Probabilistic Modelling of Brain Imaging Data
description
Transcript of Probabilistic Modelling of Brain Imaging Data
Probabilistic Modelling of Probabilistic Modelling of Brain Imaging DataBrain Imaging Data
Probabilistic Modelling of Probabilistic Modelling of Brain Imaging DataBrain Imaging Data
Will PennyWill Penny
The Wellcome Department of Imaging Neuroscience, The Wellcome Department of Imaging Neuroscience, UCLUCL
http//:www.fil.ion.ucl.ac.uk/~wpennyhttp//:www.fil.ion.ucl.ac.uk/~wpenny
OverviewOverviewOverviewOverview
1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference
2. A model of fMRI time series: The Noise2. A model of fMRI time series: The Noise
3. A model of fMRI time series: The Signal 3. A model of fMRI time series: The Signal
1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference
2. A model of fMRI time series: The Noise2. A model of fMRI time series: The Noise
3. A model of fMRI time series: The Signal 3. A model of fMRI time series: The Signal
First level of Bayesian InferenceFirst level of Bayesian InferenceFirst level of Bayesian InferenceFirst level of Bayesian Inference
)(
)()|()|(
yp
pypyp
First level of Inference: What are the best parameters ?
We have data, y, and some parameters,
Parameters are of model, M, ….
First and Second LevelsFirst and Second LevelsFirst and Second LevelsFirst and Second Levels
)|(
)|(),|(),|(
Myp
MpMypMyp
The first level again, writing in dependence on M:
)(
)()|()|(
yp
MpMypyMp
Second level of Inference: What’s the best model ?
Model SelectionModel SelectionModel SelectionModel Selection
We need to compute the Bayesian Evidence:
dpypMyp )()|()|(
We can’t always compute it exactly, but we can approximate it: Log p(y|M) ~ F(M)
Evidence = Accuracy - Complexity
Model AveragingModel AveragingModel AveragingModel Averaging
Revisiting the first level:
)|(),|()|( yMpMypypM
Model-dependent posteriors are weighted accordingto the posterior probability of each model
Multiple Levels
12112 )|()|()|( dwwwpwYpwYp
23223 )|()|()|( dwwwpwYpwYp
22211 )|(),|()|( dwYwpwYwpYwp
33322 )|(),|()|( dwYwpwYwpYwp
)(
)()|()|(
333
Yp
wpwYpYwp
w3
Y
w1
w2
333 )()|()( dwwpwYpYp
Evidence Up Posteriors Down
w3
Y
w1
w2
OverviewOverviewOverviewOverview
1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference
2. 2. A model of fMRI time series: The NoiseA model of fMRI time series: The Noise
3. A model of fMRI time series: The Signal 3. A model of fMRI time series: The Signal
1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference
2. 2. A model of fMRI time series: The NoiseA model of fMRI time series: The Noise
3. A model of fMRI time series: The Signal 3. A model of fMRI time series: The Signal
Noise sources in fMRINoise sources in fMRINoise sources in fMRINoise sources in fMRI
1. Slow drifts due to instrumentation instabilities1. Slow drifts due to instrumentation instabilities
2. Subject movement2. Subject movement
3. 3. Vasomotor oscillation ~ 0.1 HzVasomotor oscillation ~ 0.1 Hz
4. Respiratory activity ~ 0.25 Hz4. Respiratory activity ~ 0.25 Hz
5. Cardiac activity ~ 1 Hz5. Cardiac activity ~ 1 Hz
1. Slow drifts due to instrumentation instabilities1. Slow drifts due to instrumentation instabilities
2. Subject movement2. Subject movement
3. 3. Vasomotor oscillation ~ 0.1 HzVasomotor oscillation ~ 0.1 Hz
4. Respiratory activity ~ 0.25 Hz4. Respiratory activity ~ 0.25 Hz
5. Cardiac activity ~ 1 Hz5. Cardiac activity ~ 1 Hz
Remove with ICA/PCA – but non-automatic
fMRI time series modelfMRI time series modelfMRI time series modelfMRI time series model
• Use a General Linear Model at each voxel:Use a General Linear Model at each voxel:
y = X y = X + e + e
where X contains task-related regressors.where X contains task-related regressors.
• Use a General Linear Model at each voxel:Use a General Linear Model at each voxel:
y = X y = X + e + e
where X contains task-related regressors.where X contains task-related regressors.
fMRI time series modelfMRI time series modelfMRI time series modelfMRI time series model
= +
y X e
Time-seriesat one spatial
location
Putative effects ofexperimentalmanipulation
Sizeof effects
Residuals
fMRI time series modelfMRI time series modelfMRI time series modelfMRI time series model
• Use a General Linear Model at each voxel:Use a General Linear Model at each voxel:
y = X y = X + e + e
where X contains task-related regressors.where X contains task-related regressors.
• The errors are modelled as an AR(p) process.The errors are modelled as an AR(p) process. ((Parametric spectral estimationParametric spectral estimation))
• The order can be selected using Bayesian evidenceThe order can be selected using Bayesian evidence
• Use a General Linear Model at each voxel:Use a General Linear Model at each voxel:
y = X y = X + e + e
where X contains task-related regressors.where X contains task-related regressors.
• The errors are modelled as an AR(p) process.The errors are modelled as an AR(p) process. ((Parametric spectral estimationParametric spectral estimation))
• The order can be selected using Bayesian evidenceThe order can be selected using Bayesian evidence
Synthetic GLM-AR(3) DataSynthetic GLM-AR(3) DataSynthetic GLM-AR(3) DataSynthetic GLM-AR(3) Data
Map of AR model order, pMap of AR model order, pMap of AR model order, pMap of AR model order, p
p=0,1,2,3FaceData
AngiogramsAngiogramsAngiogramsAngiograms
Other subjects, aOther subjects, a11Other subjects, aOther subjects, a11
Ring ofvoxels with
highly correlatederror
Other subjects, aOther subjects, a11Other subjects, aOther subjects, a11
Unmodelledsignal
orincreasedcardiac
artifact due to increasedblood flow?
OverviewOverviewOverviewOverview
1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference
2. A model of fMRI time series: The Noise2. A model of fMRI time series: The Noise
3. 3. A model of fMRI time series: The SignalA model of fMRI time series: The Signal
1.1. Multiple levels of Bayesian InferenceMultiple levels of Bayesian Inference
2. A model of fMRI time series: The Noise2. A model of fMRI time series: The Noise
3. 3. A model of fMRI time series: The SignalA model of fMRI time series: The Signal
fMRI time series modelfMRI time series modelfMRI time series modelfMRI time series model
• Use a General Linear Model Use a General Linear Model
y = X y = X + e + e
• Priors factorise into groups:Priors factorise into groups:
p(p() = p() = p(11) p() p(22) p() p(33))
• Priors in each group may be smoothness Priors in each group may be smoothness priors or Gaussianspriors or Gaussians
• Use a General Linear Model Use a General Linear Model
y = X y = X + e + e
• Priors factorise into groups:Priors factorise into groups:
p(p() = p() = p(11) p() p(22) p() p(33))
• Priors in each group may be smoothness Priors in each group may be smoothness priors or Gaussianspriors or Gaussians
Rik’s dataRik’s dataRik’s dataRik’s data
24 Transverse Slices acquired with TR=2s
Press left key if famous, right key if not
Time series of 351 images
Part of larger study lookingat factors influencing repetition suppresion
Every face presented twice
Modelling the SignalModelling the SignalModelling the SignalModelling the Signal
Assumption: Neuronal Event Stream is Identical to the Experimental Event Stream
Convolve event-stream with basis functions to account for the HRF
FIR modelsFIR modelsFIR modelsFIR models
Timeafterevent
Sizeof
signal
FIR modelFIR modelFIR modelFIR model
Separate smoothness priors for each event type
Design matrixfor FIR model with
8 time bins in a 20-second window
Q. Is this a good prior ?
FIR basis setFIR basis setFIR basis setFIR basis set
Left occipital cortex (x=-33, y=-81, z=-24)
FIR model average responses
FIR basis setFIR basis setFIR basis setFIR basis set
Right fusiform cortex (x=45, y=-60, z=-18)
FIR model average responses
RFX-Event modelRFX-Event modelRFX-Event modelRFX-Event model
Design Matrix
97 parameters ! But only 24 effective parameters
Responses to each event of type A are randomly distributed about some typical “type A” response
Non-stationary modelsNon-stationary modelsNon-stationary modelsNon-stationary models
As RFX-event but smoothness priors
Testing for smooth temporal variations statistically …
Comparing Types of ModelsComparing Types of ModelsComparing Types of ModelsComparing Types of Models
Left Occipital Right FusiformEvidence
Model averaging to get peak post-stimulus response
RFX-Event
FIR
NonStat
RFX-Event
FIR
NonStat
SummarySummarySummarySummary
• Bayesian inference provides a framework for Bayesian inference provides a framework for
model comparison and synthesismodel comparison and synthesis
• Appropriate for fMRI as we have some prior Appropriate for fMRI as we have some prior knowledgeknowledge
• We have focussed on temporal modelsWe have focussed on temporal models
• Bayesian inference provides a framework for Bayesian inference provides a framework for
model comparison and synthesismodel comparison and synthesis
• Appropriate for fMRI as we have some prior Appropriate for fMRI as we have some prior knowledgeknowledge
• We have focussed on temporal modelsWe have focussed on temporal models