Statistical Parametric Mapping

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

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

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

4. Voxel-wise GLMs model specification parameter estimation Time 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. L g l Estimation 1. ReML-algorithm 2. Weighted Least Squares Friston et al. 2002, Neuroimage

8. Contrasts &SPMs c = 1 0 0 0 0 0 0 0 0 0 0 Q: activation during listening ? Null hypothesis:

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

10. Signal Inference for Images Noise Signal+Noise

11. Use of ‘uncorrected’ p-value, a=0.1 11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5% 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 Null Hypothesis 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. Consider a statistic image as a discretisation of a continuous underlying random field Use results from continuous random field theory Random Field Theory Discretisation

16. Topological measure threshold an image at u EC=# blobs at high u: Prob blob = avg (EC) so FWE, a = avg (EC) Euler Characteristic (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. q1 q2 a b u1 u2 l W A Y The Model r2 r1 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. L KL F Variational Bayes

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

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

30. Synthetic Data 1 : from Laplacian Prior y t x

31. F Iteration Number

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

33. Synthetic Data II : blobs Smoothing True 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. 2005 Jan 15;24(2):350-62. Global prior Laplacian Prior

37. Summary • Voxel-wise General Linear Models • Random Field Theory • Bayesian Modelling http://www.fil.ion.ucl.ac.uk/~wpenny/mbi/index.html Graph-partitioned spatial priors for functional magnetic resonance images. Harrison LM, Penny W, Flandin G, Ruff CC, Weiskopf N, Friston KJ. Neuroimage. 2008 Dec;43(4):694-707.