Learning Mixtures of Independent Component Analysers - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 67

m ¼ 1...M Þ (Eq. 3.7 ); likelihood threshold ¼ð L ð i Þ L ð i 1 ÞÞ= L ð i Þ¼ 1e 05
(i ¼ number of the current iteration and L is estimated using equation L ð X = W Þ¼
log p ð X = W Þ¼ P

N

log p ð x ð n Þ = W Þ ).

n ¼ 1

3.4.1 Performance in BSS

The ICA mixture model is a kind of nonlinear ICA technique that extends the
linear ICA method by learning multiple ICA models and weighting them in a
probabilistic manner [ 1 ]. Approaching blind source separation of ICA mixture data
using standard ICA algorithms could give an inaccurate estimation of the sources.
In this section, we investigate the performance in blind source separation of the
non-parametric Mixca and the following ICA algorithms: FastIca [ 16 ], JADE [ 17 ],
TDSEP [ 18 ], InfoMax [ 19 ], Extended InfoMax [ 14 ], Kernel-ICA [ 20 ], Npica [ 21 ],
Radical [ 22 ]. Two simulation experiments of BSS were attempted with ICA
mixture and ICA generative data models. The separation efficiency was estimated
by the mean SIR, defined as 10 log 10 P

s ð n Þ 2 P

(dB), where s ð n Þ

N N

ð s ð n Þ s ð n Þ Þ 2

n ¼ 1 n ¼ 1

is the original source signal and s ð n Þ is the reconstructed source signal.
The first experiment consisted of two ICA mixtures, each of which had three
components (M=3) composed by Laplacian with a sharp peak at the bias and
heavy tails and/or uniform distributed sources generated randomly. The observa-
tion vectors were obtained adding together the data generated from the sources of
the two ICAs, i.e., the first part of the observation vectors corresponded to data
generated by the first ICA, and the second part of the observation vectors were the
data generated by the second ICA. From the observation vectors, the ICA algo-
rithms estimated 3 sources for one theoretical underlying ICA whereas the Mixca
algorithm, configured with K=2, estimated two sets of 3 sources for two ICAs. A
total of 300 Montecarlo simulations of the first experiment were performed for
each of the following values of N = 200, 500, 1000, 2000. The mean results for the
first simulation experiment are shown in Fig. 3.1 . The Mixca algorithm that
assigns the observation vectors to different sources of two ICA models outper-
forms the standard ICA algorithms, which attempt to estimate a single set of
sources for one ICA model for all the observation vectors of the mixture. The
evolution curves of SIR values for the ICA algorithms show no improvement with
the increase in the number of observation vectors for pdf estimation, and the SIR
levels below 8-10 dB thresholds are indicative of a failure to obtaining the desired
source separation. In contrast, the match between estimated and original sources
increases significantly with a higher number of observation vectors used in pdf
estimation for the Mixca algorithm.
The second experiment consisted of one-ICA datasets with mixtures of the five
different sources shown in Table 3.4 . The SIR was used to measure algorithm