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

Mon Jun 15 15:38:16 PDT 2026

Hi Joe,

Thank you for your comments.

> It's just issuing a caution about what happens when there are different
numbers of trials in the cells (e.g., rare vs. frequent stimuli in an
oddball).

This issue is common to other methods, such as ICA, right? Theoretically,
the maximally efficient learning can be achieved when only rare event
trials are submitted to ICA/PCA.

There is unsupervised multiple-model AMICA that can partially address this
issue at least in theory, with which Shawn Hsu demonstrated his mastery and
published a couple of NeuroImage papers. But controlling the behavior of
this algorithm requires good engineering skills.

> Basically, it is saying one should apply a PCA solution to data that is
consistent with the data used to generate the PCA.

Yes, that is a conservative way to proceed. But if I understand correctly,
obtaining the PCA solution from trial-averaged ERPs and then applying the
resulting scoring coefficients to single-trial data is ok, as long as one
knows what s/he is doing, which does not have that trial imbalance problem.

Makoto

On Tue, Jun 9, 2026 at 12:38 AM Joseph Dien via eeglablist <
eeglablist at sccn.ucsd.edu> wrote:

> Hey Cedric and Makato,
>
>     the EP Toolkit also does a single-trial PCA for movement artifact
> correction (Dien, 2010; Dien, 2024).  Also, see Ouyang, Dien, and Lorenz
> (2022).  As you say, this idea has been around a while.  This article
> isn't trying to claim credit for it.  It's just issuing a caution about
> what happens when there are different numbers of trials in the cells
> (e.g., rare vs. frequent stimuli in an oddball).  In PCA based on
> subject averages, the conditions are given equal weight whereas in a PCA
> directly on the single-trial data, the cells are weighted by the number
> of trials.  You get different results when you then apply the factor
> scoring coefficients to the single-trial data.  Basically, it is saying
> one should apply a PCA solution to data that is consistent with the data
> used to generate the PCA.  I don't think the abstract conveys this
> conclusion as clearly as it could, but it is a point worth making.
>
> Cheers!
>
> Joe
>
>
> On 6/8/26 16:48, Makoto Miyakoshi via eeglablist wrote:
> > 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
> >> _______________________________________________
> >> To unsubscribe, send an empty email to
> >> eeglablist-unsubscribe at sccn.ucsd.edu or visit
> >> https://sccn.ucsd.edu/mailman/listinfo/eeglablist     .
> >>
> > _______________________________________________
> > To unsubscribe, send an empty email to
> eeglablist-unsubscribe at sccn.ucsd.edu or visit
> https://sccn.ucsd.edu/mailman/listinfo/eeglablist   .
>
> --
>
> --------------------------------------------------------------------------------
>
> Joseph Dien, PhD
> Senior Research Scientist
> Department of Human Development and Quantitative Methodology
> University of Maryland, College Park
>
> https://urldefense.com/v3/__http://joedien.com__;!!Mih3wA!H0MA__UacnPLS-gTyPeXJQcp0p7etp_g_6T_3ImJHYgbUyqJvPXFVVSYLJUuzKZ-xpIhbCgSWv0eFI1q37k$
>
> _______________________________________________
> To unsubscribe, send an empty email to
> eeglablist-unsubscribe at sccn.ucsd.edu or visit
> https://sccn.ucsd.edu/mailman/listinfo/eeglablist  .