1. Statistical Parametric Mapping
Will Penny
Wellcome Trust Centre for Neuroimaging,
University College London, UK
LSTHM, UCL, Jan 14, 2009

2. Statistical Parametric Mapping
Image time-series
Kernel
Design matrix
Realignment
Smoothing
General linear model
Statistical
inference
Random
field theory
Normalisation
p <0.05
Template
Parameter estimates

3. Outline Voxel-wise General Linear Models Random Field Theory
Bayesian modelling

4. Voxel-wise GLMs Time Time model specification parameter estimation
hypothesis
statistic
Time
Intensity
single voxel
time series
SPM

5. Temporal convolution model for the BOLD response
Convolve stimulus function with a canonical hemodynamic response function (HRF):
 HRF

6. General Linear Model = + Error Covariance Model is specified by
Design matrix X
Assumptions about e
N: number of scans
p: number of regressors

7. 2. Weighted Least Squares
Estimation
1. ReML-algorithm L l g
2. Weighted Least Squares
Friston et al. 2002, Neuroimage

8. Q: activation during listening ?
Contrasts & SPMs
c =
Q: activation during listening ?
Null hypothesis:

9. Outline Voxel-wise General Linear Models Random Field Theory
Bayesian modelling

10. Inference for Images
Noise
Signal
Signal+Noise

11. Use of ‘uncorrected’ p-value, a=0.1
11.3%
12.5%
10.8%
11.5%
10.0%
10.7%
11.2%
10.2%
9.5%
Use of ‘uncorrected’ p-value, a=0.1
Percentage of Null Pixels that are False Positives
Using an ‘uncorrected’ p-value of 0.1 will lead us to conclude on average that 10% of
voxels are active when they are not.
This is clearly undesirable. To correct for this we can define a null hypothesis for
images of statistics.

12. Family-wise Null Hypothesis
Activation is zero everywhere
If we reject a voxel null hypothesis
at any voxel, we reject the family-wise
Null hypothesis
A FP anywhere in the image
gives a Family Wise Error (FWE)
Family-Wise Error (FWE) rate = ‘corrected’ p-value

13. Use of ‘uncorrected’ p-value, a=0.1
Use of ‘corrected’ p-value, a=0.1
FWE

14. Spatial correlation
Independent Voxels
Spatially Correlated Voxels

15. Random Field Theory
Consider a statistic image as a discretisation of a continuous underlying random field
Use results from continuous random field theory
Discretisation

16. Euler Characteristic (EC)
Topological measure
threshold an image at u
EC = # blobs
at high u:
Prob blob = avg (EC) so
FWE, a = avg (EC)

17. Example – 2D Gaussian images
α = R (4 ln 2) (2π) -3/2 u exp (-u2/2)
Voxel-wise threshold, u
Number of Resolution
Elements (RESELS), R
N=100x100 voxels,
Smoothness FWHM=10,
gives R=10x10=100

18. Example – 2D Gaussian images
α = R (4 ln 2) (2π) -3/2 u exp (-u2/2)
For R=100 and α=0.05
RFT gives u=3.8

19. SPM results

20. SPM results...

21. Outline Voxel-wise General Linear Models Random Field Theory
Bayesian Modelling

22. Motivation Even without applied spatial smoothing, activation maps
(and maps of eg. AR coefficients) have spatial structure
Contrast
AR(1)
We can increase the sensitivity of our inferences by smoothing
data with Gaussian kernels (SPM2). This is worthwhile, but crude.
Can we do better with a spatial model (SPM5) ?
Aim: For SPM5 to remove the need for spatial smoothing just as
SPM2 removed the need for temporal smoothing

23. The Model Y=XW+E q1 q2 r1 r2 u1 u2 l a b A W Y Voxel-wise AR:
Spatial pooled AR:
Y=XW+E
[TxN] [TxK] [KxN] [TxN]

24. Synthetic Data 1 : from Laplacian Prior
reshape(w1,32,32) t

25. Prior, Likelihood and Posterior
In the prior, W factorises over k and A factorises over p:
The likelihood factorises over n:
The posterior over W therefore does’nt factor over k or n. It is a Gaussian
with an NK-by-NK full covariance matrix. This is unwieldy to even store,
let alone invert ! So exact inference is intractable.

26. Variational Bayes L F KL

27. Variational Bayes If you assume posterior factorises
then F can be maximised by letting
where

28. Variational Bayes
In the prior, W factorises over k and A factorises over p:
In chosen approximate posterior, W and A factorise over n:
So, in the posterior for W we only have to store and invert N K-by-K
covariance matrices.

29. Updating approximate posterior
Regression coefficients, W
AR coefficients, A
Spatial precisions for W
Spatial precisions for A
Observation noise

30. Synthetic Data 1 : from Laplacian Priorx t

31. F
Iteration Number

32. Least Squares VB – Laplacian Prior y y x x Coefficients = 1024
`Coefficient RESELS’ = 366

33. Synthetic Data II : blobs
True
Smoothing
Global prior
Laplacian prior

34. Sensitivity
1-Specificity

35. Event-related fMRI: Faces versus chequerboard
Smoothing
Global prior
Laplacian Prior

36. Event-related fMRI: Familiar faces versus unfamiliar faces
Smoothing
Penny WD, Trujillo-Barreto NJ, Friston KJ. Bayesian fMRI time series analysis with spatial priors.
Neuroimage Jan 15;24(2):
Global prior
Laplacian Prior

37. Summary Voxel-wise General Linear Models Random Field Theory
Bayesian Modelling
Graph-partitioned spatial priors for functional magnetic resonance images. Harrison LM, Penny W, Flandin G, Ruff CC, Weiskopf N, Friston KJ. Neuroimage Dec;43(4):