Mathematics in Independent Component Analysis

118 Chapter 6. LNCS 3195:726-733, 2004
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(a) source images
crosstalking error E1( Â, I)
4
3.5
3
2.5
2
1.5
1
0.5
SOBI based on multi-dimensional autocovariances 5
SOBI
SOBI transposed images
mdSOBI
mdSOBI transposed images
0
0 10 20 30 40 50 60 70
K
(b) performance comparison
Fig. 2. Comparison of SOBI and mdSOBI when applied to (unmixed) images from (a).
The plot (b) plots the number K of time lags versus the crosstalking error E1 of the
recovered matrix  and the unit matrix I; here  has been recovered by bot SOBI
and mdSOBI given the images from (a) respectively the transposed images.
after whitening of x(z1, . . . , zK). The joint diagonalizer then equals A except for
permutation, given the generalized identifiability conditions from [4], theorem
2. Therefore, also the identifiability result does not change, see [4]. In practice,
we choose the (τ (k)
1 , . . . , τ (k)
M ) with increasing modulus for increasing k, but with
the restriction τ (k)
1 > 0 in order to avoid using the same autocovariances on the
diagonal of the matrix twice.
Often, data sets do not have any substantial long-distance autocorrelations,
but quite high multi-dimensional close-distance correlations (see figure 1). When
performing joint diagonalization, SOBI weighs each matrix equally strong, which
can deteriorate the performance for large K, see simulation in section 4.
Figure 2(a) shows an example, in which the images have considerable vertical
structure, but rather random horizontal structure. Each of the two images
consists of a concatenation of stripes of two images. For visual purposes, we
chose the width of the stripes to be rather large with 16 pixels. According to
the previous discussion we expect one-dimensional algorithms such as AMUSE
and SOBI to perform well on the images, but badly (for number of time lags
≫ 16) on the transposed images. If we apply AMUSE with τ = 20 to the images,
we get excellent performance with a low crosstalking error with the unit
matrix of 0.084; if we however apply AMUSE to the transposed images, the error
is high with 1.1. This result is further confirmed by the comparison plot in
figure 2(b); mdSOBI performs equally well on the images and the transposed