voxelwise modeling: understanding brain function with
TRANSCRIPT
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Voxelwise Modeling:understanding brain function with predictive models of brain activity
Matteo Visconti di Oleggio CastelloTom Dupré la Tour
Gallant Lab
Cognitive Neuroscience Colloquium, UC BerkeleyMarch 8, 2021
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Classic GLM/SPM
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Classic GLM/SPM
Xw = Y
Not shown:- ways to account for HRF- baseline- nuisance regressors- contrasts
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Classic GLM/SPM
Xw = Y
Not shown:- ways to account for HRF- baseline- nuisance regressors- contrasts
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Classic GLM/SPM
Xw = Y
Not shown:- ways to account for HRF- baseline- nuisance regressors- contrasts
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Classic GLM/SPM
Xw = Y
w1= (0.9, 0)T
w2 = (0, 0)TNot shown:- ways to account for HRF- baseline- nuisance regressors- contrasts
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Classic GLM/SPM
Xw = Y
SE1= (0.3, 0.9)T
SE2 = (0.6, 0.5)Tw1= (0.9, 0)T
w2 = (0, 0)TNot shown:- ways to account for HRF- baseline- nuisance regressors- contrasts
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Classic GLM/SPM
Xw = Y
SE1= (0.3, 0.9)T
SE2 = (0.6, 0.5)Tw1= (0.9, 0)T
w2 = (0, 0)Tt1= (3, 0)T
t2 = (0, 0)TNot shown:- ways to account for HRF- baseline- nuisance regressors- contrasts
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Classic GLM/SPM
Xw = Y
SE1= (0.3, 0.9)T
SE2 = (0.6, 0.5)Tw1= (0.9, 0)T
w2 = (0, 0)Tt1= (3, 0)T
t2 = (0, 0)T
Effect estimate Noise estimate Statistic
Not shown:- ways to account for HRF- baseline- nuisance regressors- contrasts
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Classic GLM/SPM
Xw = Y
SE1= (0.3, 0.9)T
SE2 = (0.6, 0.5)Tw1= (0.9, 0)T
w2 = (0, 0)Tt1= (3, 0)T
t2 = (0, 0)T
Effect estimate Noise estimate Statistic
Not shown:- ways to account for HRF- baseline- nuisance regressors- contrasts
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Classic GLM/SPM
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Kanwisher, 2017
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Experimental problems with classic GLM/SPM
● Complex sensory and cognitive processes must be reduced to fit into designs that can be handled by an SPM approach
● Often this means simple factorial designs
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Methodological problems with classic GLM/SPM
● Goodness-of-fit approach based on inferential statistics○ Inferences are based on the significance of the estimated model parameters○ Effect estimates are largely ignored (Chen, Taylor, & Cox, 2017)
■ statistical significance does not imply practical significance
● No measures of whether the results (and model parameters) will generalize to new conditions or datasets○ models are fit in a single dataset (overfitting)○ variance due to the (small number of) stimuli used is largely unaccounted for
(stimulus-as-fixed-effect fallacy; Westfall, Nichols, & Yarkoni, 2017)
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Methodological problems with classic GLM/SPM
● Goodness-of-fit approach based on inferential statistics○ Inferences are based on the significance of the estimated model parameters○ Effect estimates are largely ignored (Chen, Taylor, & Cox, 2017)
■ statistical significance does not imply practical significance
● No measures of whether the results (and model parameters) will generalize to new conditions or datasets○ models are fit in a single dataset (overfitting)○ variance due to the (small number of) stimuli used is largely unaccounted for
(stimulus-as-fixed-effect fallacy; Westfall, Nichols, & Yarkoni, 2017)
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Methodological problems with classic GLM/SPM
● Classic GLM/SPM provides little guarantee that
○ the experimental results will replicate (Szucs & Ioannidis, 2017)
○ the model tested will generalize (Yarkoni, 2019; Westfall, Nichols, & Yarkoni, 2017)
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A different approach: Voxelwise Modeling
● Respect the complexity of the real world (do not reduce the elephant!)
● Avoid the goodness-of-fit approach and null-hypothesis statistical testing (data modeling culture; Breiman, 2001)
● Use methods from machine learning and data science (algorithmic modeling culture; Breiman, 2001)
○ Create models that accurately predict brain activity
○ Estimate model prediction accuracy on an independent dataset
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● low-level visual features (motion energy)● objects in the scene● facial expressions● emotions portrayed● social interactions
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● low-level visual features (motion energy)● objects in the scene● facial expressions● emotions portrayed● social interactions
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● low-level visual features (motion energy)● objects in the scene● facial expressions● emotions portrayed● social interactions
● spectral features● speech content
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● low-level visual features (motion energy)● objects in the scene● facial expressions● emotions portrayed● social interactions
● spectral features● speech content
Xw = Y
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Xw = Y
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Xw = Y
Zw
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Xw = Y
Zw
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Xw = Y
Zw
???
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Xw = Y
Zw
???
Modelselection(training set)
Modelassessment(test set)
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Example: Huth et al., 2016
Model selection(Training set)
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Example: Huth et al., 2016
Model assessment(Test set)
Model selection(Training set)
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Example: Huth et al., 2016
Model assessment(Test set)
Model selection(Training set)
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Example: Huth et al., 2016
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Model assessment(Test set)
Model selection(Training set)
Example: Huth et al., 2016
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Example: Huth et al., 2016
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Example: Deniz et al., 2019Model selection
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Example: Deniz et al., 2019Model selection Model assessment
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Example: Deniz et al., 2019
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Example: Deniz et al., 2019
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How to fit voxelwise models?
● Feature spaces describing the stimulus are high-dimensional
○ More dimensions than the number of samples available in the training set
● There is a high risk of overfitting: failure to generalize
● We need to use techniques from machine learning and data science to fit voxelwise models
○ Regularized regression
○ Cross-validation
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Regularized linear regression
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Linear regression
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Linear regression
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Linear regression
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Linear regression
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Multivariate linear regression
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Multivariate linear regression
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Multivariate linear regression
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Multivariate linear regression
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Multivariate linear regression - correlated features
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Multivariate linear regression - correlated features
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Multivariate linear regression - collinearity
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Multivariate linear regression - regularization (ridge)
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Multivariate linear regression - regularization (ridge)
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Ridge regression
Definition
Linear regression w* = argminw ||y - Xw||2
Ridge regression w* = argminw ||y - Xw||2 + 𝛼 ||w||2
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Ridge regression
Definition
Linear regression w* = argminw ||y - Xw||2
Ridge regression w* = argminw ||y - Xw||2 + 𝛼 ||w||2
Analytical solution
Linear regression w* = (XTX)-1 XTy λ0-1
Ridge regression w* = (XTX + 𝛼Id)-1 XTy (λ0+𝛼)-1
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Ridge regression
BenefitsMore robust with correlated features Fix collinearity issuesFix the case n_features > n_samples (underdetermined system)
DrawbackUnknown hyperparameter 𝛼 (theoretical link to the signal-to-noise ratio)
SolutionCross-validation
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Cross-validation
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Cross-validation
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Cross-validation
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Cross-validation
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Cross-validation
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Hyperparameter path
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Hyperparameter path
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Cross-validation - more folds
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Cross-validation - hyperparameter selection
for each hyperparameter candidatefor each split of the data
fit a model on the training foldsscore the fitted model on the validation fold
average scores over all splitsselect best hyperparameter
ExampleSelection of 𝛼 in ridge regression
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Cross-validation - model selection
for each model candidatefor each split of the data
fit a model on the training foldsscore the fitted model on the validation fold
average scores over all splitsselect best model
ExampleRidge regression versus Lasso
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Model selection example - Time delays
To model the hemodynamic response functionwe copy all the features with different time delays
but how many delays is optimal ?
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Model selection example - Time delays
To model the hemodynamic response functionwe copy all the features with different time delays
but how many delays is optimal ?Method: cross-validation
Answer: 4 (for this dataset)
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Generalization to new data
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Generalization to new data
Generalization powerEstimated with prediction on a held-out test dataset
Generalization lower-bound (i.e. significance)Estimated with permutations
Generalization upper-bound (i.e. explainable variance)Estimated with repeats of the same stimulus
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Explainable variance
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Tutorials
https://github.com/gallantlab/voxelwise_tutorialstutorials in python, notebooks stylevoxelwise modeling helper functions
https://github.com/gallantlab/himalayapython package, scikit-learn API, CPU/GPUridge-regression-like models for large number of voxels
(both repositories are still private for now)send me an email if you want an early access [email protected] much appreciated !
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Advanced Voxelwise Modeling
Advanced use of the framework include:
● use very large number of features extracted from deep neural networks
● partition the explained variance over multiple feature spaces (with banded ridge regression)
● separate features over different timescales
● ...
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Tutorials(Fit a ridge model with wordnet features)
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Association is not prediction
[Statistical Modeling: The Two Cultures, Breiman, 2001, Statistical science][Statistics versus machine learning, Bzdok et al., 2018, Nature Method]
“In the unfolding era of big data in medicine, the phrase “association is not prediction” should become as important as “correlation is not causation”.”[Bzdok et al., 2021, JAMA Psychiatry]
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1 - Voxelwise modeling vs classical fMRI analysis
ComparisonClassical: Block design, linear regression, t-testVM: Feature extraction, still a linear regression (!), but test set predictionsMain difference: association/inference vs prediction - (old debate)
(inference = interpretable) vs (prediction = black box) ?no, we can still use linear models (!= random forest or neural networks)
Prediction is about replicability, generalization to new settingsassociation can be highly dependent to particular subjects, cross-val less
Prediction estimates the effect size (explained variance)large significance (e.g. with many subjects) != large effect
Test set predictions largely reduces overfittingwith enough features, one can explains 100% variance within set
even with linear models
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2 - Voxelwise modeling
Regularized regressionReduces collinearity overfittingReduces n_features > n_samples overfittingHandles different SNR per voxel
Model selection with cross-validationhyperparameter selection - example of ridge regularizationmodel selection - example of the number of delays
Test set generalization as a final scoregeneralization lower bound (ie significance) with shufflinggeneralization upper-bound (ie explainable variance) with repeats
Interpreting feature weightsfeature importancePCA
Tutorials