Dependent Component Analysis - Biomedical Signal Analysis: Contemporary Methods and Applications - page 155

(a) ICA (b) ISA with fixed group-

size

(c) general ISA

Figure 5.4
Linear factorization models for a random vector x = As and the resulting
indeterminacies, where L denotes a one- or higher-dimensional invertible matrix
(scaling), and P denotes a permutation, to be applied only along the horizontal line
as indicated in the figures. The small horizontal gaps denote statistical
independence. One of the key differences between the models is that general ISA
may always be applied to any random vector x , whereas ICA and its generalization,
fixed-size ISA, yield unique results only if x follows the corresponding model.

the corresponding uniqueness results are illustrated in figure 5.4.
Again, we turned this uniqueness result into a separation algorithm,
this time by considering the JADE source condition based on fourth-
order cumulants. The key idea was to translate irreducibility into max-
imal block diagonality of the source condition matrices C i ( s ). Algorith-
mically, JBD was performed using JD first using theorem 5.2, followed
by permutation and block size identification, see [251].
As a short example, we consider a general ISA problem in dimen-
sion n = 10 with the unknown partition m =(1 , 2 , 2 , 2 , 3). In order
to generate two- and three-dimensional irreducible random vectors, we
decided to follow the nice visual ideas from [207] and to draw samples
from a density following a known shape - in our case 2-D letters or 3-
D geometrical shapes. The chosen source densities are shown in figure
5.5(a-d). Another 1-D source following a uniform distribution was con-
structed. Altogether, 10 4 samples were used. The sources S were mixed
by a mixing matrix A with coecients uniformly randomly sampled from
[ 1 , 1] to give mixtures X = AS . The recovered mixing matrix A was

then estimated, using the above block JADE algorithm with unknown
block size; we observed that the method is quite sensitive to the choice
of the threshold (here θ =0 . 015). Figure 5.5(e) shows the composed
mixing-separating system

−

A −1 A ; clearly the matrices are equal except