  Sparse linear regression methods including the well-known LASSO and the
Dantzig selector have become ubiquitous in the engineering practice, including
in medical imaging. Among other tasks, they have been successfully applied for
the estimation of neuronal activity from functional magnetic resonance data
without prior knowledge of the stimulus or activation timing, utilizing an
approximate knowledge of the hemodynamic response to local neuronal activity.
These methods work by generating a parametric family of solutions with
different sparsity, among which an ultimate choice is made using an information
criteria. We propose a novel approach, that instead of selecting a single
option from the family of regularized solutions, utilizes the whole family of
such sparse regression solutions. Namely, their ensemble provides a first
approximation of probability of activation at each time-point, and together
with the conditional neuronal activity distributions estimated with the theory
of mixtures with varying concentrations, they serve as the inputs to a Bayes
classifier eventually deciding on the verity of activation at each time-point.
We show in extensive numerical simulations that this new method performs
favourably in comparison with standard approaches in a range of realistic
scenarios. This is mainly due to the avoidance of overfitting and underfitting
that commonly plague the solutions based on sparse regression combined with
model selection methods, including the corrected Akaike Information Criterion.
This advantage is finally documented in selected fMRI task datasets.
