Single-trial principal components analysis of event-related potentials: Back-projection, theoretical rationale, empirical evaluation, and reliability

Temporal principal components analysis (tPCA) is widely used to identify and quantify event-related potentials (ERPs). It is less commonly applied to single-trial EEG epochs, despite concerns about the low signal-to-noise ratio and uncertainty about how single-trial components compare with those derived from averaged waveforms. Alternatively, averaged principal components (PCs) can be imposed on single trials via back-projection. This study evaluated the fidelity of single-trial and back-projected single-trial tPCA (covariance-based, unrestricted Varimax rotation) in representing meaningful averaged PCs by systematically comparing factor loadings and factor scores in two existing ERP datasets (72-channel, N=152; 67-channel, N=98), using both surface-referenced (infinity, nose) and reference-free current source density (CSD) transformations. In the 72-channel dataset with four balanced conditions (emotional hemifield paradigm), single-trial PCs closely matched those from averaged data, whereas in the 67-channel dataset with three imbalanced conditions (novelty oddball task), they did not. In contrast, across datasets and transformations, back-projected single-trial PCs consistently reproduced virtually identical averaged PCs. All single-trial tPCA solutions enabled full reconstruction of the original data matrix, confirming the methodological validity of back-projected decomposition, and critically, back-projection enabled reliable estimation of internal consistency for PCA-derived component measures at the single-trial level, with all component scores showing high or adequate reliability. These findings suggest that single-trial tPCA may approximate averaged solutions only when the variance structure is preserved, whereas back-projection provides a robust alternative by quantifying single-trial epochs relative to the averaged covariance structure, thereby maintaining component structure, integrity, and interpretability of trial-level component estimates in continuity with established ERP constructs.