Mathematics in Independent Component Analysis

100 Chapter 4. Neurocomputing 64:223-234, 2005
p
1
0
-1
1
0.8
0.6
0.4
0.6
0.2
0.4
0.2 0.4 0.6 0.8 1
1
1
2
3
0.4 0.6
4
5
0.6
0.4
0.4 0.6
0.2 0.4 0.6 0.8 1
Original
-1 0 1
Relative volume error
q
2
0
-2
0
0.8
0.6
0.4
0.2
6
5
4
3
2
1
0
�
�
− log vol(∆p,q)
Unmixing with p = 0.65, q = 0.5
2
0
-2
1
0
-1
Unmixing with p = 0.65, q = 0.65
0
Unmixing with p = 0.5, q = 0.65
-1 0 1
Fig. 3. The top-left image graphs the separation error ∆p,q by measuring the negative
logarithm of its volume. The error ∆p,q denotes the difference set of points from
either the support of the recovered source distribution or a quadrangle with the
same vertices. The other plots show the distributions of the recovered sources at
some combinations of the parameters p, q of the nonlinearities. At p = 0.5, q = 0.5
this is the original source distribution. Here light grey areas represent ∆p,q and dark
grey areas the intersection of the support and the quadrangle.
A simple estimator for mutual information based on histogram estimation of
the entropy (with 10 bins in each dimension) is used to check the independence
of the recovered sources. A more elaborate histogram-based estimator
by Moddemeijer [17] yields similar results. As shown in figure 2 the mutual
information of the recovered sources is minimal at the parameters p = q = 0.5,
which correspond to the mixing model. It can also be noticed that the minimum
is much less distinct in the second component. This indicates that in
numerical application it should be easier to detect nonlinear functions which
are bounded.
The second graph (figure 3) further illustrates that the criterion for the borders
11