64
M. Congedo
doing so, BSS makes a number of assumptions. The common one for all BSS methods is that the observed EEG potential results from an instantaneous linear mixing
of a number of cortical dipolar electric fields. Although this is an approximation of
the physical process of current generation in the brain and diffusion through the head
[70], physical and physiological knowledge support such generative model for scalp
potentials [9]. In particular, the model fits well low-frequency electrical phenomena
with low spatial resolution, which yield the strongest contribution to the recordable
EEG. The model reads
x(t) As(t),
(4.11)
where, as before, x(t) is the observed N-dimensional sensor measurement vector,
s(t) the unknown P-dimensional vector holding the true dipolar source process (with
0 < P ≤ N), and A, also assumed unknown in BSS, is named the mixing matrix. BSS
entails the estimation of a demixing matrix B allowing source process estimation
y(t) B
T x(t).
(4.12)
We say that the source process can be identified if
y(t) ≈ Gs(t),
(4.13)
where P × P matrix G = B
T A is a scaled permutation matrix, i.e., a square matrix with
only one non-null element in each row and each column. Matrix G cannot be observed
since A is unknown. It enforces a shuffling of the order and amplitude (including
possible sign switching) of the estimated source components, which cannot be solved
by BSS. Equation (4.13) means that in BSS the actual waveform of the source process
has been approximately identified, albeit the sign, scaling and order of the estimated
source process is arbitrary. Such identification is named blind because no knowledge
on the source waveform s(t) nor on the mixing process A is assumed. Fortunately,
condition (4.13) can be achieved under some additional assumptions relating to the
statistical properties of the dipolar source components (see [11, 78]).
Two important families of BSS methods operate by canceling inter-sensor second
order statistics (SOS) or higher (than two) order statistics (HOS); the latter family being better known as independent component analysis (ICA) (see [17], for an
overview). In doing so, both assume some form of independence among the source
processes, which is specified by inter-sensor statistics that are estimated from the
data. The difference between the two families resides in the assumption about the
nature of the source process; since Gaussian processes are defined exhaustively by
their mean and variance (SOS), ICA may succeed only when at most one of the components is Gaussian. On the other hand, SOS methods can identify the source process
components regardless of their distribution, i.e., even if they are all Gaussian, but
source components must have a unique power spectrum signature and/or a unique
pattern of energy variation across time, across experimental conditions or, in the case
of ERPs, across ERP classes (see [20, 24]). For HOS methods the available EEG
M. Congedo
doing so, BSS makes a number of assumptions. The common one for all BSS methods is that the observed EEG potential results from an instantaneous linear mixing
of a number of cortical dipolar electric fields. Although this is an approximation of
the physical process of current generation in the brain and diffusion through the head
[70], physical and physiological knowledge support such generative model for scalp
potentials [9]. In particular, the model fits well low-frequency electrical phenomena
with low spatial resolution, which yield the strongest contribution to the recordable
EEG. The model reads
x(t) As(t),
(4.11)
where, as before, x(t) is the observed N-dimensional sensor measurement vector,
s(t) the unknown P-dimensional vector holding the true dipolar source process (with
0 < P ≤ N), and A, also assumed unknown in BSS, is named the mixing matrix. BSS
entails the estimation of a demixing matrix B allowing source process estimation
y(t) B
T x(t).
(4.12)
We say that the source process can be identified if
y(t) ≈ Gs(t),
(4.13)
where P × P matrix G = B
T A is a scaled permutation matrix, i.e., a square matrix with
only one non-null element in each row and each column. Matrix G cannot be observed
since A is unknown. It enforces a shuffling of the order and amplitude (including
possible sign switching) of the estimated source components, which cannot be solved
by BSS. Equation (4.13) means that in BSS the actual waveform of the source process
has been approximately identified, albeit the sign, scaling and order of the estimated
source process is arbitrary. Such identification is named blind because no knowledge
on the source waveform s(t) nor on the mixing process A is assumed. Fortunately,
condition (4.13) can be achieved under some additional assumptions relating to the
statistical properties of the dipolar source components (see [11, 78]).
Two important families of BSS methods operate by canceling inter-sensor second
order statistics (SOS) or higher (than two) order statistics (HOS); the latter family being better known as independent component analysis (ICA) (see [17], for an
overview). In doing so, both assume some form of independence among the source
processes, which is specified by inter-sensor statistics that are estimated from the
data. The difference between the two families resides in the assumption about the
nature of the source process; since Gaussian processes are defined exhaustively by
their mean and variance (SOS), ICA may succeed only when at most one of the components is Gaussian. On the other hand, SOS methods can identify the source process
components regardless of their distribution, i.e., even if they are all Gaussian, but
source components must have a unique power spectrum signature and/or a unique
pattern of energy variation across time, across experimental conditions or, in the case
of ERPs, across ERP classes (see [20, 24]). For HOS methods the available EEG