Mathematics in Independent Component Analysis

Chapter 15. Neurocomputing, 69:1485-1501, 2006 221
Signal [a.u.] Signal [a.u.]
10 8 6 4 2 0 -2 -4
10 8 6 4 2 0
14
12
10
8
6
4
2
0
-2
-2 -4
14
12
10
8
6
4
2
0
-2
δ [ppm]
10 8 6 4 2 0 -2 0
10 8 6 4 2 0 -2 0
δ [ppm]
Fig. 8. The graph uncovers the differences of the LICA and KPCA denoising algorithms.
As a reference the corresponding 1D slice of the original P11 spectrum is
displayed on top. From top to bottom the three curves show: The difference of the
original and the spectrum with the GEVD-MP algorithm applied, the difference between
the original and the LICA denoised spectrum and the difference between the
original and the KPCA denoised spectrum. To compare the graphs in one diagram
the three graphs are translated vertically by 2, 4 and 6 respectively.
where
2σ 2 =
1
400 ∗ 399
is the width parameter σ, was chosen.
400
�
i,j=1
� xi − xj � 2
8
7
6
5
4
3
2
1
(24)
Finally the kernel matrix K was expressed in terms of its EVD (equation 17)
which lead to the expansion parameters α necessary to determine the principal
axes of the corresponding feature space Ω (m) :
400
�
ω = αiΦ(xi). (25)
i=1
Similar to the original data, the noisy data of the reconstructed spectra were
used to form six 400 × 1024 dimensional pattern matrices P (m) , m = 1, . . . , 6.
24