Mixture Models with Adaptive Spatial Priors Will Penny Karl Friston Acknowledgments: Stefan Kiebel and John Ashburner The Wellcome Department of Imaging Neuroscience, UCL http//:
Data transformations RealignmentSmoothing Normalisation General linear model Statistical parametric map (SPM) Image time-series Parameter estimates Design matrix Template Kernel Gaussian field theory p <0.05 Statisticalinference
Data transformations Realignment Normalisation General linear model Statistical parametric map (SPM) Image time-series Parameter estimates Design matrix Template Gaussian field theory p <0.05 Statisticalinference
Data transformations Realignment Normalisation Mixtures of Mixtures of General linear models Statistical parametric map (SPM) Image time-series Size, Position and Shape Design matrix/matrices Template Gaussian field theory p <0.05 Statisticalinference
Data transformations Realignment Normalisation Mixtures of Mixtures of General linear models Posterior Probability Map (PPM) Map (PPM) Image time-series Size, Position and Shape Design matrix/matrices Template
OverviewOverview Overall Framework - Generative model Overall Framework - Generative model Parameter estimation - EM algorithmParameter estimation - EM algorithm Inference - Posterior Probability Maps (PPMs) Inference - Posterior Probability Maps (PPMs) Model order selection - How many clusters ? Model order selection - How many clusters ? Auditory and face processing data Auditory and face processing data
Cluster-Level Analysis The fundamental quantities of interest are the properties of spatial clusters of activation
Generative Model We have ACTIVE components which describe spatially localised clusters of activity with a temporal signature correlated with the activation paradigm.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.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.At each voxel and time point fMRI data is a mixture of ACTIVE and NULL components. We have ACTIVE components which describe spatially localised clusters of activity with a temporal signature correlated with the activation paradigm.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.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.At each voxel and time point fMRI data is a mixture of ACTIVE and NULL components.
Generative Model m1m1 m2m2 11 22 r0r0 r1r1 r2r2
At each voxel i and time point t 1. Select component k with probability
Generative Model At each voxel i and time point t 1. Select component k with probability Spatial Prior
Generative Model At each voxel i and time point t 1. Select component k with probability 2. Draw a sample from component k’s temporal model Spatial Prior
Generative Model At each voxel i and time point t 1. Select component k with probability 2. Draw a sample from component k’s temporal model Spatial Prior General Linear Model
Generative Model At each voxel i and time point t 1. Select component k with probability 2. Draw a sample from component k’s temporal model Spatial Prior General Linear Model
Generative Model Scan 3
Generative Model Scan 4
Generative Model Scan 8
Generative Model Scan 9
Parameter Estimation Expectation-Maximisation (EM) algorithm
Parameter Estimation Expectation-Maximisation (EM) algorithm E-Step
Parameter Estimation Expectation-Maximisation (EM) algorithm E-Step
Parameter Estimation Expectation-Maximisation (EM) algorithm E-Step Posterior Temporal Spatial Normalizer
Parameter Estimation Expectation-Maximisation (EM) algorithm M-Step Prototype time series for component k A semi-supervised estimate of activity in clusrer k
Parameter Estimation Expectation-Maximisation (EM) algorithm M-Step Prototype time series for component k Variant of Iteratively Reweighted Least Squares
Parameter Estimation Expectation-Maximisation (EM) algorithm M-Step m k and k updated using line search Prototype time series for component k Variant of Iteratively Reweighted Least Squares
Auditory Data SPMMGLM (K=1)
Auditory Data SPMMGLM (K=2)
Auditory Data SPMMGLM (K=3)
Auditory Data SPMMGLM (K=4)
How many components ? Integrate out dependence on model parameters, This can be approximated using the Bayesian Information Criterion (BIC) Then use Baye’s rule to pick optimal model order
How many components ? KK P(D|K) BIC Log L
Auditory Data MGLM (K=2) t=15 Diffuse Activation
Auditory Data MGLM (K=3) Focal Activations t=20 t=14
Face Data This is an event-related study 60 secs Face Events BOLD Signal
Face Data SPMMGLM (K=2)
Face Data Prototype time series for cluster 60 secs GLM Estimate (solid line) (dotted line)
Smoothing can remove signal Spatial priors adapt to shape Smoothing will remove signal here
ConclusionsConclusions SPM is a special case of our modelSPM is a special case of our model We don’t need to smooth the data and risk losing signalWe don’t need to smooth the data and risk losing signal Principled method for pooling dataPrincipled method for pooling data Effective connectivityEffective connectivity Spatio-temporal clusteringSpatio-temporal clustering Spatial hypothesis testing (eg. stroke)Spatial hypothesis testing (eg. stroke) Extension to multiple subjectsExtension to multiple subjects SPM is a special case of our modelSPM is a special case of our model We don’t need to smooth the data and risk losing signalWe don’t need to smooth the data and risk losing signal Principled method for pooling dataPrincipled method for pooling data Effective connectivityEffective connectivity Spatio-temporal clusteringSpatio-temporal clustering Spatial hypothesis testing (eg. stroke)Spatial hypothesis testing (eg. stroke) Extension to multiple subjectsExtension to multiple subjects