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Soft Subspace Projection for Nuisance Removal

Source: R/soft_subspace_projection.R
soft_subspace_projection.Rd

Ridge-regularized projection to remove nuisance variance without explicitly choosing components or adding confound regressors. This is "CompCor without choosing components."

Conceptual Overview

Traditional CompCor extracts K principal components from white matter/CSF and regresses them out. This requires choosing K, which is arbitrary.

Soft subspace projection instead treats the entire WM/CSF timeseries as a nuisance basis and removes it with shrinkage. Each nuisance direction is partially removed proportional to its variance, avoiding the hard keep/delete decision of PCA truncation.

The Projection Operator

Given nuisance matrix N (time x nuisance_voxels), the soft projection is: $$P_\lambda = I - N (N^T N + \lambda I)^{-1} N^T$$

Applied to data Y and design X:

  • Y_clean = P_lambda %*% Y

  • X_clean = P_lambda %*% X (important to avoid bias)

Lambda Selection

The shrinkage parameter lambda controls aggressiveness:

  • Small lambda: aggressive removal (risk removing signal)

  • Large lambda: gentle removal (risk leaving nuisance)

Three selection methods are available:

"auto"

Singular value heuristic: lambda = median(d^2) where d are singular values of N. Fast, stable, no tuning. Components with variance below median are heavily shrunk; above median lightly shrunk.

"gcv"

Generalized cross-validation minimizes RSS/(1-df/n)^2 for ridge regression of Y on N. Finds lambda giving best leave-one-out prediction without actually doing LOO. Requires Y. Computational cost is O(k) per evaluation where k = min(n, p).

numeric

User-specified lambda value.

Typical Usage

The soft subspace projection workflow has three steps:

  1. Extract nuisance timeseries from WM/CSF mask (or provide pre-computed)

  2. Create the soft projection operator

  3. Apply to both data Y and design matrix X before GLM fitting

For use within fmri_lm(), see the convenience parameter nuisance_projection or the more detailed soft_subspace_options.

When to Use

Soft subspace projection is most beneficial when:

  • You have physiological noise from WM/CSF that isn't captured by motion parameters

  • Traditional CompCor requires arbitrary component selection

  • You want to avoid adding many confound regressors to the design

Consider alternatives when:

  • Motion parameters alone sufficiently control artifacts

  • You prefer explicit confound regressors for interpretability

  • Data has very few timepoints relative to nuisance dimensions