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

[Joseph Dien]

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
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-- 
--------------------------------------------------------------------------------

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$