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

[Makoto Miyakoshi]

Hi Cedric,

I think the idea is natural and reasonable. I would not be surprised to
find someone using this method without citing any paper.

How L is used in this algorithm reminds me of how M is used in ASR; L is a
PCA loading factor calculated from trial-averaged ERP, while M is
sqrt(cov(X_ref)) i.e., the covariance structure of clean part of the input
data. The common idea is that a canonical weight is created based on a
low-noise part of the data, then applied to each single trial/sliding
window.

Makoto

On Thu, Jun 4, 2026 at 5:08 PM Cedric Cannard via eeglablist <
eeglablist at sccn.ucsd.edu> wrote:

> I encountered this preprint that I think some may be interested in reading
> too here, in line with a recent conversation around using ICA for ERP
> analysis, differences between single trials and grand averages, etc:
> https://urldefense.com/v3/__https://osf.io/preprints/psyarxiv/d59ah_v4__;!!Mih3wA!GV5GqI8_zx4NF_22-je1lRqE2TQk6gafv2Xso_xN3ll1ddKrWNdscwMS71bNR4mo7aPnRrWOUjc4_zwF5C3vuQ$
>
> Abstract:
> 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 interpretable 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 d
>  ata, 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 estimation of internal consistency for PCA-derived
> component measures at the single-trial level, with all aggregated component
> scores showing high or adequate dependability. 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.
>
> Cedric
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