4 The Analysis of Event-Related Potentials
65
can be used directly as input of the algorithms [26]. For BSS methods, either lagged
covariance matrices or Fourier co-spectral matrices are estimated on the available
data, then the demixing matrix B is estimated as the approximate joint diagonalizer
of all these matrices [20]. Details on the application of BSS methods to ERP data
can be found in Congedo et al. [24].
Figure 4.2 shows the result of a SOS-based BSS analysis applied to P300 data; here
the ensemble averages have been aligned using the method described by Congedo
et al. [22]. Analyzing both the temporal course and spatial distribution, we see that
the BSS analysis finds two relevant source components: S7 features a topographic
map (spatial pattern) with maximum at the vertex and an ERP (temporal pattern)
with maximum at 370 ms, clearly describing the P300. S13 features a topographic
map with maximum at parietal and occipital bilateral derivations and an ERP with
the classical P100/N200 complex describing a visual ERP. Both source components
are present only in the target sweeps. Further analysis of these components will
be presented in the Sect. 4.4. time-frequency domain analysis. Clearly, BSS has
successfully separated the two ERP components.
It is worth mentioning that while traditionally only spatial BSS is performed,
a spatio-temporal BSS method for ERPs has been presented in Korczowski et al.
[49]. Just as in the case of PCA and common pattern, a spatio-temporal approach is
preferable for ERP analysis, thus it should be pursued further (Fig. 4.2).
4.4 Time-Frequency Domain Analysis
Time-Frequency Analysis (TFA) complements and expands the time domain analysis
of ERP thanks to a number of unique features. While the analysis in the time domain
allows the study of phase-locked ERP components only, TFA allows the study of
both phase-locked (evoked) and non-phase-locked (induced) ERP components. In
addition to timing, the TFA provides information about the frequency (both for evoked
and induced components) and about the phase (evoked components only) of the
underlying physiological processes. This is true for the analysis of a single time
series (univariate) as well as for the analysis of the dependency between two timeseries (bivariate), the latter not being treated here. In all cases, the time series under
analysis may be the sweeps derived at significant scalp derivations or BSS source
components with specific physiological meaning as obtained by the methods we have
discussed above. In this section we introduce several univariate TFA measures.
A time-frequency analysis (TFA) decomposes a signal in a two dimensional plane,
with one dimension being the time and the other being the frequency. Whereas several
possible time-frequency representations exist, nowadays in ERP studies we mainly
encounter wavelets [50, 89] or the analytic signal resulting from the Hilbert transform
[13, 84, 90]. Several studies comparing wavelets and the Hilbert transform have found
that the two representations give similar results [8, 53].
The example we provide below employs the Hilbert transform [37], which is easily
and efficiently computed by means of the fast Fourier transform [64]. By applying
65
can be used directly as input of the algorithms [26]. For BSS methods, either lagged
covariance matrices or Fourier co-spectral matrices are estimated on the available
data, then the demixing matrix B is estimated as the approximate joint diagonalizer
of all these matrices [20]. Details on the application of BSS methods to ERP data
can be found in Congedo et al. [24].
Figure 4.2 shows the result of a SOS-based BSS analysis applied to P300 data; here
the ensemble averages have been aligned using the method described by Congedo
et al. [22]. Analyzing both the temporal course and spatial distribution, we see that
the BSS analysis finds two relevant source components: S7 features a topographic
map (spatial pattern) with maximum at the vertex and an ERP (temporal pattern)
with maximum at 370 ms, clearly describing the P300. S13 features a topographic
map with maximum at parietal and occipital bilateral derivations and an ERP with
the classical P100/N200 complex describing a visual ERP. Both source components
are present only in the target sweeps. Further analysis of these components will
be presented in the Sect. 4.4. time-frequency domain analysis. Clearly, BSS has
successfully separated the two ERP components.
It is worth mentioning that while traditionally only spatial BSS is performed,
a spatio-temporal BSS method for ERPs has been presented in Korczowski et al.
[49]. Just as in the case of PCA and common pattern, a spatio-temporal approach is
preferable for ERP analysis, thus it should be pursued further (Fig. 4.2).
4.4 Time-Frequency Domain Analysis
Time-Frequency Analysis (TFA) complements and expands the time domain analysis
of ERP thanks to a number of unique features. While the analysis in the time domain
allows the study of phase-locked ERP components only, TFA allows the study of
both phase-locked (evoked) and non-phase-locked (induced) ERP components. In
addition to timing, the TFA provides information about the frequency (both for evoked
and induced components) and about the phase (evoked components only) of the
underlying physiological processes. This is true for the analysis of a single time
series (univariate) as well as for the analysis of the dependency between two timeseries (bivariate), the latter not being treated here. In all cases, the time series under
analysis may be the sweeps derived at significant scalp derivations or BSS source
components with specific physiological meaning as obtained by the methods we have
discussed above. In this section we introduce several univariate TFA measures.
A time-frequency analysis (TFA) decomposes a signal in a two dimensional plane,
with one dimension being the time and the other being the frequency. Whereas several
possible time-frequency representations exist, nowadays in ERP studies we mainly
encounter wavelets [50, 89] or the analytic signal resulting from the Hilbert transform
[13, 84, 90]. Several studies comparing wavelets and the Hilbert transform have found
that the two representations give similar results [8, 53].
The example we provide below employs the Hilbert transform [37], which is easily
and efficiently computed by means of the fast Fourier transform [64]. By applying