1. The SPM MfD course 12th Dec 2007 Elvina Chu
Basis Functions
The SPM MfD course
12th Dec 2007
Elvina Chu

2. Introduction What is a basis function What do they do in MRI
How are they useful in SPM

3. Basis Mathematical term to describe any point in space
Euclidian i.e. the x y z co-ordinates y
v = 4 i + 2 j 2 j i i 4 x

4. Function
Vectors are produced as each function in the function space can be represented as a linear combination of basis functions.
Linear algebra: Orthonormal i.e. same unit length with perpendicular elements

5. Uses in SPM
Spatial normalisation to register different subjects to the same co-ordinate system
Ease of reporting in standard space
Useful for reporting what happens generically to individuals in functional imaging

7. Uses in SPM
Basis functions are used to model the haemodynamic response
Finite impulse response Fourier

8. Fourier Basis % signal change with time
Fourier analysis: the complex wave at the top can be decomposed into the sum of the three simpler waves shown below.
f(t)=h1(t)+h2(t)+h3(t)
f(t)
h1(t)
h2(t)
h3(t)

9. Gamma Function
Provides a reasonably good fit to the impulse response, although it lacks an undershoot.
Fewer functions required to capture the typical range of impulse responses than other sets, thus reducing the degrees of freedom in design matrix

10. Canonical haemodynamic response function (HRF)
Typical BOLD response to an impulse stimulation
The response peaks approximately 5 sec after stimulation, and is followed by an undershoot.

11. Canonical HRF
Temporal derivative
Dispersion derivative
The canonical HRF is a “typical” BOLD impulse response characterised by two gamma functions.
Temporal derivative can capture differences in latency of peak response
Dispersion derivative can capture differences in duration of peak response

12. Design matrix 3 regressors used to model each condition
Left Right Mean
3 regressors used to model each condition
The three basis functions are:
1. Canonical HRF
2. Derivatives with respect to time
3. Derivatives with respect to dispersion

13. Comparison of the fitted response
These plots show the haemodynamic response at a single voxel. The left plot shows the HRF as estimated using the simple model. Lack of fit is corrected, on the right using a more flexible model with basis functions.

14. Summary Basis functions identify position in space
Used to model the HRF of BOLD response to an impulse stimulation in fMRI
SPM allows you to choose from 4 different basis functions

15. Hanneke den Ouden Methods for Dummies 2007 12/12/2007
Multiple Regression Analysis
&
Correlated Regressors
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.
Hanneke den Ouden
Methods for Dummies 2007
12/12/2007

16. Overview General Regression analysis Multiple regressions
Collinearity / correlated regressors
Orthogonalisation of regressors in SPM
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

17. Regression analysis
regression analysis examines the relation of a dependent variable Y to specified independent variables X:
Y = aX + b
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.
if the model fits the data well:
R2 is high (reflects the proportion of variance in Y explained by the regressor X)
the corresponding p value will be low

18. Multiple regression analysis
Multiple regression characterises the relationship between several independent variables (or regressors), X1, X2, X3 etc, and a single dependent variable, Y:
Y = β1X1 + β2X2 +…..+ βLXL + ε
The X variables are combined linearly and each has its own regression coefficient β (weight)
βs reflect the independent contribution of each regressor, X, to the value of the dependent variable, Y
i.e. the proportion of the variance in Y accounted for by each regressor after all other regressors are accounted for
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

19. Multicollinearity
Multiple regression results are sometimes difficult to interpret:
the overall p value of a fitted model is very low
i.e. the model fits the data well
but individual p values for the regressors are high
i.e. none of the X variables has a significant impact on predicting Y.
How is this possible?
Caused when two (or more) regressors are highly correlated: problem known as multicollinearity
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

20. In practice this will nearly always be the case
Multicollinearity
Are correlated regressors a problem? No
when you want to predict Y from X1 and X2
Because R2 and p will be correct
Yes
when you want assess impact of individual regressors
Because individual p values can be misleading: a p value can be high, even though the variable is important
In practice this will nearly always be the case
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

21. Correlated Regressors
General Linear Model
&
Correlated Regressors
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

22. Y = X . β + ε General Linear Model and fMRI Observed data
Y is the BOLD signal at various time points at a single voxel
Design matrix
Several components which explain the observed data Y:
Different stimuli
Movement regressors
Parameters
(or betas)
Define the contribution of each component of the design matrix to the value of Y
Error
(or residuals)
Any variance in Y that cannot be explained by the model X.β
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

23. Collinearity example Experiment:
Which areas of the brain are active in reward processing?
Subjects press a button to get a reward when they spot a red dot amongst green dots
model to be fit:
Y = β1X1 + β2X2 + ε
Y = BOLD response
X1 = button press (movement)
X2 = response to reward
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

24.  We can’t answer the question
Collinearity example
Which areas of the brain are active in reward processing?
The regressors are linearly dependent (correlated), so
variance attributable to an individual regressor may be confounded with other regressor(s)
As a result we don’t know which part of the BOLD response is explained by movement and which by response to getting a reward
this may lead to misinterpretations of activations in certain brain areas
Primary motor cortex involved in reward processing??
 We can’t answer the question
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

25. How to deal with collinearity
Avoid it:
Design the experiment so that the independent variables are uncorrelated
Use common sense
Use toolbox “Design Magic” - Multicollinearity assessment for fMRI for SPM
URL:
Allows you to assess the multicollinearity in your fMRI-design by calculating the amount of factor variance that is also accounted for by the other factors in the design (expressed in R2).
also allows you to reduce correlations between regressors through use of high-pass filters
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

26. How to deal with collinearity II
Orthogonalise the correlated regressor variables
using factor analysis (like PCA)
this will produce linearly independent regressors and corresponding factor scores.
these factor scores can subsequently be used instead of the original correlated regressor values
However, the meaning of these factors is rather unclear… so SPM does not do this
Instead SPM does something called serial orthogonalisation
(note that this is only within each condition, so for each condition and its associated parametric modulators, if there are any)
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

27. Serial Orthogonalisation
When we have only one regressor, things are simple…
Y = 1X1
1 = 1.5
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

28. Serial Orthogonalisation
When we two correlated regressors, things become difficult…
The value of 1 is now smaller, so X1 now explains less of the variance, as X2 explains some of the variance X1 used to explain
Y = 1X1 + 2X2
1 = 1
2 = 1
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

29. Serial Orthogonalisation
We now orthogonalise X2 with respect to X1, and call this X2*
- 1 now again has the original value it had when X2 was not included
- 2* is the same value as 2
- X2* is a different regressor from X2!!!
Y = 1X1 + 2*X2*
1 = 1.5
2* = 1
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

30. Serial Orthogonalisation in SPM
Regressors are orthogonalised from left to right in the design matrix
Order in which you put parametric modulators is important!!!
Put the ‘most important’ modulators first (i.e the ones whose meaning you don’t want to change)
If you add an orthogonalised regressor, the  values of the preceding regressors do not change
The regressor you orthogonalise to (X1) does not change
The regressor you are orthogonalising (X2) does change
Plot the orthogonalised regressors to see what it is you are actually estimating
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

31. Conclusions
Correlated regressors can be a big problem when analysing / interpreting your data
Try to design your experiment such that you avoid correlated regressors
Estimate how much your regressors are correlated so you know what you’re getting yourself into
If you cannot avoid them
Think about the order of the regressors in your design matrix
Look at what the regressors look like after orthogonalisation
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.

32. Sources Will Penny & Klaas Stephan
Rik Henson’s slides:
Previous years’ presenters’ slides
In my presentation I will discuss some results of a study
that we conducted in the last few years
on the vowel systems of eight regional varieties of Standard Dutch
as spoken in The Netherlands and in Flanders.