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A supervised clustering approach for fMRI-based inference of brain states

Vincent Michel, Alexandre Gramfort, Gaël Varoquaux, Evelyn Eger, Christine Keribin, Bertrand Thirion

arXiv:1104.5304v1cs.CV

TL;DR

Predicting behavior from fMRI faces a high-dimensional feature space, while common voxel-wise selection does not account for spatial structure. The paper builds connected, hierarchical parcels and prunes them using the prediction target; it reports improved prediction performance, including in inter-subject analyses.

  • Problem

    Brain decoding must localize predictive regions when the voxel count greatly exceeds the sample count, while voxel-wise selection can ignore spatial structure and select redundant features.

    Source: §1 · pp. 1–2

  • Method

    The method builds a connectivity-constrained hierarchy of voxel groups and selects a tree cut using prediction accuracy, yielding target-guided parcel averages at multiple scales.

    Source: §2.1 · p. 3, §2.1.1 · p. 4

  • Results

    The proposed methods yield better prediction accuracy than state-of-the-art approaches in inter-subject experiments, while supervised cuts achieve similar or higher accuracy than unsupervised cuts.

    Source: §5 · p. 11

  • Takeaways & Limitations

    The approach can highlight informative regions while leaving uninformative areas unsegmented, and its spatial averaging appears powerful for handling inter-subject variability.

    Source: §5 · pp. 11–12

    Internal cross-validation of both pruning and model selection is time-consuming and yields fold-specific subtrees that are difficult to combine for test-set prediction.

    Source: §2.2 · p. 5

Abstract

from arXiv · show

We propose a method that combines signals from many brain regions observed in functional Magnetic Resonance Imaging (fMRI) to predict the subject's behavior during a scanning session. Such predictions suffer from the huge number of brain regions sampled on the voxel grid of standard fMRI data sets: the curse of dimensionality. Dimensionality reduction is thus needed, but it is often performed using a univariate feature selection procedure, that handles neither the spatial structure of the images, nor the multivariate nature of the signal. By introducing a hierarchical clustering of the brain volume that incorporates connectivity constraints, we reduce the span of the possible spatial configurations to a single tree of nested regions tailored to the signal. We then prune the tree in a supervised setting, hence the name supervised clustering, in order to extract a parcellation (division of the volume) such that parcel-based signal averages best predict the target information. Dimensionality reduction is thus achieved by feature agglomeration, and the constructed features now provide a multi-scale representation of the signal. Comparisons with reference methods on both simulated and real data show that our approach yields higher prediction accuracy than standard voxel-based approaches. Moreover, the method infers an explicit weighting of the regions involved in the regression or classification task.

1. Introduction

Brain decoding uses prediction from activation images to investigate cognitive and behavioral information, but the large voxel count makes localization of predictive regions difficult. The paper introduces target-aware, multiscale clustering and reports higher prediction performance than previous approaches, including for analysis across subjects.

1. Introduction

  • Brain decoding evaluates whether a classifier predicts a behavioral target from brain activation images, helping assess the involvement of brain regions in cognitive, motor, or perceptual functions.The introduction also presents inverse inference as useful for investigating brain coding principles by linking inferred parameters to spatial organization.Source: §1 · p. 1
  • With roughly 10^5 voxels but 10^2 samples, prediction can overfit; voxel-wise ANOVA may select redundant, spatially unconstrained features that are difficult to interpret.Spatially informed predictive features can produce anatomically coherent regions within inverse inference.Source: §1 · p. 2
  • Feature agglomeration replaces voxel signals with local parcel averages, reducing the feature count from roughly 10^5 voxels to 10^2 parcels.The approach is motivated by spatial coherence and redundancy in predictive fMRI information.Source: §1 · p. 2
  • Because spatial averages can lose fine-grained information needed for accurate decoding, multiple information resolutions should be allowed.The optimal number of clusters can also be difficult to determine.Source: §1 · p. 2
  • The method uses target-aware clustering to create adaptive, multiscale brain regions and reports higher prediction performance than previous approaches, including across subjects.It is presented for both classification and regression.Source: §1 · p. 2

2. Methods

The methods represent activation maps as features for linear prediction and define parcels as connected voxel groups that partition the feature set. Parcel-based signals average voxel activity within each group, reducing the representation to one feature per parcel.

Predictive linear model

  • The linear model relates the activation-map design matrix X to the behavior variable y using weights w and an intercept b.Each row of X is a sample activation map; p is the number of features, and w can be viewed as an image.Source: Predictive linear model · p. 2
  • For classification, the linear model applies the sign function to Xw + b and uses labels in {−1, 1}.The intercept can be incorporated into w by adding a column of ones to X.Source: Predictive linear model · pp. 2–3
  • A parcel is a connected group of voxels, and a parcellation partitions the full feature set into disjoint parcels.The number of parcels is denoted δ.Source: Parcels · p. 3
  • The parcel-based signal represents each parcel by the average signal of its voxels, producing one feature for each parcel.The text notes that other representations, such as parcel medians, are possible.Source: Parcels · p. 3

Bayesian Ridge Regression

Bayesian Ridge Regression (BRR) is the article’s predictive linear model for regression, using parcel-based averages as inputs. Its formulation regularizes weights with a Gaussian prior and estimates model parameters iteratively.

Bayesian Ridge Regression

  • BRR is the predictive linear model used for regression, with implementation details also provided for parcel-based averages Xp.Source: Bayesian Ridge Regression · p. 3
  • Gaussian noise with precision α and a zero-centered Gaussian prior on weights w with precision λ define the regression model and its weight regularization.The prior constrains weight values to be small.Source: Bayesian Ridge Regression · p. 3
  • Gamma priors on α and λ complete the Bayesian framework and are added to avoid degenerate solutions.Source: Bayesian Ridge Regression · p. 3
  • The experiments set λ1, λ2, α1, and α2 to 10^-6, described as weakly informative priors.Source: Bayesian Ridge Regression · p. 3
  • BRR iteratively maximizes the log likelihood, updating µ and Σ to estimate γ, λ, and α, with convergence monitored through changes in w.The procedure starts with α = 1/var(yt) and λ = 1, and stops when the L1 difference between consecutive w values is below 10^-3.Source: Bayesian Ridge Regression · p. 3

2.1. Supervised clustering

Supervised clustering builds a hierarchical brain parcellation and selects tree cuts according to how well parcel-average signals predict the target. The resulting parcellation provides a low-dimensional representation tailored to the prediction task.

2.1. Supervised clustering

  • Ward-tree cuts are selected to maximize prediction accuracy from parcel-based averages, making parcellation a supervised learning step.Source: §2.1 · pp. 3–4

2.1.1. Bottom-Up step: hierarchical clustering

  • Ward clustering builds a nested tree from spatially connected parcels, merging parcels to minimize within-parcel sums of squared differences.Source: §2.1.1 · p. 4
  • The unsupervised cut divides the tree into its main branches, whereas the supervised cut greedily splits a parcel according to the prediction score.The supervised-cut figure contrasts a horizontal six-parcel cut with a score-selected cut focusing on more informative regions.Source: §2.1.2 · p. 4, Figure 2 · p. 4
  • From the nested parcellations, model selection retains the sub-tree with the optimal prediction score for training and testing the prediction function.Source: §2.1.2 · pp. 4–5, Figure 1 · p. 5

2.2. Algorithmic considerations

The authors use two sequential cross-validation schemes for pruning and model selection, describing this choice as empirical and potentially biased.

  • Nested internal cross-validation for pruning and model selection is very time-consuming.Source: §2.2 · p. 5
  • Internal folds produce different optimal sub-trees, which are difficult to combine into a single test-set model.Source: §2.2 · p. 5
  • The authors therefore use separate sequential cross-validation schemes for pruning and model selection, an empirical, potentially biased heuristic.Source: §2.2 · p. 5

2.3. Computational considerations

Computational cost depends on the prediction function and two cross-validation schemes; the authors report an example runtime that remained affordable for standard neuroimaging analyses.

  • O(n∆(2+α)) is the stated overall cost when fitting the prediction function costs O(δα) at step δ.The cost depends on prediction-function complexity and the two cross-validation schemes.Source: §2.3 · p. 5
  • The authors report that cost remains affordable when ∆<10^3, as in their experiments, while higher ∆ may require a less complex prediction function.Source: §2.3 · p. 5
  • 207 seconds were needed to construct the tree and 215 seconds to define parcels for one subject with approximately 7.104 voxels.The passage notes that the Python implementation was not optimized and characterizes the runtime as affordable for standard neuroimaging analyses.Source: §2.3 · pp. 5–6

2.4. Performance evaluation

Performance is evaluated with training and test data; regression uses explained variance, while classification uses a standard score denoted κ.

  • The evaluation splits available data into learning and test sets and predicts targets for the test set using parameters estimated from training.Source: §2.4 · p. 6
  • For regression, ζ is the ratio of response variability explained by the model.Source: §2.4 · p. 6
  • For classification, performance is evaluated with a standard classification score denoted κ.The number of test samples is denoted n_t, and δ is Kronecker’s delta.Source: §2.4 · p. 6

2.5. Competing methods

The section compares supervised clustering with regularized regression and classification baselines. Reference methods are evaluated after training-set Anova feature selection, with tuning and comparison procedures tailored to the task.

2.5. Competing methods

  • Elastic net regression tunes λ1 and λ2 by internal cross-validation over specified candidate ranges.λ1 is selected from fractions of ˜λ = ∥XTy∥∞, while λ2 is selected from {0.1, 0.5, 1., 10., 100.}.Source: §2.5 · p. 6
  • Linear-kernel SVR is the neuroimaging reference for regression, with C cross-validated from 10−3 to 10.For classification, SVC is optimized similarly to SVR.Source: §2.5 · p. 6
  • Classification baselines are sparse multinomial logistic regression, tuned like Elastic net, and SVC, optimized like SVR.The experiments compare supervised clustering with state-of-the-art regularization methods.Source: §2.5 · p. 6
  • Anova selects features on the training set, with the voxel count chosen from {50, 100, 250, 500} by nested cross-validation before method comparisons.The authors also report that extending the candidate range by 2,000 voxels did not improve accuracy on their real datasets.Source: §2.5 · p. 7

3. Simulated data

The simulations test supervised clustering on block-structured features and on synthetic neuroimaging data with spatially variable informative regions. The results show that supervised cuts recover informative structure that the compared alternatives miss or blur.

3.1. Simulated one-dimensional data

  • The supervised-clustering simulation uses block-structured informative features, with p = 200 features and n = 150 images, Bayesian Ridge Regression as prediction function, and four-fold cross-validation.The supervised cut uses Δ = 50.Source: §3.1 · p. 7

3.2. Simulated neuroimaging data

  • The neuroimaging simulation uses 100 images of 12×12×12 voxels, four 2×2×2 ROIs, and image-specific variation that randomly zeros half the ROI weights.Images are smoothed with a Gaussian kernel of standard deviation 2 voxels; the target has 5 dB SNR, and 100 additional simulated images form the validation set.Source: §3.2 · p. 7

3.3. Results on one-dimensional simulated data

  • The optimal parcellation focuses on two subregions, and its prediction weights clearly outline the two simulated informative regions.The predicted weights are normalized by the number of voxels in each parcel.Source: §3.3 · p. 7

3.4. Results on simulated neuroimaging data

  • Only supervised clustering extracts the simulated discriminative regions; unsupervised clustering misses part of the support, Elastic net is overly sparse, and searchlight maps are blurred.The authors attribute searchlight’s limitation to its inability to handle long-range multivariate structure.Source: §3.4 · p. 8

4. Experiments and results on real data

Real-data experiments test regression for object size and classification for object shape using fMRI data from multiple subjects. Parcel-based methods outperform voxel-based references, while the supervised cut is strongest for shape classification.

4.1. Details on real data

  • The study uses fMRI from ten subjects viewing objects at three sizes and classifies shapes in two categories, with five subjects per category.Size regression uses inter-subject leave-one-subject-out validation on 120 images; classification averages images across sizes.Source: §4.1 · pp. 8–9

4.2. Results for the prediction of size

  • For size prediction, parcel-based methods outperform voxel-based references; unsupervised cut has the best explained variance, while supervised cut is not significantly different.The supervised method yields clustered maps; informative parcels lie mainly in posterior occipital cortex, with additional anterior parcels in posterior lateral occipital cortex.Source: §4.2 · p. 9, Table 1 · p. 10, Figure 5 · p. 10

4.3. Results for the prediction of shape

  • In the object prediction analysis, supervised cut achieves the best prediction accuracy under leave-one-subject-out cross-validation; its classification score is 21% higher than voxel-based SVC and 27% higher than voxel-based SMLR.Both parcel-based approaches are significantly more accurate and more stable than voxel-based approaches, using more than ten times fewer features while yielding higher prediction accuracy.Source: Table 2 · p. 10, §4.3 · p. 10

5. Discussion

The discussion reports improved inter-subject prediction with parcel-based feature agglomeration and describes supervised tree pruning as a way to retain task-relevant, multi-scale structure. It also highlights the method’s interpretable maps and compatibility with varied prediction functions and data types.

5. Discussion

  • Feature agglomeration builds connected-voxel parcels that represent spatial structure and multivariate information, with the capacity to focus on small informative regions while leaving broad uninformative areas unsegmented.The approach is described as a multi-scale segmentation scheme.Source: §5 · pp. 11–12
  • Parcel-based spatial averaging performs well for inter-subject generalization, whereas voxel-based methods perform relatively worse under spatial variability.The discussion attributes this to parcels localizing functional activity across subjects and finding common support for neural codes; Figure 6 illustrates the contrast between weak voxel means and more informative region means.Source: §5 · p. 11, Figure 6 · p. 11
  • The method constrains Ward-based recursive parcellations to spatially connected clusters, then greedily prunes the resulting tree to optimize prediction.The greedy strategy has no guarantee of reaching the optimal cut; model selection chooses the best-generalizing available parcellation, and the method runs in a few minutes without specific optimization.Source: §5 · p. 11
  • Across inter-subject experiments, the proposed methods outperform SVR, Elastic net, SVC and SMLR; supervised cut performs better than unsupervised cut for finer-grained classification.For coarser size prediction, the unsupervised cut yields good accuracy and extracts almost all relevant information.Source: §5 · p. 11
  • The method highlights predictive regions while leaving uninformative regions unsegmented, producing contiguous, interpretable maps that incorporate non-local information.Its maps resemble searchlight maps, but the method yields one whole-brain prediction score; it is presented as intermediate between brain mapping and inverse inference.Source: §5 · pp. 11–12
  • The approach can be used with different classification and regression methods and may extend beyond brain images to datasets where multi-scale structure matters.Medical and satellite images are given as examples.Source: §5 · p. 12
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