the matrix form x1 x2 e
BOLD signal
Single voxel regression model:

9. = + + The black and white version in SPM 1 2 error Designmatrix e
n: number of scans
p: number of regressors

10. Model assumptions Designmatrix error
The design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.  Talk: Experimental Design Wed 9:45 – 10:45
Designmatrix
You want to estimate your parameters  such that you minimize:
This can be done using an Ordinary least squares estimation (OLS) assuming an i.i.d. error:
error
To define the design matrix is not enough
You need to have an assumption about your error term.
We assume that noise is randomly distributed.
Independent and identical (independent of all other noise and identical across the time series)
The designmatrix should also account for potential confounds
If the error term becomes to big this harms you in terms of statistics

11. GLM assumes identical and independently distributed errors
i.i.d. = error covariance is a scalar multiple of the identity matrix: Cov(e) = 2I
non-identity
non-independence
t t2 t1 t2
To define the design matrix is not enough
You need to have an assumption about your error term.
We assume that noise is randomly distributed.
Independent and identical (independent of all other noise and identical across the time series)
The designmatrix should also account for potential confounds
If the error term becomes to big this harms you in terms of statistics

12. = + How to fit the model and estimate the parameters? error y X
„Option 1“: Per hand = + y X

13. = + How to fit the model and estimate the parameters? error y X
OLS (Ordinary Least Squares)
error
Data predicted by our model
Error between predicted and actual data
Goal is to determine the betas such that we minimize the quadratic error = + y X

14. OLS (Ordinary Least Squares)
The goal is to minimize the quadratic error between data and model

15. OLS (Ordinary Least Squares)
The goal is to minimize the quadratic error between data and model

16. OLS (Ordinary Least Squares)
The goal is to minimize the quadratic error between data and model
This is a scalar and the transpose of a scalar is a scalar 

17. OLS (Ordinary Least Squares)
The goal is to minimize the quadratic error between data and model
This is a scalar and the transpose of a scalar is a scalar 

18. OLS (Ordinary Least Squares)
The goal is to minimize the quadratic error between data and model
This is a scalar and the transpose of a scalar is a scalar 
You find the extremum of a function by taking its derivative and setting it to zero

19. OLS (Ordinary Least Squares)
The goal is to minimize the quadratic error between data and model
This is a scalar and the transpose of a scalar is a scalar 
You find the extremum of a function by taking its derivative and setting it to zero
SOLUTION: OLS of the Parameters

20. A geometric perspective on the GLM
OLS estimates y e x2 x1
Geometric perspective
You can see the regressors as two vectors
7 cycles 8 dimensional space
The only signal that can be explained by the linear combination of the two regressors so only the gray plane
Design space defined by X

21. Correlated and orthogonal regressorsy
Design space defined by X x2
x2* x1
Same notion of geometric expression
Data are outside the plane
We want to estimate the projection of our data onto the plane
Problem the two regressors are correlated
You can orthogonalize the vectors to get more statistical power
But not both regressors change but just one will increase
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!

22. = + We are nearly there…  linear model effects estimate statistic
error estimate
Not going to talk about the statistics = +

23. What are the problems?
Design
Error
BOLD responses have a delayed and dispersed form.
The BOLD signal includes substantial amounts of low- frequency noise.
The data are serially correlated (temporally autocorrelated)  this violates the assumptions of the noise model in the GLM
What are the problems with model
bold signal need up to 5 seconds to reach peak and about 20 seconds to decay
Addictional signal from the scanner: scanner drift. Eg. From heating up because of the movement of the coils
Subjects are alive. Datat acquired at a slow paste compared to this physiological activity this is on top of the bold signal. Now the errror term is correlated

24. Problem 1: Shape of BOLD response
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).
If you mix input function and expected activation function you get the expected bold response

25. Solution: Convolution model of the BOLD response
expected BOLD response
= input function impulse response function (HRF)
 HRF
No slightly delayed and also not completely flat
blue = data
green = predicted response, taking convolved with HRF
red = predicted response, NOT taking into account the HRF

26. Problem 2: Low frequency noise
You have signals that you are not interested in at a low frequency
Build-in tool that allows you to capture these very slow oscillations
The part of the signal we are not interesetd in will be modelled in the design
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

27. Problem 2: Low frequency noise
Linear model
You have signals that you are not interested in at a low frequency
Build-in tool that allows you to capture these very slow oscillations
The part of the signal we are not interesetd in will be modelled in the design
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

28. discrete cosine transform (DCT) set
Solution 2: High pass filtering
If you have already used SPM
You have to decide on the cut-off
By default 1/128 secondsthe gray area get s modelled out
If you design something that
discrete cosine transform (DCT) set

29. Problem 3: Serial correlations
i.i.d
non-independence
non-identity
t t2 t1 t2 n
If err iid matrx with only the diagonal
Solution use an autocorrelated model (AR1 plus white noise) n
n: number of scans

30. 1st order autoregressive process: AR(1)
Problem 3: Serial correlations
with
1st order autoregressive process: AR(1)
autocovariance
function n
If err iid matrx with only the diagonal
Solution use an autocorrelated model (AR1 plus white noise) n
n: number of scans

31. Problem 3: Serial correlations
Pre-whitening: Use an enhanced noise model with multiple error covariance components, i.e. e ~ N(0,2V) instead of e ~ N(0,2I) Use estimated serial correlation to specify filter matrix W for whitening the data.
This is i.i.d
If err iid matrx with only the diagonal
Solution use an autocorrelated model (AR1 plus white noise)

32. How do we define W ? Enhanced noise model
Remember linear transform for Gaussians
Choose W such that error covariance becomes spherical
Conclusion: W is a simple function of V  so how do we estimate V ?
What sometimes worries users.
The design matrix changes after the estimation your are looking at the weighted design matrix

33. V Q1 Q2 = 1 + 2 Find V: Multiple covariance components
enhanced noise model
error covariance components Q
and hyperparameters V Q1 Q2 = 1
+ 2
Variance is modeled by a linear combination of covariance matrixes
Hyperparamters : parameters on the covariance
Way to do that is restricted maximum likelihood
Estimation of hyperparameters  with EM (expectation maximisation) or ReML (restricted maximum likelihood).

34. = + We are there…  linear model effects estimate statistic
error estimate
GLM includes all known experimental effects and confounds
Convolution with a canonical HRF
High-pass filtering to account for low-frequency drifts
Estimation of multiple variance components (e.g. to account for serial correlations)
Not going to talk about the statistics = +

35. = + We are there…  linear model effects estimate statistic
Null hypothesis:
effects estimate 
statistic
error estimate
Not going to talk about the statistics
 Talk: Statistical Inference and design efficiency. Next Talk = +

36. We are there…
Mass-univariate approach: GLM applied to > 100,000 voxels
Threshold of p<0.05  more than 5000 voxels significant by chance!
single voxel
time series
Time
Massive problem with multiple comparisons!
Solution: Gaussian random field theory
Not going to talk about the statistics

37. Outlook: further challenges
correction for multiple comparisons  Talk: Multiple Comparisons Wed 8:30 – 9:30
variability in the HRF across voxels  Talk: Experimental Design Wed 9:45 – 10:45
limitations of frequentist statistics  Talk: entire Friday
GLM ignores interactions among voxels  Talk: Multivariate Analysis Thu 12:30 – 13:30

38. Thank you!
Read me
Friston, Ashburner, Kiebel, Nichols, Penny (2007) Statistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier.
Christensen R (1996) Plane Answers to Complex Questions: The Theory of Linear Models. Springer.
Friston KJ et al. (1995) Statistical parametric maps in functional imaging: a general linear approach. Human Brain Mapping 2:
Talk Title letzte Folie
Derivative with respect to beta
Eventuell die gesamte Herleitung
Animationen überprüfen
Eventuell HGF Ableitung einbauen