4 The Analysis of Event-Related Potentials
61
however we hold here that the debate is resolved by performing a spatio-temporal
PCA, combining the advantages of both. The PCA seeks uncorrelated components
maximizing the variance of the ensemble average estimation (4.2) or (4.3); the first
component explains the maximum of its variance, while the remaining components
explain the maximum of its remaining variance, subjected to being uncorrelated
to all the previous. Hence, the variance explained by the N-P discarded components
explains the variance of the ‘noise’ that has been filtered out by the PCA. In symbols,
the PCA seeks matrices B and/or D with orthogonal columns so as to maximize the
variance of ¯
X
. Note that for any choice of 0 < P < N, the filtered ensemble average
estimator ¯
X
obtained by PCA is the best P-rank approximation to ¯
X in the leastsquares sense, i.e., for any 0 < P < N, the matrices B and/or D as found by PCA attain
the minimum variance of ¯
X − ¯
X
.
The PCA is obtained as it follows: let
¯
X U W V
T
(4.7)
be the singular-value decomposition of the ensemble average estimation, where
N × T matrix W holds along the principal diagonal the N non-null singular values in decreasing order (w 1 ≥ · · · ≥ w N ) and where N × N matrix U and T × T matrix
V hold in their columns the left and right singular vectors, respectively. Note that
the columns of U and V are also the eigenvectors of ¯
X ¯
X
T and ¯
X
T ¯
X , respectively,
with corresponding eigenvalues in both cases being the square of the singular values
in W and summing to the variance of ¯
X
. The spatial PCA is obtained filling B with
the first P column vectors of U, the temporal PCA is obtained filling D with the first
P column vectors of V and the spatio-temporal PCA is obtained filling them both.
The appropriate version of (4.4) and (4.5) then applies to obtain the components and
the sought filtered ensemble average estimation, respectively. In all cases 0 < P < N
is the chosen subspace dimension. Note that since for PCA the vectors of the spatial
and/or temporal filter matrix are all pair-wise orthogonal, (4.6) is simply verified by
setting A = B and/or E = D.
An example of spatio-temporal PCA applied to an ERP data set is shown in
Fig. 4.1, using estimator (4.2) in the second column and estimator (4.3) in the fourth
column. The ERP of this subject features a typical N1/P2 complex at occipital locations and an oscillatory process from about 50–450 ms, better visible at central and
parietal location, ending with a large positivity peaking at 375 ms (the “P300”). We
see that by means of only four components the PCA effectively compresses the ERP,
retaining the relevant signal; however, eye-related artefacts are also retained (see
traces at electrodes FP1 and FP2). This happens because the variance of these artefacts is very high, thus as long as the artefacts are somehow spatially and temporally
consistent across sweeps, they will be retained in early components along with the
consistent (time and phase-locked) ERPs, even if estimator (4.3) is used. For this
reason, artefact rejection is generally necessary before applying a PCA.
61
however we hold here that the debate is resolved by performing a spatio-temporal
PCA, combining the advantages of both. The PCA seeks uncorrelated components
maximizing the variance of the ensemble average estimation (4.2) or (4.3); the first
component explains the maximum of its variance, while the remaining components
explain the maximum of its remaining variance, subjected to being uncorrelated
to all the previous. Hence, the variance explained by the N-P discarded components
explains the variance of the ‘noise’ that has been filtered out by the PCA. In symbols,
the PCA seeks matrices B and/or D with orthogonal columns so as to maximize the
variance of ¯
X
. Note that for any choice of 0 < P < N, the filtered ensemble average
estimator ¯
X
obtained by PCA is the best P-rank approximation to ¯
X in the leastsquares sense, i.e., for any 0 < P < N, the matrices B and/or D as found by PCA attain
the minimum variance of ¯
X − ¯
X
.
The PCA is obtained as it follows: let
¯
X U W V
T
(4.7)
be the singular-value decomposition of the ensemble average estimation, where
N × T matrix W holds along the principal diagonal the N non-null singular values in decreasing order (w 1 ≥ · · · ≥ w N ) and where N × N matrix U and T × T matrix
V hold in their columns the left and right singular vectors, respectively. Note that
the columns of U and V are also the eigenvectors of ¯
X ¯
X
T and ¯
X
T ¯
X , respectively,
with corresponding eigenvalues in both cases being the square of the singular values
in W and summing to the variance of ¯
X
. The spatial PCA is obtained filling B with
the first P column vectors of U, the temporal PCA is obtained filling D with the first
P column vectors of V and the spatio-temporal PCA is obtained filling them both.
The appropriate version of (4.4) and (4.5) then applies to obtain the components and
the sought filtered ensemble average estimation, respectively. In all cases 0 < P < N
is the chosen subspace dimension. Note that since for PCA the vectors of the spatial
and/or temporal filter matrix are all pair-wise orthogonal, (4.6) is simply verified by
setting A = B and/or E = D.
An example of spatio-temporal PCA applied to an ERP data set is shown in
Fig. 4.1, using estimator (4.2) in the second column and estimator (4.3) in the fourth
column. The ERP of this subject features a typical N1/P2 complex at occipital locations and an oscillatory process from about 50–450 ms, better visible at central and
parietal location, ending with a large positivity peaking at 375 ms (the “P300”). We
see that by means of only four components the PCA effectively compresses the ERP,
retaining the relevant signal; however, eye-related artefacts are also retained (see
traces at electrodes FP1 and FP2). This happens because the variance of these artefacts is very high, thus as long as the artefacts are somehow spatially and temporally
consistent across sweeps, they will be retained in early components along with the
consistent (time and phase-locked) ERPs, even if estimator (4.3) is used. For this
reason, artefact rejection is generally necessary before applying a PCA.