Basis FunctionsBasis Functions

The SPM MfD courseThe SPM MfD course

1212thth Dec 2007 Dec 2007

Elvina ChuElvina Chu

IntroductionIntroduction

• What is a basis functionWhat is a basis function

• What do they do in MRIWhat do they do in MRI

• How are they useful in SPMHow are they useful in SPM

BasisBasis

• Mathematical term to describe any point in spaceMathematical term to describe any point in space

• Euclidian i.e. the x y z co-ordinatesEuclidian i.e. the x y z co-ordinates

x

y

4

2

i

j

i

v = 4 i + 2 j

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

FunctionFunction

Uses in SPMUses in SPM

• Spatial normalisation to register different Spatial normalisation to register different subjects to the same co-ordinate systemsubjects to the same co-ordinate system

• Ease of reporting in standard spaceEase of reporting in standard space

• Useful for reporting what happens Useful for reporting what happens generically to individuals in functional generically to individuals in functional imagingimaging

Uses in SPMUses in SPM

• Basis functions are used to model the Basis functions are used to model the haemodynamic response haemodynamic response

Finite impulse response Fourier

Fourier BasisFourier Basis

• % signal change with % signal change with timetime

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)

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

Gamma FunctionGamma Function

Canonical haemodynamic Canonical haemodynamic response function (HRF)response function (HRF)

Typical BOLD response to an impulse stimulation

The response peaks approximately 5 sec after stimulation, and is followed by an undershoot.

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

Design matrix Design matrix

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

Left Right Mean

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.

Comparison of the fitted Comparison of the fitted responseresponse

SummarySummary

• Basis functions identify position in space Basis functions identify position in space

• Used to model the HRF of BOLD response Used to model the HRF of BOLD response to an impulse stimulation in fMRIto an impulse stimulation in fMRI

• SPM allows you to choose from 4 different SPM allows you to choose from 4 different basis functionsbasis functions

Multiple Regression Analysis&

Correlated Regressors

Hanneke den OudenHanneke den Ouden

Methods for Dummies 2007Methods for Dummies 2007

12/12/200712/12/2007

Overview

GeneralGeneral

Regression analysisRegression analysis

Multiple regressionsMultiple regressions

Collinearity / correlated regressorsCollinearity / correlated regressors

Orthogonalisation of regressors in SPMOrthogonalisation of regressors in SPM

Regression analysis

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

regression analysis

examines the relation of a

dependent variable Y to

specified independent

variables X:

Y = aX + b

Multiple regression analysis

Multiple regression characterises the relationship between several Multiple regression characterises the relationship between several

independent variables (or regressors), Xindependent variables (or regressors), X11, X, X22, X, X33 etc, and a single dependent etc, and a single dependent

variable, Y:variable, Y:

Y = Y = ββ11XX1 1 + + ββ22XX22 +…..+ +…..+ ββLLXXLL + + εε

The X variables are combined linearly and each has its own regression The X variables are combined linearly and each has its own regression

coefficient coefficient ββ (weight) (weight)

ββs reflect the independent contribution of each regressor, X, to the value of s reflect the independent contribution of each regressor, X, to the value of

the dependent variable, Ythe dependent variable, Y

i.e. the proportion of the variance in Y accounted for by each regressor after i.e. the proportion of the variance in Y accounted for by each regressor after

all other regressors are accounted forall other regressors are accounted for

Multicollinearity

Multiple regression results are sometimes difficult to interpret:Multiple regression results are sometimes difficult to interpret: the overall the overall pp value of a fitted model is very value of a fitted model is very lowlow

i.e. the model fits the data welli.e. the model fits the data well

but individual but individual pp values for the regressors are values for the regressors are highhigh i.e. none of the X variables has a significant impact on predicting Y. i.e. none of the X variables has a significant impact on predicting Y.

How is this possible?How is this possible? Caused when two (or more) regressors are highly correlated: problem known Caused when two (or more) regressors are highly correlated: problem known

as as multicollinearitymulticollinearity

Multicollinearity

Are correlated regressors a problem?Are correlated regressors a problem?

NoNo when you want to predict Y from X1 and X2when you want to predict Y from X1 and X2

Because Because RR22 and and pp will be correct will be correct

YesYes when you want assess impact of individual regressors when you want assess impact of individual regressors

Because Because individual individual pp values can be misleading: a p value values can be misleading: a p value can be can be highhigh, even though the variable is important, even though the variable is important

In practice this will nearly always be the caseIn practice this will nearly always be the case

General Linear Model&

Correlated Regressors

General Linear Model and fMRI

Y = X . β + εObserved dataY is the BOLD signal at various time points at a single voxel

Design matrixSeveral 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

Collinearity example

Experiment:Experiment: Which areas of the brain are active in reward processing?Which areas of the brain are active in reward processing? Subjects press a button to get a reward when they spot a red dot Subjects press a button to get a reward when they spot a red dot

amongst green dotsamongst green dots

model to be fit:model to be fit:

Y = Y = ββ11XX1 1 + + ββ22XX22 + + εε

Y = BOLD responseY = BOLD response

X1 = button press (movement)X1 = button press (movement)

X2 = response to rewardX2 = response to reward

Collinearity example

Which areas of the brain are active in reward processing?Which areas of the brain are active in reward processing?

The regressors are linearly dependent (correlated), so The regressors are linearly dependent (correlated), so

variance attributable to an individual regressor may be confounded with other variance attributable to an individual regressor may be confounded with other regressor(s)regressor(s)

As a result we don’t know which part of the BOLD response is explained by As a result we don’t know which part of the BOLD response is explained by movement and which by response to getting a rewardmovement and which by response to getting a reward

this may lead to misinterpretations of activations in certain brain areasthis may lead to misinterpretations of activations in certain brain areas

Primary motor cortex involved in reward processing??Primary motor cortex involved in reward processing??

We can’t answer the questionWe can’t answer the question

How to deal with collinearity

Avoid it:Avoid it:

Design the experiment so that the independent variables are Design the experiment so that the independent variables are uncorrelateduncorrelated

Use common senseUse common sense Use toolbox “Use toolbox “Design Magic” - Multicollinearity assessment for fMRI for SPMDesign Magic” - Multicollinearity assessment for fMRI for SPM

URL: URL: http://www.matthijs-vink.com/tools.htmlhttp://www.matthijs-vink.com/tools.html Allows you to assess the multicollinearity in your fMRI-design by calculating the 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 amount of factor variance that is also accounted for by the other factors in the design (expressed in Rdesign (expressed in R22).). also allows you to reduce correlations between regressors through use of high-also allows you to reduce correlations between regressors through use of high-pass filterspass filters

How to deal with collinearity II

OrthogonaliseOrthogonalise the correlated regressor variables the correlated regressor variables

using factor analysis (like PCA)using factor analysis (like PCA) this will produce this will produce linearly independentlinearly independent regressors and corresponding factor regressors and corresponding factor

scores.scores. these factor scores can subsequently be used instead of the original correlated these factor scores can subsequently be used instead of the original correlated

regressor valuesregressor values

However, the meaning of these factors is rather unclear… so SPM does not do However, the meaning of these factors is rather unclear… so SPM does not do thisthis

Instead SPM does something called Instead SPM does something called serial orthogonalisationserial orthogonalisation(note that this is only within each condition, so for each condition and its (note that this is only within each condition, so for each condition and its associated parametric modulators, if there are any)associated parametric modulators, if there are any)

Serial Orthogonalisation

Y = 1X1

1 = 1.5

When we have only one regressor, things are simple…When we have only one regressor, things are simple…

Serial Orthogonalisation

Y = 1X1 + 2X2

1 = 1

2 = 1

When we two correlated regressors, things become difficult…When we two correlated regressors, things become difficult…

The value of The value of 11 is now smaller, so X is now smaller, so X11 now explains less of the variance, as X now explains less of the variance, as X22 explains some of the variance Xexplains some of the variance X11 used to explain used to explain

Serial Orthogonalisation

Y = 1X1 + 2*X2*

1 = 1.5

2* = 1

We now orthogonalise XWe now orthogonalise X22 with respect to X with respect to X11, and call this X, and call this X22**

- - 11 now again has the original value it had when X now again has the original value it had when X22 was not included was not included

- - 22* is the same value as * is the same value as 22

- X- X22* is a different regressor from X* is a different regressor from X22!!! !!!

Serial Orthogonalisation in SPM

Regressors are orthogonalised from left to right in the design matrixRegressors are orthogonalised from left to right in the design matrix

Order in which you put parametric modulators is important!!!Order in which you put parametric modulators is important!!!

Put the ‘most important’ modulators first (i.e the ones whose Put the ‘most important’ modulators first (i.e the ones whose meaning you don’t want to change)meaning you don’t want to change)

If you add an orthogonalised regressor, the If you add an orthogonalised regressor, the values of the preceding values of the preceding regressors do not changeregressors do not change

The regressor you orthogonalise to (XThe regressor you orthogonalise to (X11) does not change) does not change

The regressor you are orthogonalising (XThe regressor you are orthogonalising (X22) ) does does changechange

Plot the orthogonalised regressors to see what it is you are Plot the orthogonalised regressors to see what it is you are actually estimatingactually estimating

Conclusions

Correlated regressors can be a big problem when analysing / Correlated regressors can be a big problem when analysing / interpreting your datainterpreting your data

Try to design your experiment such that you avoid correlated Try to design your experiment such that you avoid correlated regressorsregressors

Estimate how much your regressors are correlated so you know Estimate how much your regressors are correlated so you know what you’re getting yourself intowhat you’re getting yourself into

If you cannot avoid themIf you cannot avoid them

Think about the order of the regressors in your design matrixThink about the order of the regressors in your design matrix

Look at what the regressors look like after orthogonalisationLook at what the regressors look like after orthogonalisation

Sources

Will Penny & Klaas StephanWill Penny & Klaas Stephan

Rik Henson’s slides: Rik Henson’s slides: www.mrc-cbu.cam.ac.uk/Imaging/Common/www.mrc-cbu.cam.ac.uk/Imaging/Common/rikrikSPM-GLM.pptSPM-GLM.ppt

Previous years’ presenters’ slidesPrevious years’ presenters’ slides