Mathematics in Independent Component Analysis

220 Chapter 15. Neurocomputing, 69:1485-1501, 2006
Signal [a.u.]
10 8 6 4
δ [ppm]
2 0 -2
(a) LICA denoised spectrum of P11 after the water artifact has been removed
with the algorithm GEVD-MP
Signal [a.u.]
10 8 6 4
δ [ppm]
2 0 -2
(b) KPCA denoised spectrum of P11 after the water artifact has been removed
with the algorithm GEVD-MP
Fig. 7. The figure shows the corresponding artifact free P11 spectra after the denoising
algorithms have been applied. The LICA algorithm was applied to all water
components with M, K chosen with the MDL estimator (γ = 32) between 20 and
60 and 20 and 80 respectively. The second graph shows the denoised spectrum with
a KPCA based algorithm using a gaussian kernel.
the KPCA. For each of the sample matrices X (m) the corresponding kernel
matrix K was determined by
Ki,j = k(xi, xj), i, j = 1, . . . , 400 (22)
where xi denotes the i-th column of X (m) . For the kernel function a Gaussian
kernel
k(xi, xj) = exp
�
− � xi − xj � 2
2σ 2
23
�
(23)