  Electroencephalography (EEG) data is often collected from diverse contexts
involving different populations and EEG devices. This variability can induce
distribution shifts in the data $X$ and in the biomedical variables of interest
$y$, thus limiting the application of supervised machine learning (ML)
algorithms. While domain adaptation (DA) methods have been developed to
mitigate the impact of these shifts, such methods struggle when distribution
shifts occur simultaneously in $X$ and $y$. As state-of-the-art ML models for
EEG represent the data by spatial covariance matrices, which lie on the
Riemannian manifold of Symmetric Positive Definite (SPD) matrices, it is
appealing to study DA techniques operating on the SPD manifold. This paper
proposes a novel method termed Geodesic Optimization for Predictive Shift
Adaptation (GOPSA) to address test-time multi-source DA for situations in which
source domains have distinct $y$ distributions. GOPSA exploits the geodesic
structure of the Riemannian manifold to jointly learn a domain-specific
re-centering operator representing site-specific intercepts and the regression
model. We performed empirical benchmarks on the cross-site generalization of
age-prediction models with resting-state EEG data from a large multi-national
dataset (HarMNqEEG), which included $14$ recording sites and more than $1500$
human participants. Compared to state-of-the-art methods, our results showed
that GOPSA achieved significantly higher performance on three regression
metrics ($R^2$, MAE, and Spearman's $\rho$) for several source-target site
combinations, highlighting its effectiveness in tackling multi-source DA with
predictive shifts in EEG data analysis. Our method has the potential to combine
the advantages of mixed-effects modeling with machine learning for biomedical
applications of EEG, such as multicenter clinical trials.
