Learning Mixtures of Independent Component Analysers - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 64

Table 3.3 A notation for different variants of the Mixca procedure
ICAMM parameter updating algorithm Mixca variant notation

Non-parametric estimation Mixca, non-parametric Mixca

JADE Mixca-JADE

FastIca Mixca-FastIca

TDSEP Mixca-TDSEP

InfoMax Mixca-InfoMax

In principle, the source pdf in Eq. ( 3.10 ) can also be estimated in a non-parametric
manner following

2

s km s ð n Þ

where p ð s km Þ¼ a X

N

km ð I Þ

2

p ð s k Þ¼ p ð s k1 Þ p ð s k2 Þ p ð s kM Þ e

h

ð 3 : 11 Þ

n ¼ 1

However, although independence is imposed by means of the underlying ICA
algorithm that is selected to learn the mixtures, there is no guarantee that the final
source vectors have independent elements. Some residual dependence may still
remain due to possible nonlinearities that are not accounted for in the basic model
in Eq. ( 3.1 ). If this is the case, Eq. ( 3.11 ) will produce erroneous estimates of the
source pdf, and thus incorrect computing of the probability of every class condi-
tioned to the feature vector (posterior probability) by applying Eq. ( 3.10 ). This
suggests the convenience of improving the estimation of the source pdf by con-
sidering general methods of multidimensional density estimation, preferably of the
non-parametric type [ 8 ], in order to keep the generality of the proposed frame-
work. A rather simple possibility is to consider the estimator

T

½

s k s ð n Þ
k

½

s k s ð n Þ
k

ð I Þ ð I Þ

p ð s k Þ¼ a 0 X

N

2

h 2
0

e ð 3 : 12 Þ

n ¼ 1

If this post-convergence correction is applied, the analyzers could no longer be
independent. Therefore, we have developed a procedure that analyzes mixture data
with components that may or may not be independent, which we call Mixca
(Mixture of Component Analyzers). This procedure allows all the parameters
involved in ICAMM (mixture matrices, centroids, and source probability densi-
ties) to be estimated. The Mixca procedure defines a framework with a set of
variants depending on the embedded algorithm used for ICAMM parameter
updating. A notation for different variants of the Mixca procedure is in Table 3.3 .

3.3.5 Discussion

At this point, we would like to point out that the algorithm above is intended to
extend the ICAMM to a general framework. This is done by including new