Applying ICA in EEG: Choice of the Window Length and of the Decorrelation Method - Biomedical Engineering Systems and Technologies - page 249

2.3 Optimization: Data Rotation

Second step would be finding a rotation matrix J to be applied to the decorrelated
data (whitened or sphered) in order to maximize their independence. Rotation can be
done using second order statistics (SOS) using joint decorrelations and/or using HOS
cost functions. We restrain here to the second (HOS) approach 1 . Several cost functions
and optimization techniques were described in the literature (see for example [4, 12]).
Among the most well known and used in EEG applications, we can cite FastICA (neg-
entropy maximization [6]), Extended InfoMax (mutual information minimization [7])
and JADER (joint diagonalization of fourth order cumulant matrices [9]). Another re-
cent algorithm has been proposed by Palmer et al. and is called AMICA [8]. Based on
the modeling of each source component as a sum of extended Gaussians, this method
has shown very promising results in the context of EEG data [5].
Specifically, in this paper we test the performances and the robustness of these four
ICA algorithms with respect to the sample size and the initialization step in both con-
texts of random unstructured data mixing and biologically plausible data mixing.

3 Performance Evaluation Criteria

3.1 Reliable Estimate of the Covariance: Riemannian Likelihood

As noted before, BSS model consists of decorrelation and rotation. Both steps are based
on statistical estimates. The first step is common for all algorithms and relies on the
estimation of the covariance matrix. Therefore it is necessary to have reliable estimates
of this matrix. In other words, given a known covariance matrix

, we want to evaluate
the minimum sample size N necessary to obtain a covariance matrix estimation

Σ

Σ N

close enough to the original one with respect to a distance that we have to define.
We propose here an original distance measure between the true and the estimated
covariance matrices, inspired from digital image processing and computer vision tech-
niques [15]. In the context of object tracking and texture description, a distance measure
is used to estimate whether an observed object or region corresponds to a given covari-
ance descriptor. To estimate similarity between matrices respectively corresponding to
the target model and the candidate, and knowing that covariance matrices are symmetric
positive definite, the following general 2 distance measure can be used:

)= tr log 2 Σ Σ N − 2

Σ N − 2

d 2 ( Σ N ,Σ (4)

Σ Σ N − 2 equals the
identity matrix I n and d becomes 0 ( n being the number of measured signals, equal
to the source number in our case). In real cases though, assuming that the covariance

Σ N − 2

In the ideal case of a perfect estimation, the matrix C =

1

As described in the next section, in our simulations we used random non-Gaussian station-
ary data, without any time-frequency structure. Therefore algorithms based on SOS as SOBI,
SOBI-RO and AMUSE were not used.

2

On Riemannian manifolds.