1. Mixture Models with Adaptive Spatial Priors
Will Penny
Karl Friston
Acknowledgments: Stefan Kiebel and John Ashburner
The Wellcome Department of Imaging Neuroscience, UCL
http//:

2. Data transformations 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

3. Data transformations p <0.05 Statistical parametric map (SPM)
Image time-series
Design matrix
Realignment
General linear model
Statistical
inference
Gaussian
field theory
Normalisation
p <0.05
Template
Parameter estimates

4. Data transformations p <0.05 Statistical parametric map (SPM)
Image time-series
Design matrix/matrices
Mixtures of
General linear models
Realignment
Statistical
inference
Gaussian
field theory
Normalisation
p <0.05
Template
Size, Position and Shape

5. Data transformations Image time-series Design matrix/matrices
Posterior Probability
Map (PPM)
Mixtures of
General linear models
Realignment
Normalisation
Template
Size, Position and Shape

6. Overview Parameter estimation - EM algorithm
Overall Framework - Generative model
Parameter estimation - EM algorithm
Inference - Posterior Probability Maps (PPMs)
Model order selection - How many clusters ?
Auditory and face processing data

7. Cluster-Level Analysis
The fundamental quantities of interest are the
properties of spatial clusters of activation

8. Generative Model
We have ACTIVE components which describe spatially localised clusters of activity with a temporal signature correlated with the activation paradigm.
We have NULL components which describe spatially distributed background activity temporally uncorrelated with the paradigm.
At each voxel and time point fMRI data is a mixture of ACTIVE and NULL components.

9. Generative Model S1 r0 m1 S2 r1 m2 r2

10. Generative Model At each voxel i and time point t
1. Select component k with probability

11. Generative Model At each voxel i and time point t
1. Select component k with probability
Spatial Prior

12. Generative Model At each voxel i and time point t
1. Select component k with probability
Spatial Prior
2. Draw a sample from component k’s temporal model

13. Generative Model At each voxel i and time point t
1. Select component k with probability
Spatial Prior
2. Draw a sample from component k’s temporal model
General Linear Model

14. Generative Model At each voxel i and time point t
1. Select component k with probability
Spatial Prior
2. Draw a sample from component k’s temporal model
General Linear Model

15. Generative Model
Scan 3

16. Generative Model
Scan 4

17. Generative Model
Scan 8

18. Generative Model
Scan 9

19. Expectation-Maximisation (EM) algorithm
Parameter Estimation
Expectation-Maximisation (EM) algorithm

20. Expectation-Maximisation (EM) algorithm
Parameter Estimation
Expectation-Maximisation (EM) algorithm
E-Step

21. Expectation-Maximisation (EM) algorithm
Parameter Estimation
Expectation-Maximisation (EM) algorithm
E-Step

22. Expectation-Maximisation (EM) algorithm
Parameter Estimation
Expectation-Maximisation (EM) algorithm
Temporal
E-Step
Spatial
Posterior
Normalizer

23. Expectation-Maximisation (EM) algorithm
Parameter Estimation
Expectation-Maximisation (EM) algorithm
M-Step
Prototype time series
for component k

24. Parameter Estimation Expectation-Maximisation (EM) algorithm M-Step
Prototype time series
for component k
Variant of
Iteratively Reweighted
Least Squares

25. Parameter Estimation Expectation-Maximisation (EM) algorithm M-Step
Prototype time series
for component k
Variant of
Iteratively Reweighted
Least Squares
mk and Sk updated using line search

26. Auditory Data
SPM
MGLM (K=1)

27. Auditory Data
SPM
MGLM (K=2)

28. Auditory Data
SPM
MGLM (K=3)

29. Auditory Data
SPM
MGLM (K=4)

30. How many components ? Integrate out dependence on model parameters, q
This can be approximated using the
Bayesian Information Criterion (BIC)
Then use Baye’s rule to pick optimal model order

31. How many components ?
Log L
BIC
P(D|K) K K

32. Auditory Data
MGLM (K=2)
Diffuse Activation
t=15

33. Auditory Data
MGLM (K=3)
Focal Activations
t=20
t=14

34. Smoothing can remove signal
Smoothing will remove signal here
Spatial priors adapt to shape

35. Conclusions SPM is a special case of our model
We don’t need to smooth the data and risk losing signal
Principled method for pooling data
Effective connectivity
Temporal as well as spatial clustering
Spatial hypothesis testing (eg. stroke)
Extension to multiple subjects